Seminars Sorted by Series

We study dynamical systems in several variables over a complete valued field. If $x$ is a fixed point, we show that in many cases there exist fixed analytic subvarieties through $x$. These cases include all cases in which $x$ is attracting in some...

In this talk, I will describe roughly how to define a generating function arithmetic divisors (in Arakelov sense) on a unitary Shimura variety of type $(n-1,1)$. I will then briefly explain why it is modular. If time permits, I will also talk...

In this talk I will describe some new arithmetic invariants for pairs of torus orbits on inner forms of $\mathbf{PGL}_n$ and $\mathbf{SL}_n$. These invariants allow us to significantly strengthen results towards the equidistribution of packets of...

By a result of Julia Robinson, we know that the first order theory of the field of rational numbers is undecidable, and in fact the same holds for any number field. In view of this, it is suggested by analogies studied by Vojta and others that the...

Let $G$ be a reductive group over a global field of positive characteristic. In a major breakthrough, Vincent Lafforgue has recently shown how to assign a Langlands parameter to a cuspidal automorphic representation of $G$. The parameter is a...

Based on a new decoupling inequality for curves in $\mathbb R^d$, we obtain the essentially optimal form of Vinogradov's mean value theorem in all dimensions (the case $d = 3$ is due to T. Wooley). Various consequences will be mentioned and we will...

This is joint work with Zhiwei Yun. We prove a generalization of Gross-Zagier formula in the function field setting. Our formula relates self-intersection of certain cycles on the moduli of Shtukas for $\mathrm{GL}(2)$ to higher derivatives of L...

Joint work with Jacob Tsimerman. Let $B(g,p)$ denote the number of isomorphism classes of $g$-dimensional abelian varieties over the finite field of size $p$. Let $A(g,p)$ denote the number of isomorphism classes of principally polarized $g$...

In joint work with Emmanuel Kowalski and Philippe Michel, we prove two different estimates on sums of coefficients of modular forms---one related to L-functions and another to the level of distribution. A key step in the argument is a careful...

I will give an overview of my work with Antonio Lei, David Loeffler and Guido Kings about the construction of an Euler system for Rankin-Selberg convolutions of modular forms and its arithmetic applications. I will then discuss generalisations of...

The problem of the density of squarefree discriminant polynomials is an old one, being considered by many people, and the density being conjectured by Lenstra. A proof has been out of question for a long time. The reason it was desired is that a...

I will present asymptotic upper and lower bounds for the supremum norms of Maass forms of growing Laplace eigenvalue on a wide class of semisimple groups. I will also present a general lower bound in the level aspect that improves over the trivial...

Iwasawa theory is a powerful technique for understanding the link between the special values of L-functions and arithmetic objects (such as class groups of number fields, or Mordell-Weil groups of elliptic curves). In this talk I'll discuss what...

Markoff numbers give rise to extremely low-lying reciprocal geodesics on the modular surface, but it is unknown whether infinitely many of these are fundamental, that is, the corresponding binary quadratic form has fundamental discriminant. In joint...

Waldspurger has shown that the genuine automorphic cuspidal representations of the metaplectic cover $S$ of $SL_2$ are divided naturally into packets, and that these packets are indexed by the cuspidal automorphic representations of $PGL_2$. We...

Let $G$ be a reductive group over a global function field. V. Lafforgue has recently constructed the \'automorphic-to-Galois\' direction of the global Langlands correspondence for $G$. I will discuss a potential converse to this result. This is...

For a non-degenerate integral quadratic form $F(x_1, \dots , x_d)$ in 5 (or more) variables, we prove an optimal strong approximation theorem. Fix any compact subspace $\Omega\subset\mathbb{R}^d$ of the affine quadric $F(x_1,\dots,x_d)=1$. Suppose...

A. Venkatesh asked us the question, in the context of torsion automorphic forms: Does the Standard Conjecture (of Grothendieck's) of K

Around 1990, Joe Silverman wrote a series of three articles on the variation of canonical height in families of elliptic curves. I will discuss connections between these results and dynamical systems on $\mathbb P^1$ (and an associated Berkovich...

Given a non-isotrivial elliptic curve $E$ over $K = \mathbb F_q(t)$, there is always a finite extension $L$ of $K$ which is itself a rational function field such that $E(L)$ has large rank. The situation is completely different over complex function...

This concerns recent work with P. Corvaja in which we relate the Hilbert Property for an algebraic variety (a kind of axiom linked with Hilbert Irreducibility, relevant e.g. for the Inverse Galois Problem) with the fundamental group of the variety...

Gauss famously investigated class numbers of quadratic fields, in particular characterizing the 2-divisibility of the class number for such fields. In general, it is expected that for a number field of any degree, and any rational prime $p$, the $p$...

