\section{Introduction} \nocite{*} \subsection{Background and statement of main theorem} The study of eigenvarieties began with the seminal work of Hida in the 1980s and Coleman and Mazur in the 1990s, culminating in Coleman-Mazur's construction in~\cite{cm98} of the modular eigencurve, a rigid analytic space parametrizing $p$-adic modular Hecke eigenforms. The eigencurve admits a natural projection map to \emph{weight space}, a rigid analytic space parametrizing possible weights of $p$-adic modular forms, as well as a map to $\gra_m$ corresponding to the modular forms' $U_p$-eigenvalues. Since then, further work and generalizations by numerous authors have resulted in a massive collection of eigenvarieties'' for $p$-adic automorphic forms on various groups. Particularly relevant for our purposes are the papers of Buzzard \cite{buzzard04}, \cite{buzzard07}, Chenevier \cite{chenevier04}, and Bella\"iche-Chenever \cite{bc09}, in which eigenvarieties are constructed for $p$-adic automorphic forms on definite unitary groups of all dimensions. The geometry of these eigenvarieties appears to be complicated. To discuss it, we need to introduce some notation. Let $p$ be a prime. Let $q=4$ if $p=2$ and $q=p$ otherwise. We write $v$ for the $p$-adic valuation and $|\cdot|$ for the $p$-adic norm, normalized so that $v(p)=1$ and $|p|=p^{-1}$. A \emph{weight} of a $p$-adic modular form is a continuous character of $\ints_p^\times$, and the \emph{weight space} for modular forms is the rigid analytic space $\wsc$ such that for any affinoid $\rats_p$-algebra $A$, $\wsc(A)$ is the set of continuous characters $\ints_p^\times\to A^\times$. The $T$-coordinate of an $A$-point $w\in\wsc(A)$ is the value $T(w)=w(\exp(q))-1$; the space $\wsc$ turns out to be a disjoint union of $\phi(q)$ open unit discs with parameter $T$. For $r\in(0,1)$, we write $\wsc_{>r}$ for the subset of $\wsc$ where $|T|>r$. We fix a tame level and let $\zsc$ be the corresponding eigencurve. We let $w:\zsc\to\wsc$ be the map taking an eigenform to its weight, $a_p:\zsc\to\gra_m$ be the map taking an eigenform to its $U_p$-eigenvalue, and $\zsc_{>r}$ be the preimage of $\wsc_{>r}$ in $\zsc$. One thread of approach to analyzing eigenvarieties has been to restrict analysis to the part of the eigenvariety lying over the boundary'' of weight space, in this case $\zsc_{>r}$ for $r$ sufficiently close to $1$, where it is expected to be simpler. In particular, the following folklore conjecture arose from a question of Coleman and Mazur \cite{cm98}, was suggested by a computation of Buzzard and Kilford \cite{bk05}, and is given below in the form stated by Liu, Wan, and Xiao \cite{lwx17}. \begin{conj}[Coleman-Mazur-Buzzard-Kilford, as stated by Liu-Wan-Xiao \cite{lwx17}] \label{cmbk} When $r\in(0,1)$ is sufficiently close to $1^-$, the following statements hold. \begin{enumerate} \item The space $\zsc_{>r}$ is a disjoint union of (countably infinitely many) connected components $Z_1,Z_2,\dotsc,$ such that the weight map $w:Z_n\to\wsc_{>r}$ is finite and flat for each $n$. \item\label{tend0} There exist nonnegative rational numbers $\al_1,\al_2,\dotsc\in\rats$ in non-decreasing order and tending to infinity such that, for each $n$ and each point $z\in Z_n$, we have $|a_p(z)|=|T(w(z))|^{\al_n}.$ \item The sequence $\al_1,\al_2,\dotsc$ is a disjoint union of finitely many arithmetic progressions, counted with multiplicity (at least when the indices are large enough). \end{enumerate} \end{conj} Note that Part~\ref{tend0} of Conjecture~\ref{cmbk} implies that as one approaches the boundary, the slope $v(a_p(z))$ approaches $0$ in proportion to $v(T(w(z)))$. Following various explicit computations, Liu, Wan, and Xiao \cite{lwx17}, building on the work of Wan, Xiao, and Zhang \cite{wxz17}, proved the equivalent version of this conjecture for automorphic forms on definite quaternion algebras over $\rats$. The essence of their work, from which their result follows naturally, is their strong upper and lower bounds on the Newton polygon of the characteristic power series of the $U_p$-operator. For consistency with future discussion, we state their bounds (with some imprecision, to avoid unnecessary detail) for rank-$2$ definite unitary groups over $\rats$, for which the analysis is exactly the same. Let $G$ be an algebraic group over $\rats$ such that $G(\real)\cong U_2(\real)$ and $G(\rats_p)\cong GL_2(\rats_p)$, $\usc\subset G(\aff_f)$ a compact open subgroup satisfying reasonable minor technical conditions, and $\ssc_w(G,\usc)$ the space of $p$-adic automorphic forms on $G$ of weight $w$ and level $\usc$. The basic idea is that we expect the Newton polygon of $\det(I-XU_p|\ssc_w(G,\usc))$ to be of shape approximately $y=Av(T(w))x^2$. \begin{thm}[Liu-Wan-Xiao, Corollary 3.16 and Proposition 3.20 of \cite{lwx17}] \label{lwxbounds} Fix $\usc$. \begin{enumerate} \item \label{lwxlbd} There are constants $A_1,C>0$ such that for all $w$ such that $T(w)>\frac1p$, the Newton polygon of the power series $\det(I-XU_p|\ssc_w(G,\usc))$ lies above the curve $y=(A_1x^2-C)v(T(w))$. \item \label{lwxubd} Suppose that $w(a)=a^t\chi(a)$, where $t\in\ints_{\ge0}$ and $\chi$ is a finite character of conductor $c$. Then there is a constant $h$ such that the Newton polygon of $\det(I-XU_p|\ssc_w(G,\usc))$ contains at least $h(t+1)p^{c-1}$ segments of slope at most $t+1$, hence passes below the point $\left(h(t+1)p^{c-1},h(t+1)^2p^{c-1}\right)=\left(x,A_2x^2v(T(w))\right)$ for $x=h(t+1)p^{c-1}$ and a constant $A_2$ depending on $h$. \end{enumerate} \end{thm} The lower bound, statement~\ref{lwxlbd} above, has since been re-proven by Johansson and Newton in~\cite{jn16} using a more conceptual method. In this paper, we generalize Theorem~\ref{lwxbounds} to definite unitary groups of all dimensions. Let $G$ be an algebraic group over $\rats$ such that $G(\real)\cong U_n(\real)$ and $G(\rats_p)\cong GL_n(\rats_p)$, and $\usc\subset G(\aff_f)$ a compact open subgroup satisfying reasonable minor technical conditions. The corresponding eigenvariety $\zsc$ is now a rigid analytic space of dimension $n-1$ lying over the space $\wsc$ such that for any affinoid $\rats_p$-algebra $A$, $\wsc(A)$ is the set of continuous characters $(\ints_p^\times)^{n-1}\to A^\times$. This $\wsc$ is a disjoint union of $\phi(q)^{n-1}$ open unit polydiscs of dimension $n-1$ with parameters $T_1,\dotsc,T_{n-1}$. Again let $\ssc_w(G,\usc)$ be the space of $p$-adic automorphic forms on $G$ of weight $w$ and level $\usc$. The basic idea of our main theorem is that we expect the Newton polygon of $\det(I-XU_p|\ssc_w(G,\usc))$ to be of shape approximately $y=Av(T_i(w))x^{1+\frac2{n(n-1)}}$, assuming that all the $v(T_i(w))$ are not extremely different in size. Of course, this recovers the Liu-Wan-Xiao approximation $y=Av(T_1(w))x^2$ for $n=2$. As far as we know, there is little prior work on the shape of the Newton polygon of $\det(I-XU_p|\ssc_w(G,\usc))$. The only prior result we have been able to find in the literature is the following lower bound of Chenevier, which has an exponentially smaller exponent and is valid only in the center of weight space. \begin{thm}[Chenevier, Lemma 5.1.1 of~\cite{chenevier04}] \label{chbound} Fix $\usc$. Suppose that for some $r0$ such that the Newton polygon of $\det(I-XU_p|\ssc_w(G,\usc))$ lies above the curve $y=A_rx^{1+\frac1{2^n-n-1}}-C_rx.$ \end{thm} As late as 2015, Andreatta, Iovita, and Pilloni said in Section 1.2.2 of~\cite{aip15} that there were not even any conjectures about the shape of the Newton polygon in higher dimensions in the literature. Our precise theorem is as follows. \begin{thm} \label{mybounds} Fix $\usc$. \begin{enumerate} \item \label{mylbd} There are constants $A_1,C>0$ such that for all $w$ such that each $|T_i(w)|>\frac1p$, the Newton polygon of the power series $\det(I-XU_p|\ssc_w(G,\usc))$ lies above the curve $y=\left(A_1x^{1+\frac2{n(n-1)}}-C\right)\min_iv(T_i(w)).$ \item \label{myubd} Suppose that $w(a_1,\dotsc,a_{n-1})=\prod_i a_i^{t_i}\chi_i(a_i)$, where $(t_1,\dotsc,t_{n-1})\in(\ints_{\ge0})^{n-1}$ with $t_1\ge\dotsb\ge t_{n-1}$, and each $\chi_i$ is a finite character of conductor $c_i$, satisfying technical restrictions. Let $\chi_{(1)},\dotsc,\chi_{(n-1)}$ be the characters $\chi_1,\dotsc,\chi_{n-1}$ reordered so that $\cond(\chi_{(1)})\le\cond(\chi_{(2)})\le\dotsb\le\cond(\chi_{(n-1)})$, let $c_{(i)}=\cond(\chi_{(i)})$, and let $T_{(i)}=T(\chi_{(i)})$. Then there is a constant $h$, a polynomial $d_{t_1,\dotsc,t_{n-1}}$ of total degree $\frac{n(n-1)}2$ in the $t_i$s, and a linear function $l(t_1,\dotsc,t_{n-1})$ such that the Newton polygon of $\det(I-XU_p|\ssc_w(G,\usc))$ contains at least $hp^{c_{(1)}+2c_{(2)}+\dotsb+(n-1)c_{(n-1)}-\frac{n(n-1)}2}d_{t_1,\dotsc,t_{n-1}}$ segments of slope at most $l(t_1,\dotsc,t_{n-1})$, hence passes below the point $\left(hp^{c_{(1)}+2c_{(2)}+\dotsb+(n-1)c_{(n-1)}-\frac{n(n-1)}2}d_t,hp^{c_{(1)}+2c_{(2)}+\dotsb+(n-1)c_{(n-1)}-\frac{n(n-1)}2}d_tl(t)\right)$ $=\left(x,A_2\left(v(T_{(1)})^{\frac{2}{n(n-1)}}v(T_{(2)})^{\frac{2\cdot 2}{n(n-1)}}\dotsb v(T_{(n-1)})^{\frac{2\cdot(n-1)}{n(n-1)}}\right)x^{1+\frac2{n(n-1)}}\right)$ for $x=hp^{c_{(1)}+2c_{(2)}+\dotsb+(n-1)c_{(n-1)}-\frac{n(n-1)}2}d_t$ and a constant $A_2$. Note that in particular, $v(T_{(1)})^{\frac{2}{n(n-1)}}v(T_{(2)})^{\frac{2\cdot 2}{n(n-1)}}\dotsb v(T_{(n-1)})^{\frac{2\cdot(n-1)}{n(n-1)}}\le\max_i v(T_i).