Seminars Sorted by Series

I will report on the construction of p-integral models of various algebraic compactifications of PEL-type Shimura varieties and Kuga families, allowing ramification (including deep levels) at p, with good behaviors over the loci where certain...

The trace formula has been the most powerful and mainstream tool in automorphic forms for proving instances of Langlands functoriality, including character relations. Its generalization, the relative trace formula, has also been used to prove...

In this talk, I will report on an on-going joint project with David Helm and Yichao Tian. Let p be a prime unramified in a totally real field F. The Goren-Oort strata are defined by the vanishing locus of the partial Hasse-invariants; it is an...

A few years ago Ichino-Ikeda formulated a quantitative version of the Gross-Prasad conjecture, modeled after the classical work of Waldspurger. This is a powerful local-to-global principle which is very suitable for analytic and arithmetic...

Let x_1, x_2,... be a sequence of n-tuples of roots of unity and suppose X is a subvariety of the algebraic torus. For a prime number p , Tate and Voloch proved that if the p-adic distance between x_k and X tends to 0 then all but finitely many...

The theorem of the title is that if the L-function L(E,s) of an elliptic curve E over the rationals vanishes to order r=0 or 1 at s=1 then the rank of the group of rational rational points of E equals r and the Tate-Shafarevich group of E is finite...

We will discuss some new automorphy lifting theorems for residually reducible Galois representations, and their application to proving new cases of symmetric power functoriality for elliptic modular forms. This is joint work with Laurent Clozel.

I will describe work with Yingkun Li on some arithmetic properties of the Fourier coefficients of harmonic modular forms of weight one. These are Maass forms of weight one whose eigenvalue under the Laplacian is zero and that are allowed to have...

A continuous representation of a profinite group induces a continuous pseudorepresentation, where a pseudorepresentation is the data of the characteristic polynomial coefficients. We discuss the geometry of the resulting map from the moduli formal...

Let \(\chi\) be a primitive real character. We first establish a relationship between the existence of the Landau-Siegel zero of \(L(s,\chi)\) and the distribution of zeros of the Dirichlet \(L\)-function \(L(s,\psi)\), with \(\psi\) belonging to a...

Iwasawa Theory for elliptic curves/modular forms has been traditionally in better shape at ordinary primes than at supersingular ones. After sketching the ordinary theory, we will indicate what makes the supersingular case more complicated, and then...

I will explain how to use Deligne's theory of nearby cycles over general bases to prove Thom-Sebastiani type theorems.

I will begin with a brief introduction to the deformation theory of Galois representations and its role in modularity lifting. This will motivate the study of local deformation rings and more specifically flat deformation rings. I will then discuss...

Let \(E/F\) be a quadratic extension of \(p\)-adic fields. Let \(V_n\subset V_{n+1}\) be hermitian spaces of dimension \(n\) and \(n+1\) respectively. For \(\sigma\) and \(\pi\) smooth irreducible representations of \(U(V_n)\) and \(U(V_{n+1})\) set...

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...