Person:

Mazur, Barry

We use the theory of Kolyvagin systems to prove (most of) a refined class number formula conjectured by Darmon. We show that for every odd prime (p), each side of Darmon’s conjectured formula (indexed by positive integers (n) is “almost” a (p)-adic Kolyvagin system as (n) varies. Using the fact that the space of Kolyvagin systems is free of rank one over Z(_p), we show that Darmon’s formula for arbitrary (n) follows from the case (n) = 1, which in turn follows from classical formulas.

We study the parity of 2-Selmer ranks in the family of quadratic twists of an arbitrary elliptic curve E over an arbitrary number field K. We prove that the fraction of twists (of a given elliptic curve over a fixed number field) having even 2-Selmer rank exists as a stable limit over the family of twists, and we compute this fraction as an explicit product of local factors. We give an example of an elliptic curve E such that as K varies, these fractions are dense in [0,1]. More generally, our results also apply to p-Selmer ranks of twists of 2-dimensional self-dual (F_p)-representations of the absolute Galois group of K by characters of order p.

Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit and control--thanks to ideas of Kolyvagin--the existence of rational points and other mathematical structures elsewhere. Despite all that we know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today.

We have a network of heuristics and conjectures regarding rational points, and we have massive data accumulated to exhibit instances of the phenomena. Generally, we would expect that our data support our conjectures, and if not, we lose faith in our conjectures. But here there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either. We discuss various aspects of this story, including recent heuristics and data that attempt to resolve this mystery. We shall try to convince the reader that, despite seeming discrepancy, data and conjecture are, in fact, in harmony.

Let (p) be an odd prime number, let E be an elliptic curve over a number field (k), and let (F/k) be a Galois extension of degree twice a power of p. We study the (Z_p)-corank (rk_p(E/F)) of the (p)-power Selmer group of (E) over (F). We obtain lower bounds for (rk_p(E/F)), generalizing the results in [MR], which applied to dihedral extensions.

If (K) is the (unique) quadratic extension of (k) in (F), if (G = Gal(F/K)), if (G+) is the subgroup of elements of (G) commuting with a choice of involution of (F) over (k), and if (rk_p(E/K)) is odd, then we show that (under mild hypotheses) (rkp(E/F)\ge[G:G+]).

As a very specific example of this, suppose that (A) is an elliptic curve over (Q) with a rational torsion point of order (p) and without complex multiplication. If (E) is an elliptic curve over (Q) with good ordinary reduction at (p) such that every prime where both (E) and (A) have bad reduction has odd order in (F\frac{x}{p}) and such that the negative of the conductor of (E) is not a square modulo (p), then there is a positive constant (B) depending on (A) but not on (E) or (n) such that (rk_p(E/Q(A[p^n]))/geBp^{2n}) for every (n).

This paper is about computational and theoretical questions regarding p-adic height pairings on elliptic curves over a global field K. The main stumbling block to computing them efficiently is in calculating, for each of the completions Kv at the places v of K dividing p, a single quantity: the value of the p-adic modular form E2 associated to the elliptic curve. Thanks to the work of Dwork, Katz, Kedlaya, Lauder and Monsky-Washnitzer we offer an efficient algorithm for computing these quantities, i.e., for computing the value of E2 of an elliptic curve. We also discuss the p-adic convergence rate of canonical expansions of the p-adic modular form E2 on the Hasse domain. In particular, we introduce a new notion of log convergence and prove that E2 is log convergent.

À l’image des atomes pour les molécules, les nombres premiers sont les briques élémentaires des nombres entiers. Quantité de problèmes sur les premiers sont aussi simples à énoncer que difficiles à attaquer. Un éminent mathématicien de Harvard a fait le pari de les présenter en ne faisant appel qu’à des notions élémentaires de mathématiques.

Let (K) be an arbitrary number field, and let (\rho: Gal(K \bar/K) \rightarrow GL_2(E)) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of (\rho). When (K) is totally real and rho is modular, results of Hida imply that the nearly ordinary deformation space associated to rho contains a Zariski dense set of points corresponding to "automorphic" Galois representations. We conjecture that if (K) is not totally real, then this is never the case, except in three exceptional cases, corresponding to (1) "base change", (2) "CM" forms, and (3) "Even" representations. The latter case conjecturally can only occur if the image of (\rho) is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of Leopoldt's conjecture. Second, when (K) is an imaginary quadratic field, we prove an unconditional result that implies the existence of "many" positive dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about "(p)-adic functorality", as well as some remarks on how our methods should apply to n-dimensional representations of Gal((Q \bar/Q)) when (n \lt 2).

If (V) is a commutative algebraic group over a field (k), [View the MathML] source is a commutative ring that acts on (V), and View the MathML source is a finitely generated free View the MathML source-module with a right action of the absolute Galois group of (k), then there is a commutative algebraic group [View the MathML source] over (k), which is a twist of a power of (V). These group varieties have applications to cryptography (in the cases of abelian varieties and algebraic tori over finite fields) and to the arithmetic of abelian varieties over number fields. For purposes of such applications we devote this article to making explicit this tensor product construction and its basic properties.

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field.