# Seminars & Events for Princeton University/IAS Number Theory Seminar

##### The Landau-Siegel zero and spacing of zeros of L-functions

Let χ be a primitive real character. We first establish a relationship between the existence of the Landau-Siegel zero of L(s,χ) and the distribution of zeros of the Dirichlet L-function L(s,ψ), with ψ belonging to a set Ψ of primitive characters, in a region Ω. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Pairs of p-adic L-functions for elliptic curves at supersingular primes

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 introduce ***pairs*** of objects that that are as simple as their ordinary counterparts. These pairs of objects work in tandem to shed some light on the nature of ranks of elliptic curves and the size of Sha along cyclotomic Z_p extensions.

##### Nearby cycles and local convolution

**PLEASE NOTE SPECIAL TIME AND LOCATION. ** I will explain how to use Deligne's theory of nearby cycles over general bases to prove Thom-Sebastiani type theorems.

##### G-valued flat deformations and local models

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 Kisin’s resolution of the flat deformation ring at l = p and describe conceptually the importance of local models of Shimura varieties in analyzing its geometry. Finally, I will address how to generalize these results from GL_n to a general reductive group G. If time permits, I will describe briefly the role that recent advances in p-adic Hodge theory and local models of Shimura varieties play in this situation.

##### The local Gan-Gross-Prasad conjecture for tempered representations of unitary groups

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 $m(\pi,\sigma)=dim\; Hom_{U(V_n)}(\pi,\sigma)$. This multiplicity is always less or equal to $1$ and the Gan-Gross-Prasad conjecture predicts for which pairs of representations we get multiplicity $1$. Their predictions are based on the conjectural Langlands correspondence. In this talk, I will explain a proof of the Gan-Gross-Prasad conjecture for the so-called tempered representations. This is in straight continuation of Waldspurger's work dealing with special orthogonal groups.

##### Heegner points and a B-SD conjecture

We prove a B-SD conjecture for elliptic curves (for the p^infinity 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 Kolyvagin in 1991 on p-indivisibility of (derived) Heegner points over ring class fields.

##### Independence of l and local terms

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 definition a map $u:c_1^*K\rightarrow c_2^!K$. For such an action one can define for each proper component $Z$ of the fixed point scheme of $c$ a local term $\text{lt}_Z(K, u)\in \overline {\mathbb{Q}}_\ell $. In this talk I will discuss various techniques for studying these local terms and some independence of $\ell $ results for them. I will also discuss consequences for traces of correspondences acting on cohomology

##### Genus of abstract modular curves with level l structure

To any bounded family of \F_l-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 l structure (Y_0(l), Y__1(l), Y(l) etc.). Under mild hypotheses, it is expected that the genus (and even the geometric gonality) of these curves goes to infty with l. I will sketch a purely algebraic proof of the growth of the genus - working in particular in positive characteristic.

##### Patching and p-adic local Langlands

The p-adic local Langlands correspondence is well understood for GL_2(Q_p), but appears much more complicated when considering GL_n(F), where either n>2 or F is a finite extension of Q_p. I will discuss joint work with Matthew Emerton, Toby Gee, David Geraghty, Vytautas Paskunas and Sug Woo Shin, in which we approach the p-adic local Langlands correspondence for GL_n(F) using global methods. The key ingredient is Taylor-Wiles-Kisin patching of completed cohomology. This allows us to prove many new cases of the Breuil-Schneider conjecture.

##### Complex analytic vanishing cycles for formal schemes

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 radius with center at zero. PLEASE CLICK ON SEMINAR TITLE FOR COMPLETE ABSTRACT.

##### Low-lying Fundamental Geodesics

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 of (the unit tangent bundle of) the modular surface, we say a closed geodesic is "low-lying" if it is contained in W; in particular, it does not enter "high" into the cusp. In joint work with Bourgain, we exhibit a region W which contains infinitely many fundamental geodesics, answering a question of Einsiedler-Lindenstrauss-Michel-Venkatesh.

##### The hyperbolic Ax-Lindemann conjecture

**Please note special day, time and location. **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. This is a joint work with Bruno Klingler and Andrei Yafaev.

##### Remarks on the cohomology of the Lubin-Tate tower

##### Sigel units and Euler systems

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 Q. I will explain a recent joint work with David Loeffler and Sarah Zerbes where we generalize Kato's work to construct an Euler system for the Rankin-Selberg convolution of two modular forms. I will also explain how a similar construction is possible for a modular form over ray class fields of an imaginary quadratic field.

##### Small gaps between primes

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 intervals containing $m$ primes. Moreover, this method also applies to any subset of the primes which are reasonably well-distributed in arithmetic progressions. We will also discuss more recent developments from the Polymath project which improve the numerical bounds on $\liminf_n(p_{n+1}-p_n)$.

##### Density of certain classes of potentially crystalline representations in local and global Galois deformation rings

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 representations, or potentially Barsotti--Tate representations of supercuspidal type) are Zariski dense in local or global Galois deformation space. The arguments combine methods from the p-adic representation theory of p-adic reductive groups with techniques from the p-adic Langlands program, which allow one to relate representation theory of p-adic groups to Galois representations. (Related results have been proved by Hellmann and Schraen.)

##### Eisenstein series of weight 1

Let N >= 3. In this talk, I will sketch a proof that the ring generated by Eisenstein series of weight 1 on the principal congruence subgroup Gamma(N) contains all modular forms in weights 2 and above. This means that the only forms that are not seen by polynomials in these Eisenstein series are cusp forms of weight 1. This result gives rise to a systematic way to produce equations for the modular curve X(N).

##### On a motivic method in Diophantine geometry

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 results.

##### Statistical behavior of eigenforms on quaternion algebras

**Please note special day, time and location. **I will discuss an approach to studying the limiting behavior of automorphic forms on quaternion algebras that succeeds admirably when the levels involved are large squares, and describe some new technical problems that arise in trying to relax this restriction.

##### A framework of Rogers-Ramanujan identities

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 highest class. They must be true because no one would have the imagination to invent them." -G. H. Hardy Ramanujan had a secret device, two power series identities which were independently discovered previously by L. J. Rogers. The two Rogers-Ramanujan identities are now ubiquitous in mathematics. It turns out that these identities and Ramanujan's handful of evaluations are hints of a much larger theory.