I will explain two recent results concerning non-zero coefficients of modular forms modulo a prime $p$. The first result, a joint work with K. Soundararajan, gives an asymptotic equivalent for the number of such coefficients. The second is concerned...

In this talk we will show how modular forms can be applied to energy minimization problems in Euclidean space. Namely, we will explain Cohn-Elkies linear programming method and present explicit constructions of corresponding certificate functions...

I will discuss some recent results on Serre weight conjectures in dimension $> 2$, based on the study of certain tame type deformation rings. This is joint work with (various subset of) D. Le, B. Levin and S. Morra.

We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at agiven finite set of primes. For the tame supercuspidals, we prove the limit multiplicity property with error terms. Therebywe...

Work of Kobayashi and Iovita-Pollack describes how local points of supersingular elliptic curves on ramified $\mathbb Z_p$-extensions of $\mathbb Q_p$ split into two strands of even and odd points. We will discuss a generalization of this result to...

Initiated by Langlands, the problem of computing the Hasse-Weil zeta functions of Shimura varieties in terms of automorphic L-functions has received continual study. We will discuss how recent progress in various aspects of the field has allowed the...

After beginning by giving a brief overview of how one can think of noncongruence modular curves as moduli spaces of elliptic curves with G-structures, we will discuss how these moduli interpretations fits into the greater body of knowledge...

This is a report on a work in progress in collaboration with Dinakar Ramakrishnan. A celebrated result of Mazur states that open modular curves of large enough level do not have rational points. We study analogous questions for the Picard modular...

The Cohen-Lenstra Heuristics conjecturally give the distribution of class groups of imaginary quadratic fields. Since, by class field theory, the class group is the Galois group of the maximal unramified abelian extension, we can consider the Galois...

We will explain a way how one can use moduli schemes and their natural forgetful maps in the study of certain classical Diophantine problems (e.g. finding all integral points on hyperbolic curves). To illustrate and motivate the strategy, we...

Of interest are (i) the conjecture of Bombieri (and Lang) that for any smooth projective surface $X$ of general type over a number field $k$, the set $X(k)$, of $k$-rational points is not Zariski dense, and (ii) the conjecture of Lang that $X(k)$...

I will start by explaining Quillen's notion of a superconnection, and then will use it to define some natural differential forms on period domains parametrizing Hodge structures. For hermitian symmetric domains, we will show that this construction...

We explore the comparison isomorphism of $p$-adic Hodge theory in the case of elliptic curves, and discuss some ideas which may be used to prove the S-unit theorem and the finiteness of rational points on higher-genus curves (Faltings' theorem).

We will discuss how Vinogradov's method applies to the study of the 2-part of class groups in certain thin families of quadratic number fields. We will show how the method yields a density result for the 16-rank in the family of quadratic number...

The importance of the subconvexity problem is well-known. In this talk, I will discuss a new approach to establish subconvex bounds for automorphic L-functions. The method is based on adopting the circle method to separate oscillatory factors...

The "moduli space" for principally polarized complex $n$ dimensional Abelian varieties with real structure (that is, anti-holomorphic involution) may be identified with a certain locally symmetric space for the group $\mathrm{GL}(n)$ over the real...

Let $k$ be a fixed positive integer. Myerson (and others) asked how small the modulus of a non-zero sum of $k$ roots of unity can be. If the roots of unity have order dividing $N$, then an elementary argument shows that the modulus decreases at most...

I have been developing a new bridge between number theory and symplectic geometry. The special program at the IAS and a workshop this week in Wolfensohn Hall are devoted to mirror symmetry. I will describe this bridge, explain that there are travel...

In a recent preprint with Sug Woo Shin (https://arxiv.org/abs/1609.04223) I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the...

In mod-$p$ reductions of modular curves, there is a finite set of supersingular points and its open complement corresponding to ordinary elliptic curves. In the study of mod-$p$ reductions of more general Shimura varieties, there is a "Newton...

If two motives are congruent, is it the case that the special values of their respective $L$-functions are congruent? More precisely, can the formula predicting special values of motivic $L$-functions be interpolated in $p$-adic families of motives...

Let $F$ be a field of characteristic not equal to 2. Let $r$ be a 2-dimensional even Galois representation which is induced from an $F$-valued character of odd order of the absolute Galois group of a real quadratic field $K$. After imposing some...

Given an abelian scheme over a smooth curve over a number field, we can associate two height function: the fiberwise defined Neron-Tate height and a height function on the base curve. For any irreducible subvariety $X$ of this abelian scheme, we...

This is joint work with Tasho Kaletha. The local Langlands correspondence predicts that representations of a reductive group $G$ over a $p$-adic field are related to Galois representations into the Langlands dual of $G$. A conjecture of Kottwitz (as...

The Grothendieck-Katz $p$-curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$-curvature vanishes modulo $p$...