JHU Number Theory Seminar, Spring 2006

Closed geodesics on modular curves and central values of L-functions

Special values of L-functions often encode arithmetic or geometric information about associated objects, such as in the Birch and Swinnerton-Dyer conjecture. We present such a connection, for the Rankin L-function of a cuspidal modular form f of even weight and squarefree level N, twisted by a narrow class group character of a real quadratic field K. Under a Heegner type hypothesis on K and N, the central value is related to integrals of f over certain geodesic cycles on the modular curve X_0(N), geodesics which are in one to one correspondence with narrow ideal classes in K. The formula can be used to prove the equidistribution of individual "long" closed geodesics associated to K, when the discriminant of K goes to infinity. We also discuss possible generalizations, when f is replaced by a Siegel modular form.

Shimura varieties and noncommutative geometry

About ten years ago, Bost and Connes gave a natural realization of the class field theory for Q within the operator-algebraic setting of quantum statistical mechanics. More recently, Connes and Marcolli have shown that the underlying noncommutative geometry of "Q-lattices" provides a generalization to higher dimensions, while also providing a setting for a Bost-Connes theory for imaginary quadratic fields. Their work also indicates the role of Shimura varieties: in this talk we shall bring this point to the fore and discuss a generalization which, in particular, incorporates some of the desirable features of a possible Bost-Connes theory for arbitrary number fields, and gives rise to a new procedure for attaching zeta functions to Shimura varieties.

Congruences among automorphic forms on the unitary group U(2,2)

In 1976 Ribet showed how the divisibility of a special value of a Dirichlet L-function by a prime p leads to a (mod p) congruence between a cusp form and an Eisenstein series on SL_2(Z). He used this congruence to construct non-trivial elements in class groups of cyclotomic fields. In this talk we consider a two-dimensional analogue of Ribet's theorem. The Dirichlet character is replaced with an automorphic representation $\pi$ of GL_2 and modular forms on SL_2(Z) are replaced by automorphic forms on the unitary group U(2,2). We will show that the p-divisibility of a special value of an L-function attached to $\pi$ gives rise to a (mod p) congruence between a CAP form (a substitute for Ribet's Eisenstein series) and a non-CAP cusp form on U(2,2). We then explain how this congruenceimplies that the Selmer group of a certain (mod p) Galois representation attached to $\pi$ is non-trivial. Our method of proof is different from that of Ribet.

Vanishing cycles over general bases

In the early 80's P. Deligne proposed a theory of vanishing cycles in ¨¦tale cohomology valid over general bases, generalizing the one-dimensional theory of A. Grothendieck (SGA 7). We will first briefly review the classical construction and results and then discuss a proof of the fact that, "locally" in a broad sense, the general case is equally nice (as conjectured by P. Deligne). Time permitting, an application to Lefschetz pencils in the wild case (following O. Gabber) will be given.

Heat Kernels, Ladders, and Fundamental Domains

Introduction to Drinfel'd modules and Hayes' explicit class field theory, with a view toward quantum statistical mechanics

In this introductory talk, we will review the basics of the analytic theory of Drinfel'd modules, which is an analogue for function fields of the theory of elliptic curves over C and lattices. We will review Hayes' explicit class field theory, which allows to construct the maximal abelian extension of a function field by adjoining torsion points and coefficients of certain special rank one Drinfel'd modules, called Hayes modules. We will also show how Hayes' theory allows to construct interesting endomotives, in the sense of [1], and how noncommutative objects naturally appear in the study of the arithmetic of Drinfel'd modules.
[1] A. Connes, C. Consani, M. Marcolli, "Noncommutative geometry and motives: the thermodynamics of endomotives", arXiv: math.QA/0512138

Galois theory and zeta functions of Riemannian manifolds

This talk is about an analogy going back to work of Sunada in the 1980's between lengths of closed geodesics on compact negatively curved Riemannian manifolds M and prime ideals of rings of integers of number fields. I will discuss joint work with Alan Reid, Darren Long and Emily Hamilton showing that if M is an arithmetic hyperbolic 3-manifold, the length spectrum of M determines the commensurability class of M. The proof hinges on analyzing when a number field K with one complex place is determined by its Galois closure over Q.

Hecke Orbits

Let $A_g$ be the moduli space of g-dimensional principally polarized abelian varieties. Over the complex numbers, $A_g$ has a large collection of symmetries, known as Hecke correspondences; these symmetries are systematically studied in the context of automorphic forms. Over a field of positive characteristic $p$, the Hecke symmetries are closed connected to a family of algebraic subvarieties of $A_g$ known as leaves. Each leaf is stable under prime-to-p Hecke correspondences, and the $A_g$ is the disjoint union of these leaves. Oort conjectured that every Hecke orbit is dense in the leaf containing it. This conjecture is now a theorem, and we will explain some of the methods motivated by the Hecke orbit conjecture.