## Northwestern Number Theory Seminar Winter and Spring 2016

The seminar takes place on Mondays, 4:00--5:00 PM in Lunt 107.

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Current term

## Schedule of Talks

Click on title (or scroll down) for the abstract.

 January 4 Vivek Pal (Columbia U) The Hasse principle for K3 surfaces January 11 Jeff Achter (Colorado St U) On descending cohomology geometrically January 15 Ana Caraiani (Princeton U) Locally symmetric spaces and torsion classes February 1 Vlad Serban (Northwestern U) On p-adic strengthenings of the Manin-Mumford conjecture February 8 Joel Specter (Northwestern U) Coxeter representations appearing in the cohomology of hyperplane complements February 15 Shuyang Cheng (U Chicago) Nonlinear Fourier-Deligne transforms February 22 Stefan Patrikis (U Utah) Deformations of Galois representations and exceptional monodromy February 29 Jack Shotton (U Chicago) Local deformation rings when l \neq p March 7 Xinwen Zhu (Caltech) Towards a p-adic non-abelian Hodge theory April 18 Kartik Prasanna (U Michigan) Hodge classes on products of quaternionic Shimura varieties April 25 Haoran Wang (Michigan St U) Diagrams and mod p representations May 2 Jared Weinstein (Boston U) The absolute Galois group of Q_p as a geometric fundamental group May 9 Hansheng Diao (Princeton U) Log adic spaces and overconvergent modular forms May 16 Ellen Eischen (U Oregon) p-adic families of Eisenstein series and applications

## Abstracts

• Vivek Pal (Columbia U) The Hasse principle for K3 surfaces

In this talk I will unconditionally show that certain K3 surfaces satisfy the Hasse principle. Our method involves the 2-Selmer groups of simultaneous quadratic twists of two elliptic curves. I will describe the general method and explain the tools used.

• Jeff Achter (Colorado St U) On descending cohomology geometrically

Mazur has drawn attention to the question of determining when the cohomology of a smooth, projective variety over a number field can be modeled by an abelian variety. I will discuss recent work with Casalaina-Martin and Vial which constructs such a "phantom" abelian variety for varieties with maximal geometric convieau. In the special case of cohomology in degree three, we show that the image of the (complex) Abel-Jacobi map admits a distinguished model over the base field, and that an algebraic correspondence realizes this descended intermediate Jacobian as a phantom.

• Ana Caraiani (Princeton U) Locally symmetric spaces and torsion classes

The Langlands program is an intricate network of conjectures, which are meant to connect different areas of mathematics, such as number theory, harmonic analysis and representation theory. One striking consequence of the Langlands program is the Ramanujan conjecture, which is a statement purely within harmonic analysis, about the growth rate of Fourier coefficients of modular forms. It turns out to be intimately connected to the Weil conjectures, a statement about the cohomology of projective, smooth varieties defined over finite fields. I will explain this connection and then move towards a mod p analogue of these ideas. More precisely, I will explain a strategy for understanding torsion occurring in the cohomology of locally symmetric spaces and how to detect which degrees torsion will contribute to. The main theorem is joint work with Peter Scholze and relies on a p-adic version of Hodge theory and on recent developments in p-adic geometry.

• Vlad Serban (Northwestern U) On p-adic strengthenings of the Manin-Mumford conjecture

Let $G$ be an abelian variety or a product of multiplicative groups $\mathbb{G}_m^n$ and let $C$ be an embedded curve. The Manin-Mumford conjecture (a theorem by work of Lang, Raynaud et al.) states that only finitely many torsion points of $G$ can lie on $C$ unless $C$ is in fact a subgroup of $G$. We show how these algebraic statements extend to analytic functions on open $p$-adic unit poly-disks. We use our results to study $p$-adic families of automorphic forms parametrized by weights inside these disks.

• Joel Specter (Northwestern U) Coxeter representations appearing in the cohomology of hyperplane complements

Given a finite Coxeter group W, there exists a unique irreducible, real representation V of W on which W acts as a reflection group. Let X(W) be the complement of the reflection hyperplanes in the complexification of V. Deligne showed that X(W) is a classifying space for the pure Artin-Tits group attached to W. In this talk, I will expand on work of Arnol'd, Brieskorn, Orlick and Solomon, and Lehrer to give an explicit formula for the character of the $W$ on the cohomology of $X(W)$. The proof will use etale cohomology to equate this calculation to a counting problem in elementary number theory over finite fields. This is joint work with Weiyan Chen.

