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

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

We prove, under mild hypotheses, there are no irreducible two-dimensional ordinary even Galois representations of the Galois group of Q with distinct Hodge-Tate weights, in accordance with the Fontaine-Mazur conjecture. We also show how this method can be applied to a related circle of problems.

##### An arithmetic fundamental lemma for unitary group of three variable

In this talk I'll present a relative trace formula approach to the Gross-Zagier formula and its high dimensional generalization (a derivative version of the global Gross-Prasad conjecture) for unitary group. In particular, an arithmetic fundamental lemma (AFL) is proposed. Some results proved recently will be presented, including the AFL for unitary group of three variable.

##### Volume estimates in analytic and adelic geometry

The solution to many classical counting asymptotics problems in number theory goes by comparison with an analogous volume asumptotics. In a general setting, we establish asymptotic formulae for volumes of height balls in analytic varieties over local fields and in adelic points of algebraic varieties over number fields, relating the Mellin transforms of height functions to Igusa integrals and to global geometric invariants of the underlying variety. In the adelic setting, this involves the construction of general Tamagawa measures. This is joint work with Yuri Tschinkel (Courant).

##### Torsion in the homology of arithmetic groups

##### Generalizations of the Sato-Tate conjecture

I will discuss a recent joint work with Thomas Barnet-Lamb and Toby Gee in which we prove the Sato-Tate conjecture for non-CM regular algebraic cuspidal automorphic representations of GL_2 over a totally real field.

##### Modularity lifting for n-dimensional ordinary Galois representations

I will discuss a generalization of the modularity lifting theorems of Clozel, Harris and Taylor to the case of ordinary Galois representations. The result is obtained by applying the Taylor-Wiles method (with innovations due to Kisin and Taylor) over a Hida family. A key step is to construct an appropriate ordinary lifting ring and determine its irreducible components.

##### Mean values with $GL(2)\times GL(3)$ functions

##### On the areas of rational triangles or How did Euler (and how can we) solve $xyz(x+y+z) = a$?

By Heron's formula there exists a triangle of area $\sqrt{a}$ all of whose sides are rational if and only if $a > 0$ and $xyz(x + y + z) = a$ for some rationals $x, y, z$. In a 1749 letter to Goldbach, Euler constructed infinitely many such $(x, y, z)$ for any rational $a$ (positive or not), remarking that it cost him much effort but not explaining his method. We suggest one approach, using only tools available to Euler, that he might have taken, and use it to construct several other infinite families of solutions. Then we reconsider the problem as a question in arithmetic geometry: $xyz(x+y+z) = a$ gives a K3 surface, and each family of solutions is a singular rational curve on that surface defined over ${\bf Q}$. The structure of that Néron--Severi group of that K3 surface explains why the problem is unusually hard.

##### Slope filtrations in families

In the 21st-century approach to p-adic Hodge theory, one studies local Galois representations (and related objects) by converting them into modules over certain power series rings carrying certain extra structures (Frobenius actions and derivations). A key tool in matching up the two sides is a certain classification theorem for the second class of objects, called the slope filtration theorem. A natural step in this program is to try match up analytic families of Galois representations with Frobenius-differential modules over relative power series rings (i.e., with coefficients which are functions on some base space, like an affinoid). For this, one needs to understand how the slope filtration varies in an analytic family, e.g., whether the Newton polygon is semicontinuous.

##### Hilbert modular surfaces through K3 surfaces

We describe how to use Shioda-Inose structures on K3 surfaces to write down explicit equations for Hilbert modular surfaces, which parametrize principally polarized abelian surfaces with real multiplication by the ring of integers in $Q(\sqrt{D})$. In joint work with Elkies, we have computed several of these (for fundamental discriminants less than 100), including some of general type. These techniques can be used to produce explicit examples of genus $2$ curves with real multiplication, and modular forms with coefficients in a real quadratic field.

##### An effective proof of the Oppenheim Conjecture

In the mid 80's Margulis proved the Oppenheim Conjecture regarding values of indefinite quadratic forms. I will present new work, joint with Margulis, where we quantify this statement by giving bounds on the size of integer vectors for which $|Q(x)|<\epsilon$ for an irrational indefinite quadratic form $Q$ in three variables.