$ \end{enumerate} \end{thm} We also leverage Theorem~\ref{mybounds} to prove two statements that may be more geometrically satisfying. First, we prove the following alternative version of the upper bound which provides infinitely many upper bound points on the same Newton polygon. \begin{thm} \label{unified-ubd} %There is a constant $A_2$ (depending on $n$, $p$, and $h$) such that for every locally algebraic weight $\chi t$ and radius $r>0$, there is a weight $s$ such that $|T_i(\chi t)-T_i(s)|0$, there is a weight $s$ such that $|T_i(w)-T_i(s)|2 \text{ and }c_i=1 \\ \frac{q}{p^{c_i-1}(p-1)} & \text{ if } p>2 \text{ and } c_i\ge2 \\ v(t_iq) & \text{ if }p=2 \text{ and }c_i=3 \\ \frac{q}{p^{c_i-1}(p-1)}=\frac1{2^{c_i-3}} & \text{ if }p=2 \text{ and }c_i\ge4. \end{cases} \] For completeness, we quickly prove the second case above; the others are similar. The value$\chi_i(\exp(q))=\chi_i(\exp(p))$is a primitive$p^{c_i}$th root of unity, say$\zt_{p^{c_i}}$. Let $f(X)=\frac{X^{p^{c_i}}-1}{X^{p^{c_i-1}}-1}=\prod_{a\in(\ints/p^{c_i}\ints)^\times}(X-\zt_{p^{c_i}}^a)=X^{p^{c_i}-p^{c_i-1}}+X^{p^{c_i}-2p^{c_i-1}}+\dotsb+X^{p^{c_i-1}}+1.$ Then$f(1)=p=\prod_{a\in(\ints/p^{c_i}\ints)^\times}(1-\zt_{p^{c_i}}^a)$. Each term in the product has the same valuation, since they are Galois conjugate, and there are$p^{c_i-1}(p-1)$such terms. So$v(\chi_i(\exp(q))-1)=\frac1{p^{c_i-2}(p-1)}$. The factor of$\exp(t_iq)$has no effect since it is$1\pmod{p}$. %should I include this calculation? \end{exmp} In general, if$A$is a Banach$\rats_p$-algebra, we say that a character$s:\ints_p^\times\to A^\times$is$c$-locally analytic if its restriction to$1+p^c\ints_p$is given by a convergent power series with coefficients in$A$. Every continuous character$s$is$c$-locally analytic for some$c$: let$T=s(\exp(q))-1$and choose$c$such that$\|T^{p^c/q}\|l}$with coefficients in$p\ints_p$. Similarly, $p(uz)_{ij}=Z_{j,\sg_{ij}/1}(u^{-1}\bar{N}(\un{z}))=\frac{a_{j,\sg,1_j}+\sum_{\#\tau=j,\tau\neq 1_j}a_{j,\sg,\tau}(u)Z_{j,\tau/1}(\bar{N}(\un{z}))}{a_{j,1_j,1_j}+\sum_{\#\tau=j,\tau\neq 1_j}a_{j,1_j,\tau}(u)Z_{j,\tau/1}(\bar{N}(\un{z}))}$ where$Z_{j,\tau/1}(\bar{N}(\un{z}))$is again a polynomial in the variables$\{z_{kl}\}_{l\le j,k>l}$with coefficients in$p\ints_p$. %$%e_\sg\cdot\it_j(ux)=e_\sg\cdot\it_j(u)\cdot\it_j(x)=\left(\left(\prod_{k\in\sg}u_{kk}\right)e_\sg+\sum_{\tau>\sg}a_\tau(u)e_\tau\right)\cdot\it_j(x) %e_\sg\cdot\it_j(xb)=e_\sg\cdot\it_j(x)\cdot\it_j(b)=\left(Z_{j,\sg}(x)e_1\w\dotsb\w e_j+\sum_{\tau>\{1,\dotsc,j\}}a_\tau(x)e_\tau\right) %$ \subsection{Notation for subgroups of$\iw_p$} \label{subgpnotation} Since we will work with numerous subgroups of$\iw_p$, we will introduce some notation to identify them. If$\un{c}=(c_{ij})\in\ints_{\ge0}^{n\times n}$is any$n\times n$matrix of nonnegative integers, we will write $\Gam(\un{c})=\{(x_{ij})\in GL_n(\ints_p)\mid p^{c_{ij}}\mid (x_{ij}-\del_{ij})\text{ for all }i,j\}.$ One can compute that$\Gam(\un{c})$is a group precisely when$c_{ij}\le c_{ik}+c_{kj}$for all$i,j,k$. Note that this means that if$\Gam(\un{c})$is a group, then so is$T(\ints_p)\Gam(\un{c})$. If we instead only have half a matrix of nonnegative integers$\un{c}=(c_{ij})_{n\ge i>j\ge1}\in\ints_{\ge0}^{n(n-1)/2}$, we will write $\Gam_1(\un{c})=\{(x_{ij})\in\iw_p\mid v(x_{ij})\ge c_{ij}\forall i>j;v(x_{ii}-1)\ge \min\{c_{ij}|ji\}\forall i\}%\subset\Gam_1(p)$ $\Gam_0(\un{c})=\{(x_{ij})\in\iw_p\mid v(x_{ij})\ge c_{ij}\forall i>j\}=T(\ints_p)\Gam_1(\un{c})\subset\iw_p.$ \begin{defn} We say that$\un{c}=(c_{ij})_{n\ge i>j\ge1}\in\ints_{\ge0}^{n(n-1)/2}$is \emph{group-shaped} if$c_{ij}\le c_{ik}+c_{kj}$for all$k$, where we set$c_{ab}$to be$0$if$a\le b$. \end{defn} Thus$\Gam_1(\un{c})$and$\Gam_0(\un{c})$are subgroups whenever$\un{c}$is group-shaped. \begin{defn} We call an$n(n-1)/2$-tuple$\un{c}=(c_{ij})_{n\ge i>j\ge1}\in\ints_{\ge0}^{n(n-1)/2}$\emph{compatible with} an$n$-tuple$(c_1,\dotsc,c_n)\in\ints_{\ge0}^n$if$c_i\le \min\{c_{ij}|ji\}\forall i$. Equivalently, if we define$\un{c}'\in\ints_{\ge0}^{n\times n}$by$c_{ij}'=c_{ij}$for$i>j$,$c_{ii}'=c_i$, and$c_{ij}'=0$for$i0}$is a single integer, we will write $\Gam(c)=\{(x_{ij})\in GL_n(\ints_p)\mid v(x_{ij}-\del_{ij})\ge c\text{ for all }i,j\}.$ This is always a group. We may define$\Gam_1(c),\Gam_0(c)$similarly. Finally, if$r$is a single real number in$[0,1]$, we will write$\Gam(r),\Gam_1(r),\Gam_0(r)$for the obvious final abuse of the same notation. \subsection{The sheaf of$p$-adic automorphic forms on weight space} \label{sheafpadicautforms} If$\un{c}=(c_{ij})_{n\ge i>j\ge1}\in\ints_{>0}^{n(n-1)/2}$, we say that$f\in\ssc_s$is$\un{c}$-locally analytic if, for any$\un{a}=(a_{ij})\in\ints_p^{n(n-1)/2}$, the restriction of$f$to $B(\un{a},\un{c})=\{z=(z_{ij})_{n\ge i>j\ge1}\in\ints_p^{n(n-1)/2}\mid z_{ij}\in a_{ij}+p^{c_{ij}}\ints_p\forall i,j\}$ is given by a convergent power series in the variables$z_{ij}$with coefficients in$A$. \begin{defn} We call an$n(n-1)/2$-tuple$\un{c}=(c_{ij})_{n\ge i>j\ge1}\in\ints_{\ge0}^{n(n-1)/2}$\emph{analytic-shaped} if we have$c_{(j+1)j}=c_{(j+2)j}=\dotsb=c_{nj}$for all$j$and$c_{nj}\ge c_{n(j+1)}$for all$j$. (Note that if$\un{c}$is analytic-shaped it is also group-shaped.) We call$\un{c}$\emph{compatible with} an$n$-tuple$(c_1,\dotsc,c_n)\in\ints_{\ge0}^n$if$c_j\le\min_{l\le j,k>l}c_{kl}$for all$j$. That is, for each$j_0$, all the entries of$(c_{ij})$corresponding to matrix entries appearing in or to the left of the$j_0$th column should be at least$c_{j_0}$. \end{defn} \begin{defn} If$\un{c}\in\ints_{>0}^{n(n-1)/2}$is analytic-shaped, we say that$s:(\ints_p^\times)^n\to A^\times$is$\un{c}$-locally analytic if there is$(c_1,\dotsc,c_n)$such that$s$is$(c_1,\dotsc,c_n)$-locally analytic and$\un{c}$is compatible with$(c_1,\dotsc,c_n)$. \end{defn} \begin{prop} \label{locanpres} If$s$is$(c_1,\dotsc,c_n)$-locally analytic and$f\in\ssc_s$is$\un{c}$-locally analytic for$\un{c}$analytic-shaped and compatible with$(c_1,\dotsc,c_n)$(so that$s$is$\un{c}$-locally analytic), then$uf$is also$\un{c}$-locally analytic for all$u\in\iw_p$. \end{prop} \begin{proof} By the calculations in Section~\ref{plucker}, we have$(uf)(\bar{N}(\un{z}))=s(T(u,\un{z}))f(\bar{N}(\un{uz}))$where ---$(uz)_{ij}$is a power series in the variables$\{z_{kl}\}_{l\le j,k>l}$; ---the$j$th diagonal entry of$T(u,\un{z})$, or$\frac{t_j(T(u,\un{z}))}{t_{j-1}(T(u,\un{z}))}$, is also a power series in the variables$\{z_{kl}\}_{l\le j,k>l}$. So if we restrict to$\un{z}\in B(\un{a},\un{c})$, the coefficient$(uz)_{ij}$ranges over a ball of the form$a_{ij}'+p^{\min_{l\le j,k>l}c_{kl}}\ints_p$; since$\un{c}$is analytic-shaped, we have$c_{ij}\le\min_{l\le j,k>l}c_{kl}$, and we conclude that$\un{uz}$is also restricted to a ball of the form$B(\un{a}',\un{c})$. Thus$f(\bar{N}(\un{uz}))$is analytic for$\un{z}\in B(\un{a},\un{c})$. Similarly,$\frac{t_j(T(u,\un{z}))}{t_{j-1}(T(u,\un{z}))}$ranges over a ball of the form$a_{jj}''+p^{\min_{l\le j,k>l}c_{kl}}\ints_p$; since$c_j\le\min_{l\le j,k>l}c_{kl}$and$s_j$is analytic on$a_{jj}'+p^{c_j}\ints_p$, we conclude that$s_j(T(u,\un{z}))$is analytic for$\un{z}\in B(\un{a},\un{c})$. Thus$(uf)(\bar{N}(\un{z}))$is analytic for$\un{z}\in B(\un{a},\un{c})$, as desired. \end{proof} By Proposition~\ref{locanpres}, if$s$is$\un{c}$-locally analytic with$\un{c}$group-like, the space$\ssc_{s,\un{c}}=\ind_{B(\ints_p)}^{\iw_p,\un{c}-loc.an.}(s)$, where$\un{c}-loc.an.$stands for$\un{c}$-locally analytic, is well-defined and has an action by$\iw_p$. %if$\usc_p=\iw_p$,$\ind_{B(\ints_p)}^{\iw_p,cts}(\chi)(G,\usc)$has an action by$B(\rats_p)\usc$. We let$\ssc=\ssc_{[\cdot]}=\ind_{B(\ints_p)}^{\iw_p,cts}([\cdot])$. If$\usc_p=\iw_p$, we call $\ssc(G,\usc)=\ind_{B(\ints_p)}^{\iw_p,cts}([\cdot])(G,\usc)$ the space of integral$p$-adic automorphic forms for$G$of level$\usc$; it has an action by$B(\rats_p)\usc$. This gives a sheaf on$\wsc$whose fiber over$s$is $\ssc_s(G,\usc)=\ind_{B(\ints_p)}^{\iw_p,cts}(s)(G,\usc).$ %When$A=\cplx_p$, we refer to the point of$\wsc(\cplx_p)$corresponding to$\chi:(\ints_p^\times)^n\to\cplx_p^\times$by the coordinates$(\chi(e_1),\dotsc,\chi(e_n))$, where$e_i$has$\exp(q)$in the$i$th position and$1$s elsewhere. %Note that$\ind_{B(\ints_p)}^{\iw_p,cts}$Similarly, let$\ssc_{W,\un{c}}=\ssc_{[\cdot]_W,\un{c}}=\ind_{B(\ints_p)}^{\iw_p,\un{c}-loc.an.}([\cdot]_W)$(for any$\un{c}$such that$[\cdot]_W$is$\un{c}$-locally analytic). If$\usc_p=\iw_p$, we call $\ssc_{W,\un{c}}(G,\usc)=\ind_{B(\ints_p)}^{\iw_p,\un{c}-loc.an.}([\cdot]_W)(G,\usc)$ the space of$\un{c}$-locally analytic$p$-adic automorphic forms for$G$of level$\usc$; this does \emph{not} have an action by$B(\rats_p)$, as some elements of$B(\rats_p)$do not preserve the radius of local analyticity, but we will see in the next section that it has an action by a certain submonoid. \subsection{The operators$U_p^a$} \label{upa} If$H$is any locally compact, totally disconnected topological group, we write$\hsc(H)$for the$k$-algebra of compactly supported, locally constant$k$-valued functions on$H$with the convolution product $(\phi_1\star\phi_2)(g)=\int_{h\in H}\phi_1(h)\phi_2(h^{-1}g)d\mu$ where$\mu$is a Haar measure on$H$. This algebra usually has no identity, but many idempotents. If$K$is a compact open subgroup of$H$, the idempotent$e_K=\frac{\one_K}{\mu(K)}$projects$\hsc(H)$onto the subalgebra$\hsc(H\sslash K)$of$K,K$-bi-invariant functions. If$V$is a smooth$H$-module, it is an$\hsc(H)$-module via $\phi(v)=\int_H \phi(h)(hv)dh$ and similarly$V^K$is an$\hsc(H\sslash K)$-module. In the particular case$H=B(\rats_p)\usc$,$V=\ssc_s(G,\usc)$,$K=\usc$, we can rephrase this as follows. We sometimes write$[\usc\zt\usc]$for the element$\one_{\usc\zt\usc}$of$\hsc(G(\aff_f)\sslash \usc)$. If$\zt_1,\dotsc,\zt_r$are left$\usc$-coset representatives of$\usc\zt\usc$, so that $\usc\zt\usc=\coprod_{i=1}^r\zt_i\usc,$ then for any$\phi\in \ssc_s(G,\usc)$and$x\in G(\rats)\bsl G(\aff_f)$, we have $[\usc\zt\usc](\phi)(x)=\int_{G(\aff_f)}[\usc\zt\usc](g)\cdot(g.\phi)(x)dg$ $=\int_{\usc\zt\usc}g_p\phi(xg)dg = \sum_{i=1}^r(\zt_i)_p.\phi(x\zt_i).