Seminars Sorted by Series

We consider an ordinary linear $p$-adic differential equation \[Dy=d^ny/dx^n+a_{n-1}d^{n-1}y/dx^{n-1}+\dots+a_0y=0, a_i\in\mathbb{Z}_p[[x]][p^{-1}]\] whose formal solutions in $\mathbb{Q}_p[x]$ converge in the open unit disc $|x|<1$. In 1973, Dwork proved that $y$ has a log-growth $n-1$, that is, $|y|_{\rho}=O((\log{1/\rho})^{1-n})$ as $\rho\uparrow 1$, where $|y|_{\rho}$ denotes the $\rho$-Gaussian norm of $y$. Moreover, Dwork defined the log-growth filtration of the solution space of $Dy=0$ by measuring the log-growth of $y$. Then, Dwork conjectured that the log-growth filtration can be compared with the Frobenius slope filtration when $Dy=0$ admits a Frobenius structure. Recently, some partial results on Dwork's conjecture have been obtained by André, Chiarellotto-Tsuzuki, and Kedlaya. In this talk, we discuss the rationality of breaks of the log-growth filtration.

The formal degree conjecture relates the formal degree of an irreducible square-integrable representation of a reductive group over a local field to the special value of the adjoint gamma-factor of its L-parameter. We prove the formal degree...

In 1972, Mazur showed that the Newton polygon of a crystal lies below the Hodge polygon of the associated isocrystal and the two polygons have the same end points. In 2003, Kottwitz and Rapoport showed that the converse is true, i.e., given two such...

Eigencurve was introduced by Coleman and Mazur to parametrize modular forms varying $p$-adically. It is a rigid analytic curve such that each point corresponds to an overconvegent eigenform. In this talk, we discuss a conjecture on the geometry of...

The Moebius function plays a central role in number theory; both the prime number theorem and the Riemann Hypothesis are naturally formulated in terms of the amount of cancellations one gets when summing the Moebius function. In recent joint work...

I'll describe ongoing joint work with F. Andreatta, E. Goren, and K. Madapusi Pera towards Colmez's conjecture expressing Faltings heights of CM abelian varieties in terms of values of Artin L-functions.

Kisin module is very useful to study crystalline representations. In this talk, we extend the theory of Kisin modules and crystalline representations to allow more general coefficient fields and lifts of Frobenius. In particular, we construct a...

We prove that the probability that a curve of the form $y^2 = f(x)$ over $\mathbb Q$ with $\deg f = 2g + 1$ has no rational point other than the point at infinity tends to 1 as $g$ tends to infinity. This is joint work with Michael Stoll.

In this talk I will review CM theory of complex projective K3 surfaces, and show how it can be used to construct K3 surfaces over finite fields. I will discuss work-in-progress where this is applied to describing: (1) the collection of zeta...

The André-Oort conjecture says that any subvariety of a Shimura variety with a Zariski dense set of CM points must itself be a Shimura subvariety. In recent years, this has been the subject of much work. We explain how this conjecture for the moduli...

I will talk about a general formula relating the p-adic heights of Heegner points to derivatives of p-adic L-functions. It generalizes results of Perrin-Riou and Howard to the setting of the work of Yuan-Zhang-Zhang on the complex Gross-Zagier...

I will first discuss the general conjectural picture relating algebraic cycles to L-functions and some extensions of the Gross-Zagier formula involving $p$-adic L-functions. This leads naturally to the question of constructing algebraic cycles...

We show that there is a bound $N(d,g,r)$ for the number of $K$-rational points on hyperelliptic curves $C$ of genus $g$ when the degree of the number field $K$ is $d$ and the Mordell-Weil rank $r$ of the Jacobian of $C$ is at most $g - 3$. The proof...

We investigate the shape of the reduction of certain crystalline Galois representations of integral slope 1 and of fractional slopes in (1,2). The proof uses the compatibility between the p-adic and mod p Local Langlands Correspondences with respect...

Starting from the Poisson summation formula, I discuss spectral summation formulae on GL(2) and GL(3) and present a variety of applications to automorphic forms, analytic number theory, and arithmetic.

The Colmez conjecture expresses the Faltings height of a CM abelian variety in terms of the logarithmic derivatives of certain Artin L-functions. In this talk, I will present an averaged version of the conjecture proved in my joint work with Shou-Wu...

Venkatesh has recently proposed a fascinating conjecture relating motivic cohomology with automorphic forms and the cohomology of arithmetic groups. I'll describe this conjecture, and discuss its connections with the local geometry of eigenvarieties...

The Grothendieck–Katz $p$-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo $p$ has vanishing $p$-curvatures for almost all $p$, has finite monodromy. It is known that it suffices to prove the conjecture...

Given the $p$-adic Galois representation associated to a regular algebraic polarized cuspidal automorphic representation, one naturally obtains a pure weight zero representation called its adjoint representation. Because it has weight zero, a...

Inspired by work of Masser and Zannier for torsion specializations of points on the Legendre elliptic curve, Baker and DeMarco proved that if $v$, $w$ are two points in $C$, then there are at most finitely many $t$ in $C$ such that $v$ and $w$ are...

I will speak about results contained in my article "$G$-torseurs en théorie de Hodge $p$-adique" linked to local class field theory. I will in particular explain the computation of the Brauer group of the curve and why its fundamental class is the...

Reeder and Yu gave recently a new construction of certain supercuspidal representations of $p$-adic reductive groups (called epipelagic representations). Their construction relies on the existence of stable vectors in the first Moy-Prasad filtration...

The existence of rational points on the Kummer variety associated to a 2-covering of an abelian variety $A$ over a number field can sometimes be established through the variation of the 2-Selmer group of quadratic twists of $A$. In the case when the...

For a representation of the absolute Galois group of the rationals over a finite field of characteristic $p$, we would like to know if there exists a lift to characteristic zero with nice properties. In particular, we would like it to be geometric...

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...