• Shuyang Cheng (U Chicago) Nonlinear Fourier-Deligne transforms

In the approach of Godement-Jacquet to the study of principal L-functions on GL(n), the classical Fourier transform on the vector space M(n) of n-by-n matrices has played a special role, which leads in particular to the local and global functional equations of such L-functions. More generally for automorphic L-functions on GL(n) associated with a representation r of the dual group, there exists a monoid M(r) containing GL(n) which plays the role of M(n), and one expects there to exist nonlinear analogues of Fourier transforms on M(r). Such nonlinear Fourier transforms, together with analogues over finite fields, namely nonlinear Fourier-Deligne transforms, have been constructed by Braverman-Kazhdan. In this talk I will recall the theory of such nonlinear Fourier-Deligne transforms and give a proof for a conjecture of Braverman-Kazhdan which essentially says that nonlinear Fourier-Deligne transforms commute with parabolic induction. This is joint work with B.C. Ngo.

• Stefan Patrikis (U Utah) Deformations of Galois representations and exceptional monodromy

I will explain how to realize the exceptional algebraic groups as algebraic monodromy groups of geometric (in the sense of Fontaine-Mazur) Galois representations, in particular obtaining the first such examples in types F4 and E6. The argument relies on lifting well-chosen mod p representations to characteristic zero, using a generalization (to essentially any reductive group) of a technique developed by Ravi Ramakrishna for type A1.

• Jack Shotton (U Chicago) Local deformation rings when l \neq p

Given a mod p representation of the absolute Galois group of Q_l, consider the universal framed deformation ring R parametrising its lifts. When l and p are distinct I will explain a relation between the mod p geometry of R and the mod p representation theory of GL_n(Z_l), that is parallel to the Breuil-Mézard conjecture in the l = p case. I will give examples and say something about the proof, which uses automorphy lifting techniques.

• Xinwen Zhu (Caltech) Towards a p-adic non-abelian Hodge theory

I will describe a conjectural p-adic non-abelian Hodge theory for smooth rigid analytic varieties over a p-adic field and discuss what we know so far. Then I will discuss some surprising consequences of the (known part of) theory. This is based on a joint work of Ruochuan Liu.

• Kartik Prasanna (U Michigan) Hodge classes on products of quaternionic Shimura varieties

I will discuss the relation between Langlands functoriality and the theory of algebraic cycles in one of the simplest instances of functoriality, namely the Jacquet-Langlands correspondence for Hilbert modular forms. In this case, functoriality gives rise to a family of Tate classes on products of quaternionic Shimura varieties. The Tate conjecture predicts that these classes come from an algebraic cycle, which in turn should give rise to a Hodge class that is compatible with the Tate classes. While we cannot yet prove the Tate conjecture in this context, I will outline an unconditional proof of the existence of such a Hodge class. This is joint work (in progress) with A. Ichino.

• Haoran Wang (Michigan St U) Diagrams and mod p representations

Let $p>2$ be a prime number and $L$ be a finite unramified extension of $\mathbb{Q}_p.$ To every two dimensional 'generic' local Galois representation $\overline{\rho}_p$ of $Gal(\overline{L}/L)$ over $\overline{\mathbb{F}}_p,$ Breuil and Paskunas has associated an (infinite) family of smooth admissible $\overline{\mathbb{F}}_p$-representations of $GL_2(L)$ via the 'diagrams' . When $\overline{\rho}_p$ comes from a global modular mod $p$ Galois representation $\overline{\rho}$, Emerton, Gee and Savitt showed that the $p$ component $\pi_p(\overline{\rho})$ of the mod $p$ automorphic representation associated to $\overline{\rho}$ belongs to Breuil-Paskunas' family. In this talk, I will explain how to deduce the $1+p M_2(O_L)$-invariants of $\pi_p(\overline{\rho})$ from Emerton-Gee-Savitt's work in the case where $\overline{\rho}_p$ is semisimple, hence gives a further restriction on a possible mod $p$ correspondence. This is a work in progress with Yongquan Hu.

• Jared Weinstein (Boston U) The absolute Galois group of Q_p as a geometric fundamental group

We construct an object defined over an algebraically closed field, whose fundamental group equals the absolute Galois group of Q_p. Formally, this object is a quotient of a perfectoid space, and is closely related to the "fundamental curve of p-adic Hodge theory" of Fargues and Fontaine.

• Hansheng Diao (Princeton U) Log adic spaces and overconvergent modular forms

The main objects of the talk are adic spaces with logarithmic structures. We study their Kummer etale and Kummer pro-etale topologies. In particular, we show that log adic spaces are locally "perfectoid". As an application, we establish an oveconvergent Eichler-Shimura morphism connecting modular symbols and overconvergent modular forms.

• Ellen Eischen (U Oregon) p-adic families of Eisenstein series and applications

I will discuss a construction of a p-adic family of Eisenstein series. I will also describe how it feeds into a program to construct p-adic L-functions associated to automorphic forms (in particular, on unitary groups). The latter part is joint with Michael Harris, Jian-Shu Li, and Chris Skinner.