##### Generalized modular functions

Generalized modular functions are holomorphic functions on the complex upper half-plane, meromorphic at the cusps that satisfy the usual defintion of a modular function, however with the important exception that the character need not be unitary. The theory is partly motivated from conformal field theory in physics. In my talk I will report on recent joint work with G. Mason on properties of their Fourier coefficients and characters.

##### Some Remarks on quadratic Twists of L-Functions

Suppose $E$ is a rational elliptic curve and $p$ is a given prime. It is of interest to know that there exists a square free integer $D$ such that the $D$-th quadratic twist $E_D$ of $E$ such that the p-Selmer group of $E_D$ is trivial. The existence of such a $D$ seems not yet known in general. It amounts to a nonvanishing statement in characteristic $p$ for the special parts of the L-function of $E$ and its quadratic twists.

In characteristic 0, the analogous result states that there exists a $D$ such that $E_D$ has rank zero. There are several apparently different proofs of this fact, due to Waldspurger, Bump-Friedburg-Hoffstein, and Murty-Murty.

##### Real quadratic analogues of values of the j-function at CM points

An interesting new class of modular forms has emerged in the last several years that generalize Ramanujan's mock theta functions. The generalization is based on an observation of of Zwegers who showed that mock theta functions occur as holomorphic parts of harmonic Maass forms of weight 1/2 having singularities in cusps. Their non-holomorphic parts are directly related to actual theta functions. This realization has enhanced the study of the coefficients of the mock-theta functions, which are related to various kinds of partitions.

##### On Eisenstein series and the cohomology of arithmetic groups

The automorphic cohomology of a reductive $\mathbb{Q}$--group $G$, defined in terms of the automorphic spectrum of $G$, captures essential analytic aspects of the arithmetic subgroups of $G$ and their cohomology. We discuss the actual construction of cohomology classes represented by residues or principal values of derivatives of Eisenstein series, We show that non--trivial Eisenstein cohomology classes can only arise if the point of evaluation features a 'half--integral' property. This rises questions concerning the analytic behavior of certain automorphic L--functions at half--integral arguments.

##### Geometric Overconvergent Modular Forms

We will give a geometric definition of the notion of overconvergent modular form of any p-adic weight. As a consequence, we re-obtain Coleman's theory of p-adic families of eigenforms and the eigencurve of Coleman and Mazur without using the Eisenstein family. Similar results have just been obtained independantly by Andreatta, Iovita and Stevens. We will then explain how a similar construction can be applied to construct p-adic families of Hilbert and Siegel eigenforms (over the total weight space). This last part is a work in progress with Andreatta and Iovita.

##### The average rank of elliptic curves

A *rational elliptic curve* may be viewed as the set of solutions to an equation of the form $y2=x3+Ax+B$, where $A$ and $B$ are rational numbers. It is known that the rational points on this curve possess a natural abelian group structure, and the Mordell-Weil theorem states that this group is always finitely generated. The *rank* of a rational elliptic curve measures *how many* rational points are needed to generate all the rational points on the curve.

##### An estimate for the counting function of prime chains with applications

##### On a p-adic automorphic construction of Euler systems

##### Proof, via smooth homology, of the existence of rational families of H-invariant linear forms on G-induced representations, when G/H is a symmetric, reductive, p-adic space, via smooth homology

We fix $F$ a local non archmedean field of characteristic zero, $G$ the points over $F$ of an algebraic reductive group defined over $F$ and $s$ a rational involution of $G$ defined over $F$. We note $H$ the group of fixed points of $G$ under the action of $s$ and $X(G,s)$ the connected component on the neutral element of the set of complex characters of $G$ antiinvariant under the action of $s$. Let $P$ be a $s$-parabolic subgroup of $G$, in other words the intersection $M$ of $P$ with $s(P)$ is an $s$-stable Levi subgroup, we construct from a irreducible, smooth representation $r$ of $M$, a rational family of distributions above the algebraic variety $X(G,s)$, which are $H$-invariant linear forms on tne smooth induced representation $ind(P,G; r )$. Our main trick is the use of homology of groups.