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

##### Spectral summation formulae and their applications

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.

##### On the Averaged Colmez Conjecture

**PLEASE NOTE ROOM CHANGE FOR THIS DATE ONLY: FINE 224. **The Colmez conjecture expresses the Faltings height of a CM abelian variety in terms of the logarithmic derivatives of certain Artin L-functions. In this talk, I will present an averaged version of the conjecture proved in my joint work with Shou-Wu Zhang. Combining with the recent work of Jacob Tsimerman, the Andre-Oort conjecture for Shimura varieties of abelian type is confirmed.

##### Motivic cohomology actions and the geometry of eigenvarieties

Venkatesh has recently proposed a fascinating conjecture relating motivic cohomology with automorphic forms and the cohomology of arithmetic groups. I'll describe this conjecture, and discuss its connections with the local geometry of eigenvarieties and nonabelian analogues of the Leopoldt conjecture. This is joint work with Jack Thorne.

##### Algebraic solutions of differential equations over the projective line minus three points

The Grothendieck–Katz p-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo p has vanishing p-curvatures for almost all p, has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on the projective line minus three points. In this talk, I will first focus on this case and introduce a p-adic convergence condition, which would hold if the p-curvature is defined and vanishes. Using the algebraicity criteria established by André, Bost, and Chambert-Loir, I will prove a variant of this conjecture for the projective line minus three points, which asserts that if the equation satisfies the above convergence condition for all p, then its monodromy is trivial.

##### Adjoint Selmer groups for polarized automorphic Galois representations

Given the p-adic Galois representation associated to a regular algebraic polarized cuspidal automorphic representation, one naturally obtains a pure weight zero representation called its adjoint representation. Because it has weight zero, a conjecture of Bloch and Kato says that the only de Rham extension of the trivial representation by this adjoint representation is the split extension. We will discuss a proof of this case of their conjecture, under an assumption on the residual representation. This is done by using the Taylor-Wiles patching method, Kisin's technique of analyzing the generic fibre of deformation rings, and a characterization of smooth closed points in the generic fibre of certain local deformation rings.

##### Unlikely Intersections For Two-Parameter Families of Polynomials

Inspired by work of Masser and Zannier for torsion specializations of points on the Legendre elliptic curve, Baker and DeMarco proved that if v,