Seminars Sorted by Series

The IAS recently organized an Emerging Topics Working Group on "Applications to modularity of recent progress on the cohomology of Shimura varieties". Based on ideas of Calegari and Geraghty, and using recent results of Khare-Thorne and Caraiani...

Arithmetic quantum chaos concerns the limiting behavior of a sequence of automorphic forms on spaces such as the modular surface. It is now known in many cases (by work of Lindenstrauss, Holowinsky, Soundararajan and others) that the mass...

Take \(E/Q\) to be an elliptic curve with full rational 2-torsion (satisfying some extra technical assumptions). In this talk, we will show that 100% of the quadratic twists of \(E\) have rank less than two, thus proving that the BSD conjecture...

I will discuss a comparison theorem that allows us to recover $p$-adic (pro-)etale cohomology of $p$-adic Stein spaces with semistable reduction over local rings of mixed characteristic from complexes of differential forms. To illustrate possible...

Kloosterman sums arise naturally in the study of the distribution of various arithmetic objects in analytic number theory. The 'vertical' Sato-Tate law of Katz describes their distribution over a fixed field $\mathbb F_p$, but the equivalent...

In the last two decades there has been much study of what happens when an algebraic curve in \(n\)-space is intersected with two multiplicative relations
\[x_1^{a_1} \cdots x_n^{a_n} = x_1^{b_1} \cdots x_n^{b_n} = 1 \tag{\(\times\)}\]
for \((a_1,...

The cohomology of an arithmetic subgroup of a reductive algebraic group defined over a number field is closely related to the theory of automorphic forms. We discuss in which way residues of Eisenstein series contribute non-trivially to the subspace...

The Gan-Gross-Prasad conjecture relates the non-vanishing of a special value of the derivative of an L-function to the non-triviality of a certain functional on the Chow group of a Shimura variety. Beyond the one-dimensional case, there is little...

The generalised Kato classes of Darmon-Rotger arise as $p$-adic limits of diagonal cycles on triple products of modular curves, and in some cases, they are predicted to have a bearing on the arithmetic of elliptic curves over $Q$ of rank two. In...

Let $E$ be a CM elliptic curves over rationals and $p$ an odd prime ordinary for $E$. If the $\mathbb Z_p$ corank of $p^\infty$ Selmer group for $E$ equals one, then we show that the analytic rank of $E$ also equals one. This is joint work with...

In 1880, Markoff studied a cubic Diophantine equation in three variables now known as the Markoff equation, and observed that its integral solutions satisfy a form of nonlinear descent. Generalizing this, we consider families of log Calabi-Yau...

Let $D$ be a central division algebra of degree $n$ over a field $K$. One defines the genus gen$(D)$ of $D$ as the set of classes $[D']$ in the Brauer group Br$(K)$ where $D'$ is a central division $K$-algebra of degree $n$ having the same...

A celebrated theorem of Duke states that Picard/Galois orbits of CM points on a complex modular curve equidistribute in the limit when the absolute value of the discriminant goes to infinity. The equidistribution of Picard and Galois orbits of...

Existing unconditional progress on the abc conjecture and Szpiro's conjecture is rather limited and coming from essentially only two approaches: The theory of linear forms in $p$-adic logarithms, and bounds for the degree of modular parametrizations...

$p$-adic period spaces have been introduced by Rapoport and Zink as a generalization of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic $p$-adic flag manifold. An approximation of this open subset is the...

Let $A$ be an abelian variety over a number field $K$. The number of torsion points defined over a finite extension $L$ is bounded polynomially in terms of the degree $[L:K]$. We compute the optimal exponent for this bound, in terms of the dimension...

We consider the coherent cohomology of toroidal compactifications of Shimura varieties with coefficients in the canonical extensions of automorphic vector bundles and show that they can be computed as relative Lie algebra cohomology of automorphic...

The number of integer Markoff triples below a given bound has a nice asymptotic formula with an exponent of growth of 2. The exponent of growth for the Markoff-Hurwitz equations, on the other hand, is in general not an integer. Certain elliptic K3...

The modular Jacobians decompose, up to isogeny, into the abelian varieties $X_f$ cut out by cuspforms $f$ of weight 2, and a conjecture attributed to Serre posits that $X_f$ has infinitely many ordinary primes. Similarly for the André motives in the...

The Galois group of Q acts on the homology of the complex moduli space of abelian varieties, or, equivalently, on the homology of symplectic groups Sp_{2g}(Z). (Here we take homology with finite or profinite coefficients.) In particular, the Galois...

I will discuss several results on abstract homomorphisms between the groups of rational points of algebraic groups. The main focus will be on a conjecture of Borel and Tits formulated in their landmark 1973 paper.

Our results settle this conjecture...

The representation theory of reductive groups on categories is the geometric counterpart to the representation theory of reductive groups over finite fields. I will describe, following joint work with Sam Gunningham, a geometric counterpart for...