$ The following is Lemma 4.5.2 of~\cite{chenevier04}, or Proposition 3.3.3 of~\cite{loeffler10}. \begin{lem} \label{hcoords} Fix coset representatives$x_1,\dotsc,x_h$of$G(\rats)\bsl G(\aff_f)/\usc$, and thus an isomorphism$\ssc_s(G,\usc)\cong\ssc_s^h$. Then we have $[\usc\zt\usc](\phi)(x_j)=\sum_{k=1}^r\sum_{i\mid\zt_i\in x_j^{-1}G(\rats)x_k\usc}(\zt_iu_{ij}^{-1})_p.\phi(x_k)$ for some$u_{ij}\in\usc$. That is, the action of$[\usc\zt\usc]$on$\ssc_s(G,\usc)$is of the form$\sum T_j\circ\sg_j$, where the$\sg_j$s are compositions of permutation operators on the entries of vectors in$\ssc_s^h$with projections onto one of the coordinates, and the$T_j$s are diagonal translations of$\ssc_s^h$by elements of$\usc\zt\usc$. \end{lem} \begin{proof} Write$x_j\zt_i$in the form$d_{ij}x_{k_{ij}}u_{ij}$where$d_{ij}\in G(\rats)$and$u_{ij}\in\usc$. Then $[\usc\zt\usc](\phi)(x_j)=\sum_{i=1}^r(\zt_i)_p.\phi(x_j\zt_i)$ $=\sum_{i=1}^r(\zt_i)_p.\phi(d_{ij}x_{k_{ij}}u_{ij}) =\sum_{i=1}^r(\zt_iu_{ij}^{-1})_p.\phi(x_{k_{ij}}).$ The values of$i$for which$k_{ij}=k$are those for which$\zt_i=x_j^{-1}dx_ku$for some$d\in G(\rats)$and$u\in\usc$, that is,$\zt_i\in x_j^{-1}G(\rats)x_k\usc$. \end{proof} If$a=(a_1,\dotsc,a_n)\in\ints^n$, we write $u^a=\diag(p^{a_1},\dotsc,p^{a_n})$ and define the subgroup $\Sg=\{u^a=\diag(p^{a_1},\dotsc,p^{a_n})\mid a=(a_1,\dotsc,a_n)\in\ints^n\}\subset GL_n(\rats_p)$ and its submonoids $\Sg^-=\{u^a=\diag(p^{a_1},\dotsc,p^{a_n})\mid a_1\ge a_2\ge\dotsb\ge a_n\}\subset \Sg$ $\Sg^{--}=\{u^a=\diag(p^{a_1},\dotsc,p^{a_n})\mid a_1> a_2>\dotsb> a_n\}\subset \Sg^-.$ We will frequently choose$\zt$to be an element of$\Sg^-$. Let $U_p^a=[\usc\diag(p^{a_1},\dotsc,p^{a_n})\usc].$ \begin{prop} \label{scale} If$f\in\ssc_s$and$a=(a_1,\dotsc,a_n)\in\ints^n$,$u^a$acts on$f$by$z_{ij}\mapsto p^{a_i-a_j}z_{ij}$. \end{prop} \begin{proof} We have $f((u^a)^{-1}\bar{N}(z_{ij}))=f\left( \begin{pmatrix} p^{-a_1} & \dotsb & 0 \\ 0 & \vdots & 0 \\ 0 & \dotsb & p^{-a_n} \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & \dotsb & 0 \\ pz_{21} & 1 & 0 & \dotsb & 0 \\ pz_{31} & pz_{32} & 1 & \dotsb & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ pz_{n1} & pz_{n2} & pz_{n3} & \dotsb & 1 \end{pmatrix} \right)$ $=f\begin{pmatrix} p^{-a_1} & 0 & 0 & \dotsb & 0 \\ p^{-a_2+1}z_{21} & p^{-a_2} & 0 & \dotsb & 0 \\ p^{-a_3+1}z_{31} & p^{-a_3+1}z_{32} & p^{-a_3} & \dotsb & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ p^{-a_n+1}z_{n1} & p^{-a_n+1}z_{n2} & p^{-a_n+1}z_{n3} & \dotsb & p^{-a_n} \end{pmatrix}$ $=f\left(\begin{pmatrix} 1 & 0 & 0 & \dotsb & 0 \\ p^{a_1-a_2+1}z_{21} & 1 & 0 & \dotsb & 0 \\ p^{a_1-a_3+1}z_{31} & p^{a_2-a_3+1}z_{32} & 1 & \dotsb & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ p^{a_1-a_n+1}z_{n1} & p^{a_2-a_n+1}z_{n2} & p^{a_3-a_n+1}z_{n3} & \dotsb & 1 \end{pmatrix} \begin{pmatrix} p^{-a_1} & \dotsb & 0 \\ 0 & \vdots & 0 \\ 0 & \dotsb & p^{-a_n} \end{pmatrix} \right)$ $=f(\bar{N}(p^{a_i-a_j}z_{ij}))s^0(u^a)=f(\bar{N}(p^{a_i-a_j}z_{ij})).$ \end{proof} \begin{cor} If$f\in\ssc_s$is$\un{c}$-locally analytic and$u^a\in \Sg^-$, then$u^af$is also$\un{c}$-locally analytic. So translation by$\iw_pu^a\iw_p$preserves$\ssc_{s,\un{c}}$(and hence, by Lemma~\ref{hcoords},$U_p^a$preserves$\ssc_{s,\un{c}}(G,\usc)$). \end{cor} \begin{proof} When$u^a\in \Sg^-$, we have$a_i-a_j\ge0$for all$i>j$; thus if$(z_{ij})$varies in a ball$B(\un{a},\un{c})$, so does$(p^{a_i-a_j}z_{ij})=(u^az_{ij})$. \end{proof} Let$\un{c}^0\in\ints_{>0}^{n(n-1)/2}$be minimal such that$s$is$\un{c}^0$-locally analytic. %If$c_j$is minimal such that$s_j$is$c_j$-locally analytic, this means setting$c_{ij}^0=\max_{k\le j}c_k$for all$i>j$. \begin{cor} \label{shrink} If$f\in\ssc_s$is$\un{c}$-locally analytic and$u^a\in \Sg^{--}$, then$u^af$is$\un{c}^{--}:=(\max\{c_{ij}-1,c_{ij}^0\})$-locally analytic. So translation by$\iw_pu^a\iw_p$takes$\ssc_{s,\un{c}}$into$\ssc_{s,\un{c}^{--}}$(and hence, by Lemma~\ref{hcoords},$U_p^a$takes$\ssc_{s,\un{c}}(G,\usc)$into$\ssc_{s,\un{c}^{--}}(G,\usc)$). \end{cor} \begin{proof} When$u^a\in \Sg^{--}$, we have$a_i-a_j>0$for all$i>j$; thus if$(z_{ij})$varies in a ball$B(\un{a},\un{c})$, then$(p^{a_i-a_j}z_{ij})=(u^az_{ij})$varies in a smaller ball$B(\un{a}',\un{c+1})$. \end{proof}$\ssc_{s,\un{c}}$is an orthonormalizable$A$-module, for which we choose the following orthonormal basis: for each$\un{a}\in\prod_{n\ge i>j\ge1} \ints_p/p^{c_{ij}}\ints_p$, we choose the set of monomials$\prod_{n\ge i>j\ge1}z_{ij}^{e_{ij}}$as an orthonormal basis for the restriction of$\ssc_{s,\un{c}}$to$B(\un{a},\un{c})$; then for$\ssc_{s,\un{c}}$, we may choose as orthonormal basis the set of monomials$\prod_{n\ge i>j\ge1}(z_{ij}^{\un{a}})^{e_{ij}}$, with one copy for each$\un{a}\in\prod_{n\ge i>j\ge1} \ints_p/p^{c_{ij}}\ints_p$. \begin{cor} When$a\in \Sg^{--}$, the operator of translation by$u^a$acts \emph{completely continuously} on$\ssc_{s,\un{c}}$, in the sense that it is a uniform limit of operators with finite-dimensional images. So by Lemma~\ref{hcoords},$U_p^a$is completely continuous on$\ssc_{s,\un{c}}(G,\usc)$. \end{cor} \begin{proof} %$\ssc_{s,\un{c}}$has the following orthonormal basis (oops, what norm?) over$A$: for each$\un{a}\in\prod_{n\ge i>j\ge1} \ints_p/p^{c_{ij}}\ints_p$, the restriction of$\ssc_{s,\un{c}}$to$B(\un{a},\un{c})$has as orthonormal basis the set of monomials$\prod_{n\ge i>j\ge1}z_{ij}^{e_{ij}}$; thus$\ssc_{s,\un{c}}$has as orthonormal basis the set of monomials$\prod_{n\ge i>j\ge1}(z_{ij}^{\un{a}})^{e_{ij}}$, with one copy for each$\un{a}\in\prod_{n\ge i>j\ge1} \ints_p/p^{c_{ij}}\ints_p$. By Proposition~\ref{scale},$u^a$scales$\prod_{n\ge i>j\ge1}(z_{ij}^{\un{a}})^{e_{ij}}$by$\prod_{n\ge i>j\ge1}p^{(a_i-a_j)e_{ij}}$, which goes to$\infty$as any$e_{ij}$goes to$\infty$. Furthermore, since the formulas in Section~\ref{plucker} all have integer coefficients, it is clear that translation by$\iw_p$is norm$1$. \end{proof} Since$U_p^a$is completely continuous on$\ssc_{s,\un{c}}(G,\usc)$, for any$k$, the matrix of the action of$U_p^a$(in any basis) has a finite number of nonzero rows mod$p^k$. Suppose that this matrix has$r_k$rows that are zero mod$p^k$but nonzero mod$p^{k+1}$. Then for any$N\ge r_0+r_1+\dotsb+r_k$, the coefficient of$X^N$in the characteristic power series $P_{s,\un{c}}^a(X)=\det(1-XU_p^a|\ssc_{s,\un{c}}(G,\usc))$ of$U_p^a$acting on$\ssc_{s,\un{c}}(G,\usc)$, being a linear combination of minors of size$N\ge r_0+r_1+\dotsb+r_k$, is divisible by$r_1+2r_2+\dotsb+kr_k$. Since this lower bound grows faster than any linear function of$N$,$P_{s,\un{c}}^a(X)$is an entire function of$X$. \begin{prop} \label{radindep}$P_{s,\un{c}}^a(X)$is independent of$\un{c}$. (So we will henceforth call it$P_s^a(X)$.) \end{prop} \begin{proof} This follows from applying Corollary 2 of Proposition 7 of~\cite{serre62} to the map$U_p^a:\ssc_{s,\un{c}}(G,\usc)\to\ssc_{s,\un{c}^{--}}(G,\usc)$from Corollary~\ref{shrink} and the obvious inclusion$\ssc_{s,\un{c}^{--}}(G,\usc)\inj \ssc_{s,\un{c}}(G,\usc)$. %some reorganization is going to be needed here, because I think the rigorous proof also requires Proposition~\ref{levels} below and is generally more complicated than I thought it was. But I think the fundamental idea is just that since$U_p^a$takes$\ssc_{s,\un{c}}(G,\usc)$into$\ssc_{s,\un{c}^{--}}(G,\usc)$by Corollary~\ref{shrink}, any eigenvector of$U_p^a$in$\ssc_{s,\un{c}}(G,\usc)$must already be in$\ssc_{s,\un{c}^{--}}(G,\usc)$. Note that this means the complement of$\ssc_{s,\un{c}^{--}}(G,\usc)$in$\ssc_{s,\un{c}}(G,\usc)$does not contain any finite-slope eigenforms. \end{proof} %not needed, but consider adding level-insensitivity proposition here for completeness. Let$U_p^\Sg$be the subring of$\hsc(G(\aff_f)\sslash \usc)$generated by the elements$U_p^a$for$a\in \Sg^-$. By Proposition 6.4.1 of \cite{bc09}, the map from$k[\Sg]$to$U_p^\Sg$sending$u^a$to$U_p^b(U_p^c)^{-1}$where$u^b,u^c$are any elements of$\Sg^-$such that$u^a=u^b(u^c)^{-1}$is a well-defined isomorphism of rings. So, in particular,$U_p^\Sg$is abelian. Let$\hsc$be a subalgebra of$\hsc(G(\aff_f)\sslash\usc)$given by the product of$\ints[U_p^\Sg]$at$p$and some commutative subalgebra of$\hsc(G(\aff_f^p)\sslash\usc^p)$away from$p$. We write$u_i$for the image of$\diag(1,\dotsc,1,p,1,\dotsc,1)\in k[\Sg]$in$U_p^\Sg$. If$f\in\ssc_{s,\un{c}}(G,\usc)$is a simultaneous eigenvector for$\hsc$, let$u_i(f)=\lam_i f$. We call these the$\lam$-values associated to$f$. Unless otherwise specified, we will generally set$\usc$to be a compact open subgroup of$G(\aff_f)$given by the product of$\iw_p$at$p$and a fixed tame level structure away from$p$chosen so that$x^{-1}G(\rats)x\cap U_0(p)=1$for all$x$. Call this subgroup$U_0(p)$. (Note that for the same reason as in Proposition~\ref{levels} below, our choice of$\iw_p$as the wild level structure does not actually affect$P_s^a(X)$.) \subsection{The eigenvariety} \label{spectralvariety} Given our setup so far, the eigenvariety is easy to define. For a given$u^a\in\Sg^{--}$, let$\zsc^a$be the subvariety of$\wsc\times\gra_m$which, in any subset$W\times\gra_m$where$W\subset\wsc$is open affinoid, is cut out by the characteristic power series$P_W^a(X)$of$U_p^a$acting on$\ssc_W(G,U_0(p))$. Let$w:\zsc^a\to\wsc$be the first projection (weight) map, and$s^a:\zsc^a\to\gra_m$the \emph{inverse} of the second projection (slope) map. Then for any point$z\in\zsc^a$,$s^a(z)$is a nonzero eigenvalue of$U_p^a$acting on$\ssc_{w(z)}(G,U_0(p))$, and for any$w\in\wsc$, all nonzero eigenvalues of$U_p^a$acting on$\ssc_w(G,U_0(p))$can be found in the fiber of$\zsc^a$over$w$. We call$\zsc^a$the spectral variety associated to$U_p^a$. It is convenient to fix a particular choice of$u^a\in\Sg^{--}$; we will choose$a=(n-1,n-2,\dotsc,1,0)$. From now on, we will write$U_p=U_p^{(n-1,n-2,\dotsc,1,0)}$and$\zsc=\zsc^{(n-1,n-2,\dotsc,1,0)}$. We call an eigenform$f\in\ssc_w(G,U_0(p))$finite-slope