Seminars Sorted by Series

We prove a B-SD conjecture for elliptic curves (for the \(p^\infty\) Selmer groups with arbitrary rank) a la Mazur-Tate and Darmon in anti-cyclotomic setting, for certain primes \(p\). This is done, among other things, by proving a conjecture of...

Let \(k\) be an algebraically closed field and let \(c:C\rightarrow X\times X\) be a correspondence. Let \(\ell \) be a prime invertible in \(k\) and let \(K\in D^b_c(X, \overline {\mathbb Q}_\ell )\) be a complex. An action of \(c\) on \(K\) is by...

To any bounded family of \(\mathbb F_\ell\)-linear representations of the etale fundamental of a curve \(X\) one can associate families of abstract modular curves which, in this setting, generalize the `usual' modular curves with level \(\ell\)...

The \(p\)-adic local Langlands correspondence is well understood for \(\mathrm{GL}_2(\mathbb Q_p)\), but appears much more complicated when considering \(\mathrm{GL}_n(F)\), where either \(n>2\) or \(F\) is a finite extension of \(\mathbb Q_p\). I...

Let \(R={\cal O}_{{\bf C},0}\) be the ring of power series convergent in a neighborhood of zero in the complex plane. Every scheme \(\cal X\) of finite type over \(R\) defines a complex analytic space \({\cal X}^h\) over an open disc \(D\) of small...

It is classical that an element of the class group of a real quadratic field corresponds to a closed geodesic on the modular surface, but not every closed geodesic arises this way; we call those that do "fundamental." Given a fixed compact subset W...

The hyperbolic Ax Lindemann conjecture is a functional transcendental statement which describes the closure of "algebraic flows" on Shimura varieties. We will describe the proof of this conjecture and its consequences for the André-Oort conjecture...

An Euler system is a family of cohomology classes that satisfy some compatibility condition under the corestriction map. Kato constructed an Euler system for a modular form over the cyclotomic extensions of \(\mathbb{Q}\). I will explain a recent...

We will introduce a refinement of the `GPY sieve method' for studying small gaps between primes. This refinement will allow us to show that \(\liminf_n(p_{n+m}-p_n) \infty\) for any integer \(m\), and so there are infinitely many bounded length...

In this talk I will explain some results (joint with Vytas Paskunas) showing that certain classes of potentially crystalline representations (e.g. in the case of two-dimensional representations: crystabelline potentially Barsotti--Tate...

By studying the variation of motivic path torsors associated to a variety, we show how certain nondensity assertions in Diophantine geometry can be translated to problems concerning K-groups. Then we use some vanishing theorems to obtain concrete...

In his first letter to G. H. Hardy, Ramanujan hinted at a theory of continued fractions. He offered shocking evaluations which Hardy described as: "These formulas defeated me completely...they could only be written down by a mathematician of the...

The work of Green, Tao and Ziegler can be used to prove existence and approximation properties for rational solutions of the Diophantine equations that describe representations of a product of norm forms by a product of linear polynomials. One can...

Reeder and Yu have constructed in a uniform way certain supercuspidal representations of \(p\)-adic groups called "epipelagic representations", using invariant theory studied by Vinberg et al. In the function field case, we will realize these...

Let \[ f(t)=\sum_{N n 2N}a_nn^{-it} \] be a Dirichlet polynomial. We consider the weighted square mean value \[ I=\int_{-\infty}^{\infty}|f(t)|^2\exp\{-\Delta^{-2}(t-T)^2\}\,dt, \] where \(T\) is a large paremeter and \[ \Delta = \frac{T}{\log T}...

In this talk we will explain how to attach Galois representations to certain Hecke eigenforms in the coherent cohomology of good reduction Shimura varieties. As an application we will give a new proof of cases of a recent theorem of Scholze...

Let \(G\) be a connected split reductive \(p\)-adic group. Examples are \(\mathrm{GL}(n,F)\), \(\mathrm{SL}(n, F )\), \(\mathrm{SO}(n, F)\), \(\mathrm{Sp}(2n, F )\), \(\mathrm{PGL}(n, F )\) where \(n\) can be any positive integer and \(F\) can be...

Given an elliptic curve \(E\) over a field \(k\), its \(p\)-torsion \(E[p]\) gives a 2-dimensional representation of the Galois group \(G_k\) over \(\mathbb F_p\). The Frey-Mazur conjecture asserts that for \(k= \mathbb Q\) and \(p > 13\), \(E\) is...

I will discuss conjectures, theorems, and experiments concerning the moments, at the central point, of zeta functions associated to hyperelliptic curves over finite fields of odd characteristic. Let \(q\) be an odd prime power, and \(H_{d,q}\)...

We will describe a new strategy to prove the plus-minus main conjecture for elliptic curves having good supersingular reduction at \(p\). It makes use of an ongoing work of Kings-Loeffler-Zerbes on explicit reciprocity laws for Beilinson-Flach...

We consider a Rankin-Selberg integral representation of a cuspidal (not necessarily generic) representation of the exceptional group \(G_2\). Although the integral unfolds with a non-unique model, it turns out to be Eulerian and represents the...

We construct a new Euler system from a collection of special 1-cycles on certain Shimura 3-folds associated to \(U(2,1) \times U(1,1)\) and appearing in the context of the Gan--Gross--Prasad conjectures. We study and compare the action of the Hecke...

The Arthur-Selberg trace formula is a powerful tool in the theory of automorphic forms. Roughly speaking, it expresses the character of the regular representation on the automorphic spectrum in terms of distributions indexed by conjugacy classes...

We give a geometric theory of vector-valued modular forms attached to Weil representations of rank 1 lattices. More specifically, we construct vector bundles over the moduli stack of elliptic curves, whose sections over the complex numbers...

In the first part of the talk we will survey some recent results on representations of finite (simple) groups. In the second part we will discuss applications of these results to various problems in number theory and algebraic geometry.

We formulate a conjectural identity relating the Fourier--Jacobi periods on unitary groups and the central value of certain Rankin--Selberg \(L\)-functions. This refines the Gan--Gross--Prasad conjecture. We give some examples supporting this...

One of the major themes of the analytic theory of automorphic forms is the connection between equidistribution and subconvexity. An early example of this is the famous result of Duke showing the equidistribution of Heegner points on the modular...

We prove a level raising mod $p = 2$ theorem for elliptic curves over $\mathbb Q$, generalizing theorems of Ribet and Diamond-Taylor. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families. We will begin by...

In 2008, Zeev Dvir gave a surprisingly short proof of the Kakeya conjecture over finite fields: a finite subset of $F_q^n$ containing a line in every direction has cardinality at least $c_n q^n$. The "polynomial method" introduced by Dvir has led to...

The Bloch-Kato Conjecture, which generalizes the B-SD Conjecture to higher dimensional varieties, predicts a relation between certain Selmer group and L-function. The famous works of Gross-Zagier and Kolyvagin give results for elliptic curves when...

The endoscopy theory provides a large class of examples of Langlands functoriality, and it also plays an important role in the classification of automorphic forms. The central part of this theory are some conjectural identities of Harish-Chandra...

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.