Consider a pair $(X,\widehat{L})$ of a regular and projective algebraic variety over $\mathbb{Q}$ and a semipositive adelically metrized line bundle $\widehat{L}$ over $X$. Assume that all of Zhang's successive minima of the pair $(X,\widehat{L})$...

The classical conjectures of Ramanujan-Petersson and Sato-Tate on the Fourier coefficients of modular forms, or more generally on the Satake parameters of automorphic representations, are highly sensitive to questions of functoriality. For example...

We prove the modularity of some two-dimensional residually reducible p-adic Galois representations over Q under certain hypothesis on the residual representation at p. To do this, we generalize Emerton's local-global compatibility result and devise...

A basic but difficult question in the analytic theory of automorphic forms is: given a reductive group G and a representation r of its L-group, how many automorphic representations of bounded analytic conductor are there? In this talk I will present...

In this talk, we construct the so-called Fourier-Jacobi cycles on unitary Shimura varieties. The height pairing of these cycles can be regarded as the arithmetic analogue of classical Fourier-Jacobi periods for the pair of unitary groups of equal...

Braverman and Kazhdan have conjectured the existence of summation formulae that are essentially equivalent to the analytic continuation and functional equation of Langlands L-functions in great generality. Motivated by their conjectures and related...

Deformation spaces of representations of Galois groups of p-adic fields with p-adic Hodge theoretic conditions play a central role in many arithmetic questions, such as the weight part of Serre's conjecture and modularity lifting theorems, yet they...

I will try to explain what the Fourier expansion of a "modular form" on an exceptional group looks like, from the point of view of the archimedean place. In more detail, Gross-Wallach and Gan-Gross-Savin have singled out what a modular form on an...

This talk will be an exposition of a circle of ideas that concerns the cohomology of arithmetic groups and the special values of certain automorphic L-functions. I will explain some recent results about the critical values of (1) Rankin-Selberg L...

The analytic theory of Poincaré series and Maass cusp forms and their L-functions for $SL(3,Z)$ has, so far, been limited to the spherical Maass forms, i.e. elements of a spectral basis for $L^2(SL(3,Z)\PSL(3,R)/SO(3,R))$. I will describe the Maass...

S-operators were originally introduced by V. Lafforgue as certain operators acting on the cohomology of moduli of Shtukas. I will discuss their analogues and generalizations in the Shimura variety setting. They induce Hecke equivariant maps between...

The Faltings height is a useful invariant for addressing questions in arithmetic geometry. In his celebrated proof of the Mordell and Shafarevich conjectures, Faltings shows the Faltings height satisfies a certain Northcott property, which allows...

The Langlands-Rapoport conjecture gives a description of the mod $p$ points of suitable integral models of Shimura varieties. Such results are of use, for example, in computing the local factor of the (semi-simple) Hasse-Weil zeta function of the...

At the November workshop, I described a new algorithm to cover compact, congruence locally symmetric spaces by balls. I’ll discuss how to compute the nerve of such a covering and Hecke actions on its cohomology. Joint work with Aurel Page.At the...

I will describe recent work with Ngaiming Mok and Jacob Tsimerman giving analogues of "Ax-Schanuel" for Shimura varieties. I will sketch the role of "o-minimal structures" in the proof, in particular point counting and powerful algebraicity results...

Let $E$ be a CM elliptic curve over a totally real number field $F$ and $p$ an odd ordinary prime. If the ${p^{\infty}\mbox{-}\mathrm{Selmer}}$ group of $E$ over $F$ has ${\mathbb{Z}_{p}\mbox{-}\mathrm{corank}}$ one, we show that the analytic rank...

We explain the magic of the entire function $\Psi_p \in \mathbb{Z}[x] \cap \mathbb{Q}_p\{x\}$ defined by the functional equation \[x = \sum_{j=0}^{\infty} p^{-j}\Psi_p (p^{j} x)^{p^{j}}\] which satisfies $\Psi_p (\mathbb{Q}_p) \subset \mathbb{Z}_p$...

We start by showing that for any 1-parameter family of genus $g>2$ curves, the number of rational points is bounded by $g$, degree of the field, and the Mordell-Weil rank. Apart from the classical Faltings-Vojta-Bombieri method, the new input is a...

Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever $E$ has a rational 3-isogeny...

We introduce a new $p$-adic Maass-Shimura operator acting on a space of "generalized $p$-adic modular forms" (extending Katz's notion of $p$-adic modular forms) defined on the $p$-adic (preperfectoid) universal cover of Shimura curves. Using this...

In the late 1950’s, Burgess developed a novel method for bounding short sums of a multiplicative Dirichlet character. This resulted in a subconvexity bound for Dirichlet $L$-functions on the critical line. In recent work, we have adapted the Burgess...

In his influential paper "Modular curves and the Eisenstein ideal", Mazur studied congruences modulo p between cusp forms and the Eisenstein series of weight 2 and prime level N. In particular, he defined the Eisenstein ideal in the relevant Hecke...

The seminal work of Deligne and Lusztig on the representations of finite reductive groups has influenced an industry studying parallel constructions in the same theme. In this talk, we will discuss recent progress on studying analogues of Deligne-...