if$U_pf\neq0$(i.e. the slope of the$U_p$-eigenvalue is finite, and$f$appears on the eigenvariety), and infinite-slope otherwise. %Given our setup so far, the eigenvariety is easy to define. For a given$a\in\Sg^{--}$, let$\zsc^a$be the subvariety of$\wsc\times\gra_m$which, in any subset$W\times\gra_m$where$W\subset\wsc$is open affinoid, is cut out by the characteristic power series$P_W^a(X)$of$U_p^a$acting on$\ssc_W(G,U_0(p))$. Let$w:\zsc^a\to\wsc$be the first projection (weight) map, and$s^a:\zsc^a\to\gra_m$the \emph{inverse} of the second projection (slope) map. Then for any point$z\in\zsc^a$,$s^a(z)$is a nonzero eigenvalue of$U_p^a$acting on$\ssc_{w(z)}(G,U_0(p))$, and for any$w\in\wsc$, all nonzero eigenvalues of$U_p^a$acting on$\ssc_w(G,U_0(p))$can be found in the fiber of$\zsc^a$over$w$. We call an eigenform$f\in\ssc_w(G,U_0(p))$finite-slope if$U_pf\neq0$(i.e. the slope of the$U_p$-eigenvalue is finite, and$f$appears on the eigenvariety), and infinite-slope otherwise. We call$\zsc^a$the spectral variety associated to$U_p^a$. Let$\zsc=\zsc^{(n-1,n-2,\dotsc,1,0)}$. %\\$P_{(n-1,n-2,\dotsc,1,0)}(X)$of$U_p^{(n-1,n-2,\dotsc,1,0)}$acting on$\ssc(U_0(p))$. Since$\hsc$is commutative, we can construct the space$\dsc$whose points correspond to systems of eigenvalues of all Hecke operators in$\hsc$, including in particular all$U_p^a$s simultaneously, by simply taking$\dsc$to be the finite cover of$\zsc$which, over an affinoid$W\subset\wsc$, is given by the MaxSpec of the image of$\hsc\ten\Lam^n$in the endomorphism ring of$\ssc_W(G,U_0(p))$. Then$\dsc$inherits the weight map$w:\zsc^a\to\wsc$and each slope map$s^a:\dsc\to\gra^m$. Note that$\dsc\to\zsc^a$is degree$1$away from multiple roots of$P_W^a(X)$, hence degree$1$away from a Zariski-closed subset of$\wsc$of lower dimension. So in general, the bounds and geometric properties we get for$\zsc^a$will also apply to$\dsc$. For most of this paper, we will focus on the properties of$\zsc^a$for a fixed$a$. For additional details on properties of$\zsc^a$and$\dsc$and their proofs, see~\cite{chenevier04} or~\cite{buzzard07}. %We remark that one says that an eigenform$f\in\ssc_w(G,U_0(p))$is finite-slope if$U_pf\neq0$(i.e. the slope of the$U_p$-eigenvalue is finite), and that$f$is infinite-slope if$U_pf=0$. %may want more details here? %The sheaf$\ssc(G,V,U_0(p))$on$V$MaxSpec of the image of$\hsc$elaborate. %summarize claims about classical points and uniqueness to be proven below. \section{Locally algebraic weights} \label{localgweights} In this section, we analyze classical automorphic forms of locally algebraic weights and their associated automorphic representations. In Section~\ref{localgrepdef}, we define these spaces of classical forms and check their basic properties, including that they embed into the infinite-dimensional spaces of Section~\ref{padicautforms}. In Section~\ref{classicality}, we reproduce Bella\"iche-Chenevier's slope criterion guaranteeing that a given form is classical, phrased to work for locally algebraic weights instead of just algebraic weights; while this is not directly needed for our purposes, it is useful to give a sense of where classical forms fit in among the world of all$p$-adic automorphic forms. In Section~\ref{assocautrep}, we explain the standard translation between classical forms and automorphic representations. In Section~\ref{smstruct}, we analyze certain Iwahori subrepresentations that may appear in the local component at$p$of such an automorphic representation, including a particularly important irreducible subrepresentation. In Section~\ref{rochecalc}, we apply the work of Roche to a calculation of Hecke eigenvalues in ramified principal series. In Section~\ref{repstructure}, we identify a subspace of the classical forms whose associated automorphic representations have ramified principal series as their local components at$p$, and compute their$U_p$-eigenvalues in terms of the parameters of the corresponding principal series. \subsection{$p$-adic automorphic forms of locally algebraic weights} \label{localgrepdef} In Section~\ref{padicautforms}, we defined classical forms of algebraic weights via the algebraic representation$S_t(k)$of$GL_n(\rats_p)$. This construction may be generalized to locally algebraic weights as follows. Let$\chi=\chi_1\dotsb\chi_n$be a finite character of$(\ints_p^\times)^n$. Then$t\chi$is a locally algebraic character of$(\ints_p^\times)^n$, in the sense that it is algebraic upon restriction to$\prod_{i=1}^n(a_i+p^{m_i}\ints_p)$for some choice of$m_i$s and any nonzero$a_i$s. Similarly to earlier notation, for a positive integer$c$, let $B(\un{a},c)=\{z=(z_{ij})_{n\ge i>j\ge1}\in\ints_p^{n(n-1)/2}\mid z_{ij}\in a_{ij}+p^{c}\ints_p\forall i,j\}$ Then there are two equivalent definitions of the space $S_{t\chi,c}=\ind_{B(\ints_p)}^{\iw_p,c-loc. alg.}(t\chi)$ where$c-loc.alg.$stands for \emph{$c$-locally algebraic}. The first is through the usual induction operator above, as follows. We say that$f\in\ind_{B(\ints_p)}^{\iw_p}(t\chi)$is \emph{$c$-locally algebraic} if it has an algebraic extension to$B(\un{a},c)$for all$\un{a}\in\ints_p^{n(n-1)/2}$of degree bounded as follows: writing$f$as a polynomial in the variables$Z_{i,k/1}$as in Section~\ref{plucker}, we require that for each fixed$i$, the degree of$f$as a polynomial in all the variables$Z_{i,k/1}$should be at most$t_i-t_{i+1}=:m_i$. As in Proposition~\ref{locanpres}, one can see using the formulas in Section~\ref{plucker} that assuming$\cond(\chi_i)\le c$for all$i$, this condition is invariant under right translation by$\iw_p$. %check this again, bad at algebra right now. %check if the following sentence is still worded correctly. Also, this condition requires$\chi_i$to extend to an algebraic function on the$j_i(g)$s for all$g\in\iw_p$and$Z_{i,k/1}$varying over matrices in$B(\un{a},c)$, that is,$\chi$must extend to$T\Gam(c)$with the extension trivial on$\Gam(c)$. The second definition, coming from the perspective of Loeffler (Section 2.5 of~\cite{loeffler10}), is $\left(\ind_{B(\ints_p)}^{\iw_p,alg}t\right)\ten\left(\ind_{B(\ints_p)/B(\ints_p)\cap \Gam(c)}^{\iw_p/\Gam(c)}\chi\right).$ Note that$\Gam(c)$is normal in$\iw_p$because it is the kernel of the reduction map from$\iw_p$to the corresponding group with coefficients in$\ints_p/p^c\ints_p$. %where$J_1$is a compact open subgroup to which$\chi$extends and is trivial (figure out dependence of$J_1$on$r$). Except for an annoying technical distinction which we will discuss at the end of this subsection, the space$\ind_{B(\ints_p)}^{\iw_p,alg}t$is the same (as an$\iw_p$-representation) as the space$S_t(k)$defined in Section~\ref{padicautforms}, since$\iw_p$is Zariski-dense in$GL_n$. Let$d_t=\dim\ind_{B(\ints_p)}^{\iw_p,alg}t$. %Mark thinks this might be a lie before inverting p. Check. \begin{prop} The natural map $\left(\ind_{B(\ints_p)}^{\iw_p,alg.}t\right)\ten\left(\ind_{B(\ints_p)/B(\ints_p)\cap \Gam(c)}^{\iw_p/\Gam(c)}\chi\right)\to \ind_B^{\iw_p,c-loc. alg.}(t\chi)$ $f\ten g\mapsto fg$ is an isomorphism. \end{prop} \begin{proof} To construct an inverse, let$\phi\in \ind_{B(\ints_p)}^{\iw_p,c-loc.alg.}\chi$. Let$\phi_{alg}:\iw_p\to\cplx$be defined by $\phi_{alg}(b\bar{n})=t(b)\phi'(\bar{n})$ for all$b\in B,\bar{n}\in\bar{N}\cap \iw_p$, where$\phi'$is the unique algebraic extension of$\phi|_{\bar{N}\cap \Gam(c)}$to$\bar{N}\cap \iw_p$. Let$\phi_{sm}:\iw_p/\iw_p\cap \Gam(c)\to\cplx$be defined by $\phi_{sm}(\bar{b}\bar{\bar{n}})=\chi(b)(\phi/\phi')(\bar{n})$ where$b,\bar{n}$are any lifts of$\bar{b}\in B/B\cap \Gam(c)$,$\bar{\bar{n}}\in(\bar{N}\cap \iw_p)/(\bar{N}\cap \Gam(c))$. This suffices to prove surjectivity. %We showed that this map was surjective and$\iw_p$-equivariant in Proposition~\ref{tendecomp}. Injectivity follows from dimension counting: both sides have dimension$d_tp^{c\binom n2}$. \end{proof} \begin{rem} There is a simple isomorphism of$\iw_p$-representations $\ind_{B(\ints_p)/B(\ints_p)\cap \Gam(c)}^{\iw_p/\Gam(c)}\chi\isom\ind_{\Gam_0(c)}^{\iw_p}\chi$ so we could just as easily have phrased this section in terms of$\ind_{\Gam_0(c)}^{\iw_p}\chi$. For now, we have no particular reason to do this, but it may be more convenient for future work. \end{rem} We call $S_{t\chi,c}(G,\usc)=\ind_B^{\iw_p,c-loc. alg.}(t\chi)(G,\usc)$ the space of classical$p$-adic automorphic forms on$G$of weight$t\chi$, radius$c$, and level$\usc$. By the definitions, it embeds into$\ssc_{t\chi}(G,\usc)$, and we call its image a classical subspace of$\ssc_{t\chi}(G,\usc)$. The following proposition is a quick generalization of part 4 of Lemma 4 of~\cite{buzzard04}. %Let$t=(m_1,\dotsc,m_n)\in\nats^{n-1}\times\ints$. \begin{prop} \label{levels} For any positive integers$c$,$d$, and$e$with$d\le e$and$c+d-e\ge1$, we have a natural vector space isomorphism $S_{t\chi,c}(G,\usc^p\Gam_0(d))\cong S_{t\chi,c+d-e}(G,\usc^p\Gam_0(e)).$ such that systems of$\hsc$-eigenvalues (where$\hsc$is obtained with respect to$\usc^p\Gam_0(d)$) on the left go to identical systems of$\hsc$-eigenvalues on the right (where$\hsc$is obtained with respect to$\usc^p\Gam_0(e)$). %of$\hsc$-modules. %Furthermore, \end{prop} \begin{proof} %it is likely that at least one sign below has been flipped. Also, find a better name than$\osc$. For the purposes of this proposition, let$X=G(\rats)\bsl G(\aff_f)$. The left-hand side is the subset of (*) $(\hom(X,\cplx_p)\ten\ind_B^{\iw_p,c-loc. alg.}(t\chi))^{\Gam_0(e)}$ that remains invariant under a set of coset representatives$A$for$\Gam_0(e)\bsl\Gam_0(d)$. This subset has a map by restriction of the second factor to $(\hom(X,\cplx_p)\ten \osc)^{\Gam_0(e)}$ where$\osc$is the space of functions on$B((p^{e-d}\ints_p)^{n(n-1)/2},c)$that are algebraic on each ball$B(\un{a},c)$. The map is an isomorphism: if$\phi\in (\hom(X,\cplx_p)\ten \osc)^{\Gam_0(e)}$, its inverse$\psi$may be defined by $\psi(x)(z)=\phi(xa^{-1})(\bar{N}^{-1}(\bar{N}(z)a^{-1}))\text{ for }a\in A\text{ such that }za^{-1}\in B\left((p^{e-d}\ints_p)^{n(n-1)/2},c\right).