Person:

Elkies, Noam

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.

In an earlier paper we showed that any integral unimodular lattice L of rank n which is not isometric with Z^n has a characteristic vector of norm at most n-8. [A "characteristic vector" of L is a vector w in L such that 2|(v,w-v) for all v in L; it is known that the characteristic vectors all have norm congruent to n mod 8 and comprise a coset of 2L in L.] Here we use modular forms and the classification of unimodular lattices of rank <24 to find all L whose minimal characteristic vectors have norm n-8. Along the way we also obtain congruences and a lower bound on the kissing number of unimodular lattices with minimal norm 2. We then state and prove analogues of these results for self-dual codes, and relate them directly to the lattice problems via "Construction A".

The ABC conjecture is a central open problem in modern number theory, connecting results, techniques and questions ranging from elementary number theory and algebra to the arithmetic of elliptic curves to algebraic geometry and even to entire functions of a complex variable. The conjecture asserts that, in a precise sense that we specify later, if (A,B,C) are relatively prime integers such that (A + B = C) then (A,B,C) cannot all have many repeated prime factors. This expository article outlines some of the connections between this assertion and more familiar Diophantine questions, following (with the occasional scenic detour) the historical route from Pythagorean triples via Fermat’s Last Theorem to the formulation of the ABC conjecture by Masser and Oesterl´e. We then state the conjecture and give a sample of its many consequences and the few very partial results available. Next we recite Mason’s proof of an analogous assertion for polynomials (A(t),B(t), C(t)) that implies, among other things, that one cannot hope to disprove the ABC conjecture using a polynomial identity such as the one that solves the Diophantine equation (x^2 + y^2 = z^2). We conclude by solving a Putnam problem that predates Mason’s theorem but is solved using the same method, and outlining some further open questions and fragmentary results beyond the ABC conjecture.