# Seminars & Events for Princeton University/IAS Number Theory Seminar

##### An integral Eisenstein-Sczech cocycle on $GL_n(Z)$ and $p$-adic L-functions of totally real fields

In 1993, Sczech defined an $n-1$ cocycle on $GL_n(Z)$ valued in a certain space of distributions. He showed that specializations of this cocyle yield the values of the partial zeta functions of totally real fields of degree $n$ at nonpositive integers. In this talk, I will describe an integral refinement of Sczech's cocycle. By introducing a "smoothing" prime $l$, we define an $n-1$ cocycle on a congruence subgroup of $GL_n(Z)$ valued in a space of $p$-adic measures. We prove that the specializations analogous to those considered by Sczech produce the $p$-adic L-functions of totally real fields. We also consider certain other specializations that conjecturally yield the Gross-Stark units defined over abelian extensions of these fields. This is joint work with Pierre Charollois.

##### Even Galois Representations and the Fontaine-Mazur conjecture

Fontaine and Mazur have a remarkable conjecture that predicts which (p-adic) Galois representations arise from geometry. In the special case of two dimensional representations with distinct Hodge-Tate weights, they further conjecture that these "geometric" representations are also modular. Kisin has proven this conjecture in almost all cases under the assumption that the action of complex conjugation has determinant -1. We remove this restriction. In the case of even Galois representations, this generalizes previous work in which the representation was assumed to be ordinary.

##### Expanding the scope of Hilbert irreducibility

If $K$ is the rational function field $K=Q(t)$, then a polynomial $f$ in $K[x]$ can be regarded as a one-parameter family of polynomials. If $f$ is irreducible, then a basic form of Hilbert's irreducibility theorem states that there are infinitely many parameters in $Q$ for which the corresponding polynomial is also irreducible. Moreover, if one exploits conjectures of Mordell and Lang (proved by Faltings) one can frequently prove stronger statements, e.g. that there are only finitely many reducible specializations where the parameter lies in a quadratic extension of $Q$.

##### Constructing Abelian varieties over Qbar not isogenous to a Jacobian

We discuss the following question of Nick Katz and Frans Oort: Given an Algebraically closed field K, is there an Abelian variety over K of dimension g which is not isogenous to a Jacobian? For K the complex numbers its easy to see that the answer is yes for g>3 using measure theory, but over a countable field like Qbar new methods are required. Building on work of Chai-Oort, we show that, as expected, such Abelian varieties exist for K=Qbar and g>3. We will explain the proof as well as its connection to the Andre Oort conjecture.

##### Periods of special cycles and derivatives of L-series

In this talk, I will state some conjectures and examples concerning the central derivatives of automorphic L-series in terms of heights of special cycles on Shimura varieties.

##### Endoscopic transfer of depth-zero supercuspidal L-packets

In a recent paper, DeBacker and Reeder have constructed a piece of the local Langlands correspondence for pure inner forms of unramified $p$-adic groups and have shown that the corresponding L-packets are stable. In this talk we are going to discuss the endoscopic transfer of these L-packets: the theory of endoscopy -- an instance of the broad principle of functoriality -- predicts precise relationships between the L-packets on a group $G$ and its endoscopic groups $H$. This relationship is encoded in the so called endoscopic character identities. We will motivate and state these identities, paying attention to the precise normalization of all objects involved. If time permits, we will then discuss their proof and the extension of the correspondence to non-pure inner forms via the theory of isocrystals with additional structure.

##### The Iwasawa Main Conjectures for Modular Forms

##### Parahoric subgroups and supercuspidal representations of $p$-adic groups

This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is that this induced representation can (in certain situations) have finite length. I will describe the parahorics and characters for which this occurs, and what the Langlands parameters of the corresponding irreducible summands must be.

##### Weyl's sums for roots of quadratic congruences

It is known that the roots of congruences for a fixed irreducible quadratic polynomial are equidistributed. This statement translates to getting cancellation in the corresponding sum of Weyl's sums. In a recent work by W. Duke, J. Friedlander and H. Iwaniec we succeeded to get cancellation (so also the equidistribution) in very short sums of Weyl's sums relatively to the discriminant. The spectral theory of metaplectic automorphic forms is the basic tool, of which some special aspects will be the subject of this talk. Numerous applications of the result will be also discussed.

##### Impossible intersections for elliptic curves

We proved with Umberto Zannier that there are at most finitely many complex numbers $\lambda \neq {0,1}$ such that two points on the Legendre elliptic curve $y^2 = x(x-1)(x-\lambda)$ with coordinates $x=2$ and $x=3$ both have finite order. However we still do not know how to find these $\lambda$ effectively (there are probably none). We discuss various extensions of this result, some of them effective. For example some comments of Serre encouraged us to show that there are no $\lambda$ at all for coordinates $x=u$ and $x=v$ when the complex numbers $u,v$ are not both algebraic with $1,u,v$ linearly dependent over the field of rationals and $uv(u-1)(v-1)(u-v) \neq 0$. The proof needs a close study of what may be called "bicyclotomic polynomials."

