Person:

Elkies, Noam

A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in (R^3), and his classification is equivalent to the converse for (R^3). In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.

We show that, if (L) is an extremal Type II lattice of rank 40 or 80, then (L) is generated by its vectors of norm (min(L)+2). This sharpens earlier results of Ozeki, and the second author and Abel, which showed that such lattices L are generated by their vectors of norms (min(L)) and (min(L)+2).

We give explicit equations for the simplest towers of Drinfeld modular curves over any finite field, and observe that they coincide with the asymptotically optimal towers of curves constructed by Garcia and Stichtenoth.

Cut the unit circle (S^1 = \mathbb{R}/\mathbb{Z}) at the points ({\sqrt{1}}, {\sqrt{2}}, . . ., {\sqrt{N}}), where ({x} = x mod 1), and let (J_1, . . . , J_N) denote the complementary intervals, or gaps, that remain. We show that, in contrast to the case of random points (whose gaps are exponentially distributed), the lengths (\mid J_i\mid/N) are governed by an explicit piecewise real-analytic distribution (F(t)dt) with phase transitions at (t=\frac{1}{2}) and (t=2). The gap distribution is related to the probability (p(t)) that a random unimodular lattice translate (\Lambda \subset \mathbb{R}^2) meets a fixed triangle (S_t) of area (t); in fact (p^"(t) = -F(t)). The proof uses ergodic theory on the universal elliptic curve: (E = (SL_2(\mathbb{R}) ⋉ \mathbb{R}^2) / (SL_2(\mathbb{Z}) ⋉ \mathbb{Z}^2))

We give a general recipe for explicitly constructing asymptotically optimal towers of modular curves such as ({X_0(l^n)}_{n>1}). We illustrate the method by giving equations for eight towers with various geometric features. We conclude by observing that such towers are all of a specific recursive form, and speculate that perhaps every tower of this form that attains the Drinfeld-Vladut bound is modular.

The (2,3,7) triangle group is known to be associated with a quaternion algebra A/K ramified at two of the three real places of K=Q(cos2π/7) and unramified at all other places of K. This triangle group and its congruence subgroups thus give rise to various Shimura curves and maps between them. We study the genus-1 curves X_0(3), X_1(3) associated with the congruence subgroups Γ_0(3), Γ_1(3). Since the rational prime 3 is inert in K, the covering X_0(3)/X(1) has degree 28, and its Galois closure X(3)/X(1) has geometric Galois group PSL2(F27). Since X(1) is rational, the covering X_0(3)/X(1) amounts to a rational map of degree 28. We compute this rational map explicitly. We find that X_0(3) is an elliptic curve of conductor 147=3·72 over Q, as is the Jacobian J_1(3) of X_1(3); that these curves are related by an isogeny of degree 13; and that the kernel of the 13-isogeny from J_1(3) to X_0(3) consists of K-rational points. We also use the map X_0(3) --> X(1) to locate some complex multiplication (CM) points on X(1). We conclude by describing analogous behavior of a few Shimura curves associated with quaternion algebras over other cyclic cubic fields.

We study a construction of the Mathieu group M-12 using a game reminiscent of Loyd's "15-puzzle." The elements of M-12 are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There is a natural extension to a "pseudogroup" M-13 acting on all 13 points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, akin to that induced by a Cayley graph, on both M-12 and M-13. We develop these results, and extend them to the double covers and automorphism groups of M-12 and M-13, using the ternary Golay code and 12 x 12 Hadamard matrices. In addition, we use experimental data on the quasi-Cayley metric to gain some insight into the structure of these groups and pseudogroups.

A Sylvester-Gallai (SG) configuration is a finite set S of points such that the line through any two points in S contains a third point of S. According to the Sylvester-Gallai theorem, an SG configuration in real projective space must be collinear. A problem of Serre (1966) asks whether an SG configuration in a complex projective space must be coplanar. This was proved by Kelly (1986) using a deep inequality of Hirzebruch. We give an elementary proof of this result, and then extend it to show that an SG configuration in projective space over the quaternions must be contained in a three-dimensional flat.

For (r = 6, 7, . . . , 11) we find an elliptic curve (E/Q) of rank at least (r) and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for (r = 6)) to over 100 (for (r = 10) and (r=11)). We describe our search methods, and tabulate, for each (r = 5, 6, . . . , 11), the five curves of lowest conductor, and (except for (r = 11)) also the five of lowest absolute discriminant, that we found.

We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c_q with the following property: for every non-negative integer g, there is a genus-g curve over F_q with at least c_q * g rational points over F_q. Moreover, we show that there exists a positive constant d such that for every q we can choose c_q = d * (log q). We show also that there is a constant c > 0 such that for every q and every n > 0, and for every sufficiently large g, there is a genus-g curve over F_q that has at least cg/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ)^r for some r > cg/n.