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

##### Iwasawa Main Conjecture for Supersingular Elliptic Curves

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 elements to reduce to another main conjecture of Greenberg type, which can in turn be proved using Eisenstein congruences on the unitary group U(3,1).

##### The standard L-function for G_2: a "new way" integral

We consider a Rankin-Selberg integral representation of a cuspidal (not necessarily generic) representation of the exceptional group G2. Although the integral unfolds with a non-unique model, it turns out to be Eulerian and represents the standard L-function of degree 7. We discuss a general approach to the integrals with non-unique models. The integral can be used to describe the representations of G2 for which the (twisted) L-function has a pole as functorial lifts. This is a joint work with Avner Segal.

##### Euler Systems from Special Cycles on Unitary Shimura Varieties and Arithmetic Applications

We construct a new Euler system from a collection of special 1-cycles on certain Shimura 3-folds associated to U(2,1) x U(1,1) and appearing in the context of the Gan--Gross--Prasad conjectures. We study and compare the action of the Hecke algebra and the Galois group on these cycles via distribution relations and congruence relations obtain adelically using Bruhat--Tits theory for the corresponding buildings. If time permits, we explain some potential arithmetic applications in the context of Selmer groups and the Bloch--Kato conjectures for Galois representations associated to automorphic forms on unitary groups.

##### TBA - Chaudouard

##### An algebro-geometric theory of vector-valued modular forms of half-integral weight

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 correspond to vector-valued modular forms attached to rank 1 lattices. The key idea is to construct vector bundles of Schrodinger representations and line bundles of half-forms over appropriate `metaplectic stacks' and then show that their tensor products descend to the moduli stack of elliptic curves. We discuss extensions to the cusp at infinity and give an algebraic notion of q-expansions of vector-valued modular forms.

##### Representations of finite groups and applications

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.

##### Fourier--Jacobi periods on unitary groups

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 conjecture.

##### Weyl-type hybrid subconvexity bounds for twisted L-functions and Heegner points on shrinking sets

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 surface, a problem that boils down to the subconvexity problem for the quadratic twists of Hecke-Maass L-functions. It is interesting to understand if the Heegner points also equidistribute on finer scales, a question that leads one to seek strong bounds on a large collection of central values. Work of Conrey and Iwaniec from 2000 gives the best-known subconvexity bound for twisted L-functions, but lacks the uniformity required for the more advanced equidistribution problems. I will discuss recent work that resolves these problems..

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##### Level raising mod 2 and arbitrary 2-Selmer ranks

We prove a level raising mod p=2 theorem for elliptic curves over 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 explaining our motivation from W. Zhang's approach to the p-part of the BSD conjecture. Explicit examples will be given to illustrate different phenomena compared to odd p. This is joint work with Bao V. Le Hung.

##### The polynomial method

**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 a wave of activity in applications of algebraic and arithmetic geometry to extremal problems in combinatorial geometry, including a theme semester at IPAM in spring 2014. I'll give a general talk about this line of work, including results of Guth and Katz, Kollar, and some of my own (joint with Oberlin-Tao and Hablicsek) and talk about some open questions and ideas for further progress.**

##### Selmer groups, automorphic periods, and Bloch-Kato Conjecture

**Double-header seminars: actual time to be determined. **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 analytic rank is at most 1. In this talk, we will discuss new results toward the Conjecture for cases of more complicated motives. We will also discuss how automorphic periods show up in the picture, which is a crucial step in the proof.

##### Endoscopy theory for symplectic and orthogonal similitude groups

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 characters between a reductive group and its endoscopic groups. Such identities are established in the real case by Shelstad, but they are still largely unknown in the p-adic case due to our limited knowledge of characters in this case. Arthur uses certain (twisted) stable trace formulas to establish the character identities for symplectic groups and orthogonal groups in the p-adic case. In this talk, I will explain how to extend Arthur's character identities to the similitude groups by using some other cases of the twisted stable trace formula.

##### On the rationality of the logarithmic growth filtration of solutions of $p$-adic differential equations

**Please note special day.** 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[p^{-1}] whose formal solutions in $\mathbb{Q}_p$ 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 Andr\'e, Chiarellotto-Tsuzuki, and Kedlaya.

##### On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups

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 conjecture for odd special orthogonal and metaplectic groups in the generic case, which combined with Arthur's work on the local Langlands correspondence implies the conjecture in full generality. This is joint work with Erez Lapid and Zhengyu Mao.

##### Kottwitz-Rapoport conjecture on crystals with additional structure

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 polygons, there exists a crystal with given polygons as its Hodge polygon and Newton polygon respectively. Kottwitz and Rapoport conjectured a similar statement for crystals with additional structure. This conjecture plays an important role in the study of reduction of Shimura varieties. In this talk, I will explain this conjecture, its relation to the Shimura varieties, and I will discuss some ideas of the proof.

##### Eigencurve over the boundary of the weight space

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 eigencurve: over the boundary annuli of the weight space, the eigencurve breaks up into infinite disjoint union of connected components and the weight map is finite and flat on each component. This was first verified by Buzzard and Kilford by an explicit computation in the case of p = 2 and tame level 1. We will explain a generalization to the definite quaternion case with no restriction on p (except p > 2) or the tame level. This is a joint work with Ruochuan Liu and Daqing Wan, based on an idea of Robert Coleman.

##### Around the Moebius function

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 with K. Matomaki we have shown that the sum of the Moebius function exhibits cancellations in ``almost all intervals'' of increasing length. This goes beyond what was previously known conditionally on the Riemann Hypothesis and allows us to settle a conjecture on correlations of consecutive values of the Liouville function (a close cousin of Moebius function). Our result holds in fact in greater generality. Exploiting this generality we show that between a fixed number of consecutive squares there is always an integer composed of only ``small'' prime factors.

##### Faltings heights of CM abelian varieties

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.

##### F-crystalline representations and Kisin modules

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 general class of totally wildly ramified extensions K _\infty/K so that the functor V↦V|_{G_\infty} is fully-faithfull on the category of crystalline representations. We also establish a new classification of Barsotti-Tate groups via Kisin modules of height 1. This is a joint work with Bryden Cais.

##### Most odd degree hyperelliptic curves have only one rational point

We prove that the probability that a curve of the form y^2 = f(x) over 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.