$ In$\bar{N}(z)a^{-1}$,$a$should be interpreted as a coset representative for$\Gam_0(e-d+1)\bsl\iw_p$. Note that this inverse depends on the choice of coset representatives$A$. Now \\$B((p^{e-d}\ints_p)^{n(n-1)/2},c)$is isomorphic to$B(\ints_p^{n(n-1)/2},c+d-e)$via multiplication by$p^{d-e}$, so$(\hom(X,\cplx_p)\ten \osc)^{\Gam_0(e)}$is the desired right-hand side. To check that the Hecke operator action is preserved, it suffices to note that the Hecke operator action on the left-hand side can be calculated on its inclusion into (*). %make this more explicit? \end{proof} %This proposition only addresses subgroups that are only subject to congruence conditions on the entries below the diagonal. Subgroups subject to congruence conditions on both sides of the diagonal may be obtained by conjugating everything by diagonal matrices with$1$s and$p$s on the diagonal as appropriate. Subgroups of the form$\Gam_0(blah)$may be obtained by summing over possible finite torus characters. \begin{cor} \label{emb1} For all positive integers$c$and group-like$\un{d}\in\ints_{\ge0}^{n(n-1)/2}$, we have a vector space embedding %an embedding of$\hsc$-modules $S_{t\chi,c}(G,\usc^p\Gam_0(\un{d}))\inj \ssc_{t\chi}(G,U_0(p))$ preserving systems of$\hsc$-eigenvalues. \end{cor} \begin{proof} Let$d=\max d_{ij}$. Then we have an embedding $S_{t\chi,c}(G,\usc^p\Gam_0(\un{d}))\inj S_{t\chi,c}(G,\usc^p\Gam_0(d)).$ By Proposition~\ref{levels}, we have an isomorphism $S_{t\chi,c}(G,\usc^p\Gam_0(d))\cong S_{t\chi,c+d-1}(G,\usc^p\Gam_0(1))=S_{t\chi,c+d-1}(G,\usc^p\iw_p).$ The space on the right certainly embeds into$\ssc_{t\chi}(G,\usc^p\iw_p)=\ssc_{t\chi}(G,U_0(p))$as discussed above. \end{proof} For future reference, it will be important to note the following distinction between the space$S_t(G,1,U_0(p))$defined above and the space$S_t(k)(G,U_0(p))$of classical algebraic automorphic forms defined in Section~\ref{padicautforms}, which is that they are identical except for the normalization of the action of the$U_p$-operator. This is because, as in the beginning of Section~\ref{plucker}, the action of$u^a$on$S_t=\ind_{B(\ints_p)}^{\iw_p,alg}t=\ind_{B(\rats_p)}^{B(\rats_p)\iw_p,alg}t^0$implicitly arises from the extension of$t$to$t^0:(\rats_p^\times)^n\to\cplx$where$t^0(u^a)=1$, whereas the action of$u^a$on$S_t(k)$arises from the algebraic character$t:(\rats_p^\times)^n\to\cplx$, for which we can compute$t(u^a)=p^{\sum_i a_it_i}$. Thus we have $U_p^a|S_t(G,1,U_0(p))=p^{\sum_i a_it_i}U_p^a|S_t(k)(G,U_0(p)).$ %not happy with wording of this explanation. Especially notation needs improvement. \subsection{A classicality theorem following Bella\"iche-Chenevier} \label{classicality} This is essentially Proposition 7.3.5 of \cite{bc09}. We will just summarize the proof with modifications so that it also works for locally algebraic weights. \begin{thm} Let$f\in\ssc_{t\chi}(G,\usc)$where$t\chi=(t_1\chi_1,\dotsc,t_n\chi_n)$, in which the$t_i$are integers such that$t_1\ge\dotsb\ge t_n$and the$\chi_i$are finite, such that$f$is an eigenform for all operators$U_p^{(a_1,\dotsc,a_n)}$. Let$\lam_1,\dotsc,\lam_{n-1}$be the$\lam$-values associated to$f$as defined at the end of Section~\ref{upa}. If $v(\lam_1\lam_2\dotsb\lam_i)1. Thus it suffices to check that U_i has norm \le1 on Q_i'(G,\usc), which follows from the claim that any element of the form \[ \frac{g\left(\prod_{j=1}^iu_i\right)g'}{p^{m_i+1}}$ for$g,g'\in\iw_p$has norm$\le 1$on$Q_i'$. This follows from Lemma 7.3.6 of \cite{bc09}. \end{proof} \subsection{Automorphic representations associated to automorphic forms of locally algebraic weights} \label{assocautrep} Fix an isomorphism$\it_p:\bar{\rats}_p\isom\cplx$. Let$f\in\ssc_{t\chi}(G,U_0(p))$be a$p$-adic automorphic form coming from some classical subspace$S_{t\chi,c}(G,U_0(p))$. Let$W=\ind_{B(\ints_p)}^{\iw_p,c-loc. alg.}(t\chi)$, so that$f$is a function$G(\rats)\bsl G(\aff_f)\to W$. Following the proof of Proposition 3.8.1 of \cite{loeffler10}, let$W=W^{sm,c}(\chi)\ten S_t(\cplx)$, where, as in Section~\ref{localgrepdef}, $W^{sm,c}(\chi)=\ind_{B(\ints_p)/B(\ints_p)\cap \Gam(c)}^{\iw_p/\iw_p\cap \Gam(c)}\chi,$ $S_t(\cplx)=\ind_{B(\ints_p)}^{\iw_p,alg}t,$ and let$\rho_{sm},\rho_{alg}$denote the actions of$\iw_p$on$W^{sm,c}(\chi)\ten S_t(\cplx)$given by acting on only the first factor and only the second factor respectively. Then we can define a function$f_\infty:G(\aff)\to W$by$f_\infty(g)=\rho_{alg}(g_\infty^{-1}\it_p(g_p))f(g_f)$which satisfies the relation $f_\infty(gu)=\rho_{sm}(u_p)^{-1}\rho_{alg}(u_\infty)^{-1}f_\infty(g)$ for all$u\in G(\real)U_0(p)$. Equivalently,$f_\infty$can be viewed as the function $f_\infty^\vee:(W^{sm,c}(\chi)^\vee\ten S_t(\cplx)^\vee)\times G(\rats)\bsl G(\aff)\to \cplx$ $(\phi,x)\mapsto \phi(f_\infty(x))$ which satisfies $f_\infty^\vee(\phi,xu)=\phi(f_\infty(xu))=\phi(\rho_{sm}(u_p)^{-1}f_\infty(x))=f_\infty^\vee(u_p\phi,x)$ for all$u\in U_0(p)$. Thus for each$\phi\in W^{sm,c}(\chi)^\vee\ten S_t(\cplx)^\vee$, the function$f_\infty^\vee(\phi,\cdot)$is an element of$C(G(\rats)\bsl G(\aff),\cplx)$which generates under right translation by$\iw_p$a representation containing an irreducible component of$W^{sm,c}(\chi)^\vee$. The right translates of$f_\infty^\vee(\phi,\cdot)$under$G(\aff)$generate an automorphic representation$\pi_f$of$G(\aff)$which decomposes as a tensor product$\bigotimes_p'\pi_{f,p}$. We are interested in describing the structure of$\pi_{f,p}$. Note that this process is reversible, in that given$\psi\in C(G(\rats)\bsl G(\aff),\cplx)$which generates a representation containing an irreducible component of$W^{sm,c}(\chi)^\vee$under right translation by$\iw_p$, we get a unique$f_\psi\in S_{t\chi,c}(G,U_0(p))$. %Note that action of the Hecke operator$[U_0(p)\zt U_0(p)]$on$f$corresponds to the action of a \emph{different} Hecke operator on the image of$f_\infty^\vee(\phi,\cdot)$in$\pi_f$: the one associated to the function$G(\aff)\to\cplx$which is$\chi(x)$if$x\in G(\real)U_0(p)$and$0$otherwise. This is because the Hecke action on$S(G,t\chi,c,U_0(p))$comes from the group action$u(f)(x)=u_pf(xu)$, whereas the one on$\pi_f$comes from right translation. \subsection{Structure of$W^{sm,c}(\chi)$} \label{smstruct} We are interested in the representation $W^{sm,c}(\chi)=\ind_{B(\ints_p)/B(\ints_p)\cap \Gam(c)}^{\iw_p/\Gam(c)}\chi$ of$\iw_p$. Note that there is an obvious embedding$W^{sm,c}(\chi)\inj W^{sm,c+1}(\chi)$which takes$f\in W^{sm,c}(\chi)$to the composition of$f$with the reduction map$\iw_p/\Gam(c+1)\to\iw_p/\Gam(c)$. Let$J$be the compact open subgroup of$GL_n(\rats_p)$corresponding to$\chi$defined in Section 3 of \cite{roche98}; we have$J=\Gam(\un{c})$where$c_{ii}=0$,$c_{ij}=\flr{\cond(\chi_i\chi_j^{-1})/2}$if$ij$. Then$\chi$extends to a character of$J$which we will also call$\chi$; it is defined by the equation$\chi(j^-jj^+)=\chi(j)$when$j^-\in J\cap \bar{N}(\ints_p)$,$j\in T(\ints_p)$, and$j^+\in J\cap N(\ints_p)$. %Let$U^{sm}(\chi):=\ind_J^{\iw_p}\chi$. Now note that$W^{sm,c}(\chi)$contains the vector $f(\bar{x})=\begin{cases} \chi(j)\chi(b) & \text{ if }\bar{x}=\bar{j}\bar{b}\text{ with }j\in J\text{ and }b\in B(\ints_p),\\ 0 & \text{otherwise.} \end{cases}$ Note furthermore that for any$j\in J$and$\bar{x}\in \iw_p/\Gam(c)$, we have $(jf)(\bar{x})=f(j^{-1}\bar{x})=\chi(j^{-1})f(\bar{x})=\chi^{-1}(j)f(\bar{x})$ so that$f$is$(J,\chi^{-1})$-isotypic. \begin{prop} \label{irred} Assume$\chi=(\chi_1,\dotsc,\chi_n)$satisfies \begin{enumerate} \item \label{jbig} for all$i\neq j$,$\cond(\chi_i\chi_j^{-1})=\max(\cond(\chi_i),\cond(\chi_j))$; and \item \label{charclose} for all$i\neq j$with$i,j\neq n$,$\cond(\chi_i)<2\cond(\chi_j)$. \end{enumerate} Then$U^{sm}(\chi):=\ind_J^{\iw_p}\chi$is irreducible. \end{prop} \begin{proof} %Assume for the sake of simplicity that$\cond(\chi_i\chi_j^{-1})=\max(\cond(\chi_i),\cond(\chi_j))$for all$i\neq j$(for example, because all the$\cond(\chi_i)$s are different; or they're all the same but no pair of them has a character of smaller conductor as a quotient). Recall that$J=\Gam(\un{c})$where$c_{ij}=\flr{[\cond(\chi_i\chi_j^{-1})+1]/2}$if$i>j$, or \\$\flr{[\cond(\chi_i\chi_j^{-1})]/2}$if$ic_{jk}+c_{kj}-v(t_{jk}). \] This is just because we chose $i,j$ such that $c_{ij}+c_{ji}-v(t_{ji})\ge c_{jk}+c_{kj}-v(t_{jk})$, and such that if equality holds then $k>i$, in which case $v(s_{ki})\ge1$ since $s\in\iw_p$. So if we choose $b$ such that $v(b)=c_{ij}+c_{ji}-v(t_{ji})-1\ge c_{ij}$, then we have $v(bs_{ki}t_{jk})\ge c_{jk}+c_{kj}=\cond(\chi_k\chi_j^{-1})$ for all $k\neq i,j$, hence % %We need to choose $b$ such that $p^{c_{ij}+c_{ji}}$ does not divide $bt_{ji}$, that is, $v(b)c_{jk}+c_{kj}-v(t_{jk})$ or $c_{ij}+c_{ji}-v(t_{ji})=c_{jk}+c_{kj}-v(t_{jk})$ and $k>i$ in which case $v(s_{ki})\ge1$ since $s\in\iw_p$. For such a $b$, we have $\chi_k(1+bs_{ki}t_{jk})=1.$ Then we have $\chi_1(1+bs_{1i}t_{j1})\dotsb\chi_i(1+bs_{ii}t_{ji})\dotsb\chi_j(1+bs_{ji}t_{jj})\dotsb\chi_n(1+bs_{ni}t_{jn})$ $=\chi_i(1+bs_{ii}t_{ji})\chi_j(1+bs_{ji}t_{jj})=\chi_i(1+bs_{ii}t_{ji})\chi_j\left(1+b\left(\sum_{k\neq i}s_{ki}t_{jk}\right)\right) %\chi_j\left(\prod_{k\neq i}(1+bs_{ki}t_{jk})\right).