##### Heuristics for lambda invariants

The $\lambda$-invariant is an invariant of an imaginary quadratic field that measures the growth of class numbers in cyclotomic towers over the field. It also measures the number of zeroes of an associated $p$-adic L-function. In this talk, I will discuss the following question: How often is the $p$-adic $\lambda$-invariant of an imaginary quadratic field equal to $m$? I'll explain how one can model this question by statistics of $p$-adic random matrices, and show one can test this model by computing $\lambda$-invariants rapidly.

This is joint work with Jordan Ellenberg and Akshay Venkatesh.

##### Whittaker Functions and Demazure Characters

It is well-known that there are connections between the representation theory of a reductive p-adic

group and the topology of the flag variety of the Langlands L-group. We will discuss Whittaker functions in this light. The Casselman-Shalika formula shows that the values of the spherical Whittaker function are the same as the characters of irreducible representations of the L-group. The Borel-Weil-Bott theorem identifies these same characters as cohomology groups of line bundles on the flag variety. Generalizing the Borel-Weil-Bott theorem, cohomology groups of line bundles on Schubert varieties are "Demazure characters". We will show how Demazure characters also appear in Iwahori Whittaker functions. This is joint work with Ben Brubaker and Anthony Licata.

##### Periods of quaternionic Shimura varieties

In the early 80's, Shimura made a precise conjecture relating Peterssoninner products of arithmetic automorphicforms on quaternion algebras over totally real fields, up to algebraic factors. This conjecture (which is a consequence of the Tate conjecture on algebraic cycles) was proved a few years later by Michael Harris. In the first half of my talk I will motivate and describe an integral version of Shimura's conjecture i.e. up to p-adic units for a good prime p. In the second half I will describe work in progress (joint with Atsushi Ichino) that makes some progress in understanding this refined conjecture.

##### Affine sieve and expanders

I will talk about the fundamental theorem of affine sieve (joint with Sarnak). The main black box in the proof of this result will be also explained. It is a theorem on a necessary and sufficient condition for a finitely generated subgroup of SL(n,Q) under which the Cayley graphs of such a group modulo square free integers form a family of expanders (joint with Varju).

##### Niebur Integrals and Mock Automorphic Forms

Among the bounty of brilliancies bequeathed to humanity by Srinivasa Ramanujan, the circle method and the notion of mock theta functions strike wonder and spark intrigue in number theorists fresh and seasoned alike. The former creation was honed to perfection for its original purpose of counting partitions by Hans Rademacher. The latter ingenuity, despite receiving considerable scrutiny, remained largely enigmatic for decades. In 2002 Sander Zwegers ascertained the essential properties characterizing Ramanujan's mock theta functions. This breakthrough has triggered an avalanche of activity (in mathematics and physics) associated with mock automorphic forms. In 1968 Douglas Niebur, acting upon a suggestion made originally by Atle Selberg, uncovered a natural generalization of automorphic forms.

##### Random maximal isotropic subspaces and Selmer groups

We show that the $p$-Selmer group of an elliptic curve is naturally the intersection of two maximal isotropic subspaces in an infinite-dimensional locally compact quadratic space over $F_p$. By modeling this intersection as the intersection of a random maximal isotropic subspace with a fixed compact open maximal isotropic subspace, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2-Selmer groups in certain families of quadratic twists, and the average size of 2- and 3-Selmer groups as computed by Bhargava and Shankar. The random model is consistent with Delaunay's heuristics for Sha[p], and predicts that the average rank of elliptic curves is at most $1/2$. This is joint work with Eric Rains.

##### Algebraic cycles and Euler systems for real quadratic fields

I will discuss some possible applications of algebraic cycles and p-adic families of modular forms to the arithmetic of elliptic curves over abelian extensions of real quadratic fields. This is a report on work in progress with Victor Rotger and Ignacio Sols, and on earlier work with Massimo Bertolini and Kartik Prasanna.

##### Serre´s conjectures on the number of rational points of bounded height

We give a survey of recent results on conjectures of Heath-Brown and Serre on the asymptotic density of rational points of bounded height. The main tool in the proofs is a new global determinant method inspired by the local real and p-adic determinant methods of Bombieri-Pila and Heath-Brown.

##### Relative Homotopy type and obstructions to the existence of rational points

In 1969 Artin and Mazur defined the etale homotopy type $Et(X)$ of scheme $X$, as a way to homotopically realize the etale topos of a $X$. In the talk I shall present for a map of schemes $X\rightarrow S$ a relative version of this notion. We denoted this construction by $Et(X/S)$ and call it the homotopy type of $X$ over $S$. It turns out that the relative Homotopy type, can be especially useful in studying the sections of the map $X\rightarrow S$. In the special case where $S=Spec K$ is the spectrum of a field, the set of sections are just the set of rational points $X(K)$ and then the relative homotopy type $Et(X/Spec K)$ can be used to define obstructions to the existence of a rational point on $X$.