$ since $v(bs_{ki}t_{jk})\ge\cond(\chi_j)$, but this is $\chi_i(1+bs_{ii}t_{ji})\chi_j(1-bs_{ii}t_{ji})$ since $\sum_{k}s_{ki}t_{jk}=\sum_k t_{jk}s_{ki}=(ts)_{ji}=0$, and this can be rewritten as $\frac{\chi_i}{\chi_j}(1+bs_{ii}t_{ji})\chi_j(1-b^2s_{ii}^2t_{ji}^2)=\frac{\chi_i}{\chi_j}(1+bs_{ii}t_{ji})$ because if $i>j$ then $v(b^2)\ge 2c_{ij}\ge c_{ij}+c_{ji}$ and if $ij$. \begin{lem} The $(J,\chi)$-isotypic piece of $\pi(\chi)$ is $1$-dimensional. \end{lem} \begin{proof} By Theorem 6.3 of \cite{roche98}, $\hsc(GL_n(\rats_p)\sslash J,\chi)$ is abelian. (To be precise, the theorem gives an isomorphism between $\hsc(GL_n(\rats_p)\sslash J,\chi)$ and $\hsc(W_\chi^0,S_\chi^0)\ten\cplx[\Om_\chi]$, where by our assumption that $\chi_i(p)\neq \chi_j(p)p$ for $i\neq j$, we have $W_\chi^0=S_\chi^0=1$, $\hsc(W_\chi^0,S_\chi^0)=\cplx$, and $\Om_\chi=\ints^n$.) Thus the $(J,\chi)$-isotypic piece of $\pi(\chi)$ decomposes as a representation of $\hsc(GL_n(\rats_p)\sslash J,\chi)$ into $1$-dimensional pieces. But by Theorem 9.2 of \cite{roche98}, because $\pi(\chi)$ is irreducible, the $(J,\chi)$-isotypic piece of $\pi(\chi)$ is irreducible as a representation of $\hsc(GL_n(\rats_p)\sslash J,\chi)$. Thus it is itself $1$-dimensional. \end{proof} \begin{lem} \label{psevals} If $a=(a_1,\dotsc,a_n)$ is such that $a_1\ge a_2\ge\dotsb\ge a_n$, the action of the element $[Ju^aJ]$ of $\hsc(GL_n(\rats_p)\sslash J,\chi)$ corresponding to $u^a=\diag(p^{a_1},\dotsc,p^{a_n})$ on the $(J,\chi)$-isotypic piece of $\pi(\chi)$ is multiplication by $\chi(u^a)=\chi_1(p^{a_1})\dotsb\chi_n(p^{a_n}).$ \end{lem} \begin{proof} %In the model of $\pi(\chi_1,\dotsc,\chi_n)$ given by functions $f:GL_n(\rats_p)\to\cplx$ such that $f(bg)=(\chi\del)(b)f(g)$ for all $b\in B(\rats_p)$ and $g\in GL_n(\rats_p)$, with $GL_n(\rats_p)$-action by right translation, The $(J,\chi)$-isotypic piece is generated by $f(g)=\begin{cases} (\chi\del)(b)\chi(j) & \text{ if }g=bj\text{ with }b\in B(\rats_p)\text{ and }j\in J,\\ 0 & \text{otherwise.} \end{cases}$ This is just because this function $f$ satisfies the $(J,\chi)$-isotypic condition by construction, and is well-defined because $(\chi\del)(b)=\chi(b)$ for any $b\in B\cap J$. We claim that (*) $%f(j_1uj_2)=\chi(j_1)\chi(u)\chi(j_2)=\chi(j_1)\chi(u)\chi(j_2)f(1)\text{ for any }j_1uj_2\in JuJ. f(ju^a)=\chi(j)\chi(u^a)\del(u^a)=\chi(j)\chi(u^a)\del(u^a)f(1)\text{ for any }j\in J.$ The lemma follows from this, because if $Ju^aJ=\coprod_{i=1}^rj_iu^aJ$, then $([Ju^aJ]f)(1)=\sum_{i=1}^r\chi(j_i)^{-1}f(j_iu^a)=\sum_{i=1}^r\chi(u^a)\del(u^a)f(1)=\chi(u^a)f(1)$ because $r=\del(u^a)^{-1}$ (reason: the same calculation as in Proposition~\ref{scale} shows that the index of $J$ in $[(u^a)^{-1}Ju^a]J$ is $p^{\sum_{i0}^{n(n-1)/2} associated to \chi defined immediately before Corollary~\ref{shrink} satisfies c_{ij}^0=c_i for all i>j. We claim that the intersection of \ssc_{t\chi,\un{c}^0}(G,U_0(p)) with S_{t\chi,c}(G,U_0(p)) is precisely (U^{sm}(\chi)\ten S_t)(G,U_0(p)). To show that (U^{sm}(\chi)\ten S_t)(G,U_0(p)) is contained in \ssc_{t\chi,\un{c}^0}(G,U_0(p)), it suffices to note that U^{sm}(\chi)\ten S_t is contained in \ssc_{t\chi,\un{c}^0}. This is because f\ten\phi\in U^{sm}(\chi)\ten S_t is clearly contained in \ssc_{t\chi,\un{c}^0} for the vector f\in U^{sm}(\chi) defined at the beginning of Section~\ref{smstruct} and any \phi\in S_t, and U^{sm}(\chi)\ten S_t is irreducible. To show that (U^{sm}(\chi)\ten S_t)(G,U_0(p)) exhausts \ssc_{t\chi,\un{c}^0}(G,U_0(p)) \cap S_{t\chi,c}(G,U_0(p)), we simply note that the latter space also has dimension hd_tp^{j(\chi)}, since as a vector space it is h copies of the locally algebraic vector subspace of \ssc_{t\chi,\un{c}^0}. By Proposition~\ref{radindep}, in order for U_pf to be nonzero, f must lie in \ssc_{t\chi,\un{c}^0}(G,U_0(p)); this completes the proof. If the c_i are not in decreasing order, by the beginning of Section~\ref{myubdproof}, the finite-slope subspace of S_{t\chi,c}(G,U_0(p)) has the same dimension as that of S_{t\chi^w,c}(G,U_0(p)) where \chi^w is \chi with the components rearranged so that the c_i are in decreasing order. This completes the argument for all \chi simple. \end{proof} The combination of Propositions~\ref{manyps} and~\ref{notfs-v2} gives us the following precise version of Theorem~\ref{classifyps-vague}. \begin{thm} \label{classifyps} If \chi is simple, then the finite-slope classical subspace of \ssc_{t\chi,c}(G,U_0(p)) is precisely (U^{sm}(\chi)^\perp\ten S_t)(G,U_0(p)). \end{thm} In the following, for convenience, we will sometimes refer to the algebraic weight (t_1,\dotsc,t_{n-1},0), t_1\ge\dotsb\ge t_{n-1}, by its successive differences m_1=t_1-t_2, m_2=t_2-t_3, ..., m_{n-1}=t_{n-1}. \begin{prop} \label{lamtopsi} Suppose that \chi is simple and f=f_x is an eigenform in (U^{sm}(\chi)\ten S_t)(G,U_0(p))\subset S_{t\chi,c}(G,U_0(p)). Suppose that we have \pi_{f,p}=\pi(\psi_1,\dotsc,\psi_n) (note that this is an equality because when \chi is simple, \pi(\psi_1,\dotsc,\psi_n) is irreducible). The \lam-values associated to x as in Section~\ref{upa} satisfy \[ \lam_i=p^{(n-1)/2-i+1-m_n-m_{n-1}-\dotsb-m_{n-i+1}}\psi_i(p).$ \end{prop} \begin{proof} %Follows from Lemma~\ref{psevals} and explain power of $p$. We are given that for all $u^a\in\Sg^-$, we have $U_p^af=\lam_1^{a_1}\dotsb\lam_n^{a_n}f$. Since any eigenvector of $U_p^a=[U_0(p)u^aU_0(p)]$ is also an eigenvector of $[Ju^aJ]$, we can calculate its eigenvalue using $[Ju^aJ]$ instead. Let $Ju^aJ=\coprod_{i=1}^r\zt_iJ.$ Then for any $\phi\in U^{sm}(\chi)\ten S_t$, we have $(U_p^af)_\infty^\vee(\phi,x)=\phi(\rho_{alg}(x_\infty^{-1}\it_p(x_p))(U_p^af)(x_f))$ $=\del^{1/2}(u^a)p^{-\sum a_it_i}\phi\left(\rho_{alg}(x_\infty^{-1}\it_p(x_p(\zt_i)_p))\sum_{i=1}^r\rho_{sm}((\zt_i)_p)f(x\zt_i)\right).$ Choose $\phi=\phi_{sm}\ten\phi_{alg}$ so that $\phi_{sm}$ is a $(J,\chi)$-isotypic vector in $U^{sm}(\chi)$. Then by definition $\phi(\rho_{sm}((\zt_i)_p)f(x\zt_i))=\psi((\zt_i)_p)\phi(f(x\zt_i))$ so $(U_p^af)_\infty^\vee(\phi,x)=\del^{1/2}(u^a)p^{-\sum a_it_i}\sum_{i=1}^r\psi((\zt_i)_p)\phi\left(\rho_{alg}(x_\infty^{-1}\it_p(x_p(\zt_i)_p))f(x\zt_i)\right)$ $=\del^{1/2}(u^a)p^{-\sum a_it_i}\sum_{i=1}^r\psi((\zt_i)_p) f_\infty^\vee(\phi,x\zt_i).$ That is, we have $\sum_{i=1}^r\psi((\zt_i)_p) f_\infty^\vee(\phi,x\zt_i)=\del^{-1/2}(u^a)p^{\sum a_it_i}\lam_1^{a_1}\dotsb\lam_n^{a_n}f_\infty^\vee(\phi,x).$ So the image of $f_\infty^\vee(\phi,\cdot)$ in $\pi_{f,p}$ is a $J$-new vector ($\hsc(J,\psi)$-module). By Lemma~\ref{psevals}, we have $\del^{-1/2}(u^a)p^{\sum a_it_i}\lam_1^{a_1}\dotsb\lam_n^{a_n}=\psi_1(p^{a_1})\dotsb\psi_n(p^{a_n}).$ The proposition follows. \end{proof} One application of this structure theory is the following comparison theorem between Chenevier's and Emerton's eigenvarieties. This is similar to Proposition 3.10.3 of \cite{loeffler10}. % What Chenevier is really computing is the Hecke eigenvalues of all eigenforms of all levels at $p$, with respect to whatever subgroup gives their level. %(but does Loeffler's version work for the entire eigenvariety, not just the center of weight space?) \begin{prop} There is a natural isomorphism from $\dsc$ as constructed in Section~\ref{construction} to Emerton's eigenvariety, the space $E(0,\usc^p)$ in Definiton 0.6 of~\cite{emerton06}. \end{prop} \begin{proof} By Proposition~\ref{jacquet}, over a locally algebraic weight, the automorphic representations that appear as classical points of $\dsc$ are precisely those whose $p$-parts have nonzero Jacquet module. But Emerton's eigenvariety also has this property built into its construction. In particular, by Proposition 2.3.3(iii) of \cite{emerton06}, $E(0,\usc^p)$ has a map to the space $\hat{T}$ of locally analytic characters of $T(\rats_p)$ such that points in the fiber over a character $\chi$ of $T(\rats_p)$ correspond to $\chi$-eigenspaces of Emerton's locally analytic Jacquet functor $J_B(H^0(\usc^p)_{\rats_p-loc.an.})$. For $\chi$ locally algebraic, by Section 0.13 of \cite{emerton06jac}, this is just the usual Jacquet module of $H^0(\usc^p)_{\rats_p-loc.an.}$. So under the natural map $E(0,\usc^p)\to\wsc$ given by composing $E(0,\usc^p)\to\hat{T}$ with the projection $\hat{T}\to\wsc$, the classical points in the fiber over a locally algebraic $w\in\wsc$ indeed correspond to automorphic representations of weight $w$ whose $p$-parts have nonzero Jacquet module. But Proposition 7.2.8 of~\cite{bc09} says that a space satisfying this property for automorphic forms on $G$ is unique up to unique isomorphism. \end{proof} \section{Bounds on the Newton polygon} \label{boundproofs} In this section, we prove Theorem~\ref{mybounds}. We prove Part~\ref{mylbd} in Section~\ref{mylbdproof} and Part~\ref{myubd} in Section~\ref{myubdproof}. In Section~\ref{combine}, we prove a modified version of Part~\ref{myubd} which generates infinitely many upper bound points for the same Newton polygon. Fix a character of $\Del^n$, and thus a particular polydisc in $\wsc^n$. Over the subset of this polydisc where $T_n=0$, we have $\det(I-XU_p)=\sum_{N\ge0}c_N(T_1,\dotsc,T_{n-1})X^N\in\ints_p\ps{T_1,\dotsc,T_{n-1}}\ps{X}$ with $c_0(T_1,\dotsc,T_{n-1})=1$. \subsection{A lower bound on the Newton polygon} \label{mylbdproof} The following is Part~\ref{mylbd} of Theorem~\ref{mybounds}. \begin{thm} There are constants $A_1$, $C$ (depending on $n$, $p$, and $h$) such that for all $T_1,\dotsc,T_{n-1}$ such that all $|T_j|>\frac1p$, the Newton polygon of $\sum_{N\ge0}c_N(T_1,\dotsc,T_{n-1})X^N$ lies above the points $\left(x, \left(A_1x^{1+\frac2{n(n-1)}}-C\right)\cdot\min v(T_j)\right)$ for all $x$. \end{thm} %TODO: I'd still like to rephrase this in a way that doesn't require citing all the machinery of Johansson-Newton but still uses their avoidance of Liu-Wan-Xiao's computations. Will try to do that again before finalizing. \begin{proof} We use the language of~\cite{jn16}. Fix an index $a$, and restrict to the subset $|T_a|\ge|T_j|$ for all $j\neq a$. Let %Fix a character $\eta:(\ints_p^\times)^n\to\fld_p^\times$, and let $R^\circ$ be the $T_a$-adic completion of $\ints_p\ps{T_1,\dotsc,T_{n-1}}\left[\frac{p}{T_a},\frac{T_1}{T_a},\dotsc,\frac{T_{n-1}}{T_a}\right]$ and let $R=R^\circ[1/T_a]$. Give $R$ the norm $|r|=\inf\{p^{-n}\mid r\in T_a^n R_\eta^\circ\}$. Let $[\cdot]_R:(\ints_p^\times)^n\to R^\times$ be the universal character with values in $R$. Let $\dcal$ be the continuous $R$-dual of $\ind_{B(\ints_p)}^{\iw_p,cts}[\cdot]_R$. $\dcal$ is orthonormalizable with the following norm: choose topological generators $\bar{n}=(\bar{n}_1,\dotsc,\bar{n}_{n(n-1)/2})$ %$\bar{\nfr}=(\bar{\nfr}_1,\dotsc,\bar{\nfr}_{n(n-1)/2})$ for $\bar{N}$, for example the matrix coefficients $pz_{21},pz_{31},pz_{32},\dotsc,pz_{n(n-1)}$ of Section~\ref{plucker}. Let $\bar{\nfr}_i\in\dcal$ be the Dirac distribution at $\bar{n}_i$ on $\bar{N}$. For $\eta=(\eta_1,\dotsc,\eta_{n(n-1)/2})\in\ints_{\ge0}^{n(n-1)/2}$, write $\bar{\nfr}^{\eta}:=\prod_{i=1}^{n(n-1)/2}\bar{\nfr}_i^{\eta_i}$ and $|\eta|=\sum_{i=1}^{n(n-1)/2}\eta_i$ for short. Then $\{\bar{\nfr}^\eta\}_{\eta\in\ints^{n(n-1)/2}}$ is a basis for $\dcal$, and the norm is $\left\|\sum_\eta d_\eta\bar{\nfr}^\eta\right\|_r=\sup_\eta|d_\eta|r^{|\eta|}.$ Let $\dcal^r$ be the completion of $\dcal$ with respect to this norm. By Corollary 4.1.5 of~\cite{jn16}, $\sum_{N\ge0}c_N(T_1,\dotsc,T_{n-1})X^N$ can be computed by the action of $U_p$ on the space $\dcal^{1/p}(G,U_0(p))$. By Section 3.3 of~\cite{jn16}, $\dcal^r$ has a potential orthonormal basis given by the elements $e_{r,\eta}:=T_a^{-n(r,T_a,\eta)}\bar{\nfr}^\eta$, where $n(r,T_a,\eta)=\left\lfloor\frac{|\eta|\log_pr}{\log_p|T_a|}\right\rfloor,$ and correspondingly $\dcal^r(G,U_0(p))$ has a potential orthonormal basis given by the elements $e_{r,\eta}^t:=(0,\dotsc,0,e_{r,\eta},0,\dotsc,0)\subset\bigoplus_{t=1}^h \dcal^r\cong\dcal^r(G,U_0(p))$ where the $e_{r,\eta}$ is in the $t$th position. By Lemma 6.2.1 of~\cite{jn16}, we have $U_p(e_{r,\eta}^t)=\sum_{u,\mu}a_\mu^u e_{r,\mu}^u$ with $|a_\mu^u|\le|T_a|^{n(r,T_a,\mu)-n(r^{1/p},T_a,\mu)}.$ We have $n(p^{-1},T_a,\mu)=|\mu|$ and $n(p^{-1/p},T_a,\mu)=\flr{|\mu|/p}$. So whenever $|\mu|=N$, every matrix entry of $U_p$ in the row $e_{r,\mu}^u$ has coefficient $a_\mu^u$ divisible by $|T_a|^{N-\flr{N/p}}$. There are %So for each integer $N\ge0$, the matrix of $U_p$ has $h\binom{N+n(n-1)/2-1}{n(n-1)/2-1}$ choices of $u$ and $\mu$ such that $|\mu|=N$, and hence that many rows which we can guarantee are divisible by $T_a^{N-\flr{N/p}}$ (not counting rows which we can guarantee are divisible by higher powers of $T_a$). We conclude that $\np\left(\sum_{N\ge0}c_N(T_1,\dotsc,T_{n-1})X^N\right)$ passes above the point $\left(h\sum_{N=0}^M\binom{N+n(n-1)/2-1}{n(n-1)/2-1},h\sum_{N=0}^M\binom{N+n(n-1)/2-1}{n(n-1)/2-1}(N-\flr{N/p})v(T_a)\right)$ for every integer $M\ge0$. Since the $x$-coordinate of the above expression is a polynomial in $M$ of degree $n(n-1)/2$ and the $y$-coordinate is $v(T_a)$ times a polynomial in $M$ of degree $n(n-1)/2+1$, the claim follows. \end{proof} \subsection{Systems of eigenvalues associated to classical points} \label{myubdproof} A refined principal series'' is a principal series representation $\pi$ of $GL_n(\rats_p)$ together with an ordered sequence of characters $(\psi_1,\dotsc,\psi_n):(\rats_p^\times)^n\to\cplx^\times$ such that $\pi\cong\pi(\psi_1,\dotsc,\psi_n)$. So there are $n!$ possible refinements of each $\pi$. The language comes from Galois representation theory. Assume $\chi$ is simple. From our setup so far, it is easy to see that an eigenform $f\in (U^{sm}(\chi)\ten S_t)(G,U_0(p))$ is naturally associated to a particular refined principal series: the principal series $\pi_{f,p}$, together with, if $f$ has $\lambda$-values $\lam_1,\dotsc,\lam_n$, the ordered sequence $(\psi_1,\dotsc,\psi_n):(\rats_p^\times)^n\to\cplx^\times$ such that $\pi\cong\pi(\psi_1,\dotsc,\psi_n)$ and $\lam_i=p^{(n-1)/2-i+1-m_n-m_{n-1}-\dotsb-m_{n-i+1}}\psi_i(p)$. Also note that this refined principal series depends only on the point $x$ on $\dsc$ that $f$ is associated to. For a character $\chi:(\ints_p^\times)^n\to\cplx^\times$ or $\psi:(\rats_p^\times)^n\to\cplx^\times$, and for any $w\in S_n$, we write $\chi^w=(\chi_{w(1)},\dotsc,\chi_{w(n)})$, and $\psi^w$ similarly. Now note that if $f_x\in (U^{sm}(\chi)\ten S_t)(G,U_0(p))\subset S_{t\chi,c}(G,U_0(p))$ is an eigenform associated to a point $x$ on $\dsc$ with associated refined principal series $(\pi(\psi),\psi^{\id})$, then the refined principal series $(\pi(\psi),\psi^w)$ is also associated to a point $x^w$ on $\dsc$ and a form $f_x^w\in (U^{sm}(\chi^w)\ten S_t)(G,U_0(p))\subset S_{t\chi^w}(G,c,U_0(p))$ (arising from the unique $(J,\chi^w)$-vector in $\pi(\psi)$). The forms $f_x^w$ are called companion forms of $f_x$. Having defined these companion forms, it is straightforward to show that the slopes appearing in $(U^{sm}(\chi)\ten S_t)(G,U_0(p))$ are not only finite but bounded above by a linear function of $t$, as follows. \begin{prop} \label{ubd} If $f\in (U^{sm}(\chi)\ten S_t)(G,U_0(p))\subset S_{t\chi,c}(G,U_0(p))$ is a $U_p^a$-eigenform with eigenvalue $a_p^{\id}$, %there is for each $w\in S_n$ a form $f^w\in (U^{sm}((\chi^w)^{-1})\ten S_t)(G,U_0(p))\subset S(G,t\chi^w,c,U_0(p))$, where $\chi^w=(\chi_{w(1)},\dotsc,\chi_{w(n)})$, and each companion form $f^w$ has $U_p^a$-eigenvalue $a_p^w$, then we have $\sum_{w\in S_n}v(a_p^w)=l^a(t)$ where $l^a(t)$ is a linear function of $t_1,\dotsc,t_n$. In particular, let $l^{(n-1,n-2,\dotsc,0)}(t)=l(t)$. Then for each $w$, the Newton polygon of $\sum_{N\ge0}c_N(T_1(t\chi^w),\dotsc,T_{n-1}(t\chi^w))X^N$ contains $hp^{j(\chi)}d_t$ slopes of size at most $l(t)$, hence in particular passes below the point $\left(hp^{j(\chi)}d_t, hp^{j(\chi)}d_tl(t)\right).$ \end{prop} %TODO: planning to eventually add a second way of getting this proposition with potentially better constants. \begin{proof} Let $\pi_{f,p}=\pi(\psi_1,\dotsc,\psi_n)$. By Proposition~\ref{lamtopsi}, we have $\prod_i\lam_i=p^{-(nm_n+(n-1)m_{n-1}+\dotsb+m_1)}\prod_i\psi_i(p).$ The $\lam$-values of $x^w$ are given by $\lam_i^w=p^{(n-1)/2-i+1-m_n-m_{n-1}-\dotsb-m_{n-i+1}}\psi_{w(i)}(p)$ for each $w\in S_n$. Then for $a=(a_1,\dotsc,a_n)$, the $U_p^a$-eigenvalue associated to $x^w$ is $\prod_i(\lam_i^w)^{a_{n-i+1}}=\prod_ip^{a_{n-i+1}[(n-1)/2-i+1-m_n-m_{n-1}-\dotsb-m_{n-i+1}]}\psi_{w(i)}(p)^{a_{n-i+1}}$ so the product of the $U_p^a$-eigenvalues associated to all the $x^w$s is $p^{(n-1)!\sum_ia_{n-i+1}[(n-1)/2-i+1-m_n-m_{n-1}-\dotsb-m_{n-i+1}]}\left(\prod_i\psi_i(p)\right)^{(n-1)!\sum_ia_i}$ $=p^{(n-1)!\sum_ia_{n-i+1}[(n-1)/2-i+1-m_n-m_{n-1}-\dotsb-m_{n-i+1}]}\left(p^{nm_n+(n-1)m_{n-1}+\dotsb+m_1}\prod_i\lam_i\right)^{(n-1)!\sum_ia_i}.$ But $\prod_i\lam_i$ is the eigenvalue associated to the operator $U_p^{(1,1,\dotsc,1)}$, which is just right translation by the central matrix $\diag(p,p,\dotsc,p)$, which preserves $f$, so $\prod_i\lam_i=1$. So the sum of the valuations of the $U_p^a$-eigenvalues associated to the companion points is %If we choose $x$ of trivial central character, then $\prod_i\lam_i=1$, $(n-1)!\sum_ia_{n-i+1}[(n-1)/2-i+1-m_n-m_{n-1}-\dotsb-m_{n-i+1}]$ $+(nm_n+(n-1)m_{n-1}+\dotsb+m_1)(n-1)!\sum_ia_i$ $=(n-1)!\left(\sum_ia_{n-i+1}((n-1)/2-i+1)-\sum_j m_j(a_1+\dotsb+a_j)+\sum_j jm_j\left(\sum_i a_i\right)\right).$ Defining $l^a(t)$ to be this last expression, we find that $\sum_{w\in S_n}v(a_p^w)=l^a(t)$ as desired. The conclusion that each individual $v(a_p^w)$ is bounded above by $l^a(t)$ follows because all the $a_p^w$s are algebraic integers. \end{proof} Let $c_i=\cond(\chi_i)$, let $\chi_{(1)},\dotsc,\chi_{(n-1)}$ be the characters $\chi_1,\dotsc,\chi_{n-1}$ reordered so that $\cond(\chi_{(1)})\le\cond(\chi_{(2)})\le\dotsb\le\cond(\chi_{(n-1)})$, let $c_{(i)}=\cond(\chi_{(i)})$, and let $T_{(i)}=T(\chi_{(i)})$. To get from Proposition~\ref{ubd} to the statement of Theorem~\ref{mybounds}, we just need to check that for all $t$ and $\chi$, $\left(hp^{j(\chi)}d_t, hp^{j(\chi)}d_tl(t)\right)$ lies below the curve $y=A_2\left(v(T_{(1)})^{\frac{2}{n(n-1)}}v(T_{(2)})^{\frac{2\cdot 2}{n(n-1)}}\dotsb v(T_{(n-1)})^{\frac{2\cdot(n-1)}{n(n-1)}}\right)x^{1+\frac2{n(n-1)}}$ for a fixed constant $A_2$ (depending only on $n$, $p$, and $h$). Note that by the formula stated in Example~\ref{localgtcoords}, we have $v(T_{(i)})=v(T(\chi_{(i)}))=Ap^{-c_{(i)}}$ for a constant $A$ (depending on $p$). Thus we have $p^{j(\chi)}=p^{c_{(1)}+2c_{(2)}+\dotsb+(n-1)c_{(n-1)}-\frac{n(n-1)}2}=A'v(T_{(1)})^{-1}v(T_{(2)})^{-2}\dotsb v(T_{(n-1)})^{-(n-1)}.$ Next we check the size of $d_t$. \begin{prop} The dimension $d_t$ is a polynomial of total degree $\frac{n(n-1)}2$ in $m_1,\dotsc,m_{n-1}$. \end{prop} \begin{proof} By Corollary 14.9 of~\cite{ms04}, $\ind_{B(\ints_p)}^{GL_n(\ints_p),alg.}t$ has a basis indexed by chains in the poset described in Section 14.2 of~\cite{ms04}. For a subset $\sg$ of $\{1,\dotsc,n\}$, let $f(\sg)=\sum_{k\notin\sg}(n+1-k)$. We claim that when you take one step down the poset, $f(\sg)$ goes down by $1$. This is because, if $\sg$ is one step below $\tau$, there are two possibilities. The first is that $|\tau|=|\sg|$ and there is some $i$ for which $\sg_i=\tau_i-1$ and $\sg_j=\tau_j$ for all $j$ with $j\neq i$; in this case the complements $\sg^c$ and $\tau^c$ are the same except for $\sg_i\in\tau^c$ and $\sg_i+1=\tau_i\in\sg^c$, which contribute $n-\sg_i$ and $n-\sg_i-1$ to the sums $f(\sg)$ and $f(\tau)$, so $f(\sg)=f(\tau)-1$. The second is that $|\sg|=|\tau|+1$ and $\sg$ contains $n$ and $\tau$ does not, so again $f(\sg)=f(\tau)-1$. So a maximal chain in this poset starts with $\{n\}$, which has $f$-value $2+\dotsb+n=\frac{n(n+1)}2-1$, and ends with $\{1,2,\dotsc,n-1\}$, which has $f$-value $1$; its length is therefore $\frac{n(n+1)}2-1$. A leading term of $d_{m_1,\dotsc,m_{n-1},0}$ comes from distributing $m_1,\dotsc,m_{n-1}$ among corresponding variables in a maximal chain. So it is a product $\prod\binom{m_i+c_i}{c_i}$ where the $c_i+1$s sum to $\frac{n(n+1)}2-1$; that is, the $c_i$s sum to $\frac{n(n+1)}2-1-(n-1)=\frac{n(n-1)}2$. \end{proof} Fix $\eps>0$ and assume that for all $i\neq j$, we have $m_i\ge\eps m_j$. Then we can find some $A_{\eps}$ such that $l(t)\le A_{\eps}d_t^{\frac2{n(n-1)}}$ for all such $m_1,\dotsc,m_{n-1}$. So if we let $x=hp^{j(\chi)}d_t$ and $y=hp^{j(\chi)}d_tl(t)$, we have $y=\left(hp^{j(\chi)}d_t\right)^{1+\frac2{n(n-1)}}\left(hp^{j(\chi)}d_t\right)^{-\frac2{n(n-1)}}l(t)$ $=Ax^{1+\frac2{n(n-1)}}\left(p^{j(\chi)}\right)^{-\frac2{n(n-1)}}d_t^{-\frac2{n(n-1)}}l(t)$ $\le AA_{\eps}x^{1+\frac2{n(n-1)}}\left(v(T_{(1)})^{-1}v(T_{(2)})^{-2}\dotsb v(T_{(n-1)})^{-(n-1)}\right)^{-\frac2{n(n-1)}}$ $=A'\left(v(T_{(1)})^{\frac{2}{n(n-1)}}v(T_{(2)})^{\frac{2\cdot 2}{n(n-1)}}\dotsb v(T_{(n-1)})^{\frac{2\cdot(n-1)}{n(n-1)}}\right)x^{1+\frac2{n(n-1)}}$ as desired. This proves Part~\ref{myubd} of Theorem~\ref{mybounds} for all $t\chi$ such that $m_i\ge\eps m_j$ for all $i\neq j$ and $\chi$ is simple. \subsection{Combining upper bound points} \label{combine} We show that Theorem~\ref{unified-ubd} is a natural consequence of Part~\ref{myubd} of Theorem~\ref{mybounds}. First we need the following lemma of Wan, which is stated in~\cite{wan98} with $\ints_p$-coefficients but works identically with $\osc_{\cplx_p}$-coefficients. \begin{lem}[Wan 1998] \label{np-coincide} Let $Q_1(X),Q_2(X)$ be two elements in $\osc_{\cplx_p}\ps{X}$ with $Q_1(0)=Q_2(0)=1$. Let $N_i(x)$ be the function on $\real_{\ge0}$ whose graph is the Newton polygon of $Q_i(X)$. Assume that $\nu(x)$ is a strictly increasing continuous function on $\real_{\ge0}$ such that $\nu(0)\le0$, $N_i(x)\ge x\nu(x)$ for $1\le i\le 2$ and $x\ge1$, and $\lim_{x\to\infty}\nu(x)=\infty$. Assume further that the function $x\nu^{-1}(x)$ is increasing on $\real_{>0}$, where $\nu^{-1}(x)$ denotes the inverse function of $\nu(x)$ defined at least on $\real_{\ge0}$. For $x\ge0$, we define the integer-valued increasing function $m_\nu(x)=\flr{x\nu^{-1}(x)}$. If the congruence $Q_1(X)\equiv Q_2(X)\pmod{p^{m_\nu(\al)+1}}$ holds for some $\al\ge0$, then the two Newton polygons $N_i(x)$ coincide for all the sides with slopes at most $\al$. \end{lem} %need to edit to clarify that the approximate relative size of the $m_i$s is not messed up by this procedure. \begin{proof}[Proof of Theorem~\ref{unified-ubd}] By Corollary~\ref{ubd}, $\np(t\chi)$ passes below the point $\left(hp^{j(\chi)}d_t, hp^{j(\chi)}d_tl(t)\right).$ Note that the slope of $\np(t\chi)$ at $x$-coordinate $hp^{j(\chi)}d_t$ is at most $l(t)$. We may apply Lemma~\ref{np-coincide} with $\nu(x)=A_1x^{\frac2{n(n-1)}}\min_iv(T(\chi_i))$, so that $m_\nu(x)\asymp \frac{x^{1+\frac{n(n-1)}2}}{\left(\min_iv(T(\chi_i))\right)^{\frac{n(n-1)}2}}.$ Let $t_i^{(1)}=t_i+(n-i)p^{m_\nu(l(t))+1}$. By Lemma~\ref{np-coincide} applied to $P(X,t\chi)$ and $P(X,t^{(1)}\chi)$, we find that $\np(t^{(1)}\chi)$ also passes below this point. However, by Corollary~\ref{ubd}, $\np(t^{(1)}\chi)$ also passes below $\left(hp^{j(\chi)}d_{t^{(1)}}, hp^{j(\chi)}d_{t^{(1)}}l(t^{(1)})\right).$ Repeating this, we find a sequence $t=t^{(0)},t^{(1)},t^{(2)},\dotsc$ of dominant algebraic weights such that $\np(t^{(k)}\chi)$ passes below $\left(hp^{j(\chi)}d_{t^{(0)}}, hp^{j(\chi)}d_{t^{(0)}}l(t^{(0)})\right),\dotsc,\left(hp^{j(\chi)}d_{t^{(k)}}, hp^{j(\chi)}d_{t^{(k)}}l(t^{(k)})\right).$ Evidently the $t^{(k)}$ approach a limit $t^\infty$, and $\np(t^\infty\chi)$ passes below $\left(hp^{j(\chi)}d_{t^{(k)}}, hp^{j(\chi)}d_{t^{(k)}}l(t^{(k)})\right)$ for all $k$. The result follows as in the end of Section~\ref{myubdproof}. (Note that since $m_i^{(k)}=m_i^{(k-1)}+p^{m_\nu(l(t^{(k-1)}))+1}$, if $m_i^{(k-1)}\ge\eps m_j^{(k-1)}$ for all $i\neq j$, the same is true for the $m_i^{(k)}$.) \end{proof} %TODO: consider adding example for $n=3$ to give more specific formulas and show difference in constants (about $3/5$ I think). \section{Geometry of the eigenvariety over the boundary of weight space} \label{geometry} Fix an index $a$, and let $\wsc_{<\nu}$ be the subset of characters $w$ such that $v(T_a(w))<\nu$ and $v(T_a(w))<\nu v(T_j(w))$ for all $j\neq a$ (so in particular $v(T_a)=\min_i v(T_i)$). let $\zsc_{<\nu}$ be the preimage of $\wsc_{<\nu}$ in the eigencurve $\zsc$. For any real number $\al$, let $X(<\al)$ be the subset of $\zsc$ of points $x$ for which $v(a_p(x))<\al v(T_a(w(x)))$, and define $X(=\al)$, $X(>\al)$ similarly. As in the previous section, fix a polydisc in $\wsc$. For $T=(T_1,\dotsc,T_{n-1})$ in the polydisc and $m=(m_1,\dotsc,m_{n-1})\in\ints_{\ge0}^{n-1}$, write $T^m=T_1^{m_1}\dotsb T_{n-1}^{m_{n-1}}$ for short. Let $\det(1-XU_p)=\sum_{N\ge0}c_N(T)X^N$, where $c_N(T)=\sum_{m=(m_1,\dotsc,m_{n-1})\in\ints_{\ge0}^{n-1}}b_{N,m}T^m\in\ints_p\ps{T_1,\dotsc,T_{n-1}}$. Let $y=\np(T)(x)$ be the Newton polygon of $\sum_{N\ge0}c_N(T)X^N$. For the following theorem, the only input we need is a lower bound for $y=\np(T)(x)$ of the form $y=v(T_a)f(x)$ where $f(x)$ is a convex function, which we have (with $f(x)=A_1x^{1+\frac2{n(n-1)}}$) from Part~\ref{mylbd} of Theorem~\ref{mybounds}. \begin{thm} \label{disconnect} For every $\al\in\real_{\ge0}$, there is some valuation $\nu(\al)>0$ such that $X(=\al)_{<\nu(\al)}$ is disconnected from its complement in $\zsc_{<\nu(\al)}$. \end{thm} %TODO: this is probably not the only thing I can prove of this nature using the same technique, but it's the one that sounds nicest. May add others later. \begin{proof} Let $d(\al,T)$ be the number of slopes in $y=\np(T)(x)$ of value strictly less than $\al v(T_a)$ (so the dimension of $\ssc_T(G,U_0(p))^{<\al v(T_a)}$). Assume $v(T_a)<1$. We claim that the point $(d(\al,T),\np(T)(d(\al,T)))$ lies inside the region bounded by the line $y=\al v(T_a)x$ and the function $y=v(T_a)f(x)$. It lies below $y=\al v(T_a)x$ because all slopes of $NP(T)$ up to $d(\al,T)$ are less than $\al v(T_a)$. It lies above $y=v(T_a)f(x)$ because this is a lower bound for $y=\np(T)(x)$. This region lies inside the box whose lower left corner is $(0,0)$ and whose upper right corner is $(d(\al),\al d(\al)v(T_a))$, where $d(\al)$ is the nonzero solution to $\al x=f(x)$. We have $(d(\al,T),\np(T)(d(\al,T)))=(j,v(c_j(T)))$ for some $j$. This is a vertex of $y=\np(T)(x)$. The vertex immediately preceding it is of the form $(i,v(c_i(T)))$ for some $i$. The slope between the two is $\frac{v(c_j(T))-v(c_i(T))}{j-i}.$ This is the largest slope of $y=\np(T)(x)$ less than $\al v(T_a)$. We have $1\le j-i\le d(\al)$. But $c_j(T)=\sum_{m\ge0}b_{j,m}T^m$ is a sum of terms $b_{j,m}T^m$ where $v(b_{j,m})$ is an integer and $v(T^m)=m_1v(T_1)+\dotsb+m_{n-1}v(T_{n-1})$. Thus $v(c_j(T))=\mu_j+\lam_j^1v(T_1)+\dotsb+\lam_j^{n-1}v(T_{n-1})$ where $\mu_j,\lam_j^k$ are integers in the range $[0,\al d(\al)]$ (since $v(c_j(T))\le \al d(\al)v(T_a)$). Similarly $v(c_i(T))=\mu_i+\lam_i^1v(T_1)+\dotsb+\lam_i^{n-1}v(T_{n-1})$ where $\mu_i,\lam_i^k\in[0,\al d(\al)]$ as well. Assume that $v(T_a)<\frac1{\al d(\al)}$, so that $\al d(\al) v(T_a)<1$, and furthermore that $v(T_a)<\frac1{\al d(\al)}v(T_j)$ for all $j\neq a$. Then in order to have $v(c_i(T)),v(c_j(T))\le \al d(\al) v(T_a)$, we must have $\mu_i=\mu_j=0$ and $\lam^k=0$ for all $k\neq i$. So the largest slope of $y=\np(T)(x)$ less than $\al v(T_a)$ is of the form $\frac{\lam_j-\lam_i}{j-i}v(T_a)$, where $\lam_j-\lam_i\in[0,\al d(\al)]$ and $j-i\in[1,d(\al)]$. This is a finite, discrete set of points. So the ratio of the largest slope of $y=\np(T)(x)$ less than $\al v(T_a)$ to $v(T_a)$ is bounded away from $\al$ independently of $T_a$. Setting $\nu(\al)<\frac1{\al d(\al)}$, we conclude that $X(<\al)_{<\nu(\al)}$ is disconnected from its complement in $\zsc_{<\nu(\al)}$. This argument goes through exactly the same way if $X(<\al)$ is replaced by $X(\le\al)$: either the smallest slope greater than $\al$ is at least $\al+1$, or, if not, the next endpoint is again trapped in a box whose area is at most linear in $v(T)$, and the same argument applies. So we can choose $\nu(\al)$ such that $X(=\al)_{<\nu(\al)}$ is disconnected from its complement in $\zsc_{<\nu(\al)}$. \end{proof} As Liu-Wan-Xiao do in Theorem 3.19 of~\cite{lwx17}, we can also use Part~\ref{mylbd} of Theorem~\ref{mybounds} to give a simple proof of the fact that the ordinary part of $\zsc$ is finite and flat over $\wsc$ and disconnected from its complement. \begin{thm} $X(=0)$ is finite and flat over $\wsc$ and is a union of connected components of $\zsc$. \end{thm} \begin{proof} The proof of Theorem 3.19 of~\cite{lwx17} goes through almost word-for-word. By Part~\ref{mylbd} of Theorem~\ref{mybounds}, there is some maximal $N$ such that $c_N(T_1,\dotsc,T_{n-1})$ is a unit in $\ints_p\ps{T_1,\dotsc,T_{n-1}}$, or equivalently, the constant term of $c_N(T_1,\dotsc,T_{n-1})$ is a unit in $\ints_p$. Then for each $(T_1,\dotsc,T_{n-1})$, the Newton polygon of $\sum_{n=0}^\infty c_N(T_1,\dotsc,T_{n-1})X^N$ starts with $N$ segments of slope $0$ followed by a segment of slope at least \\ $\max(1,B\min_j v(T_j))$ for some constant $B$. Since $\max(1,B\min_j v(T_j))$ is uniformly bounded away from $0$ over any affinoid subdomain, $X(=0)$ is disconnected from its complement, and it is finite and flat of degree $N$. \end{proof} %\begin{thm} %At a weight $s$ of the form given in Theorem~\ref{unified-ubd}, if $r=\frac{\min T_j}{\max T_j}$, the fiber of $X(\le\al)$ over $s$ has size at least fill in. %\end{thm} %\begin{thm} %As $\al$ goes to infinity, the maximum degree of the fibers of $X(<\al)_{>r(\al)}$ over $\wsc_{>r(\al)}$ also goes to infinity. Thus in particular there exists an infinite sequence of rational numbers $\al_n$ going to infinity such that $X(=\al_n)_{>r(\al_n)}$ is nonempty. %\end{thm} %\begin{proof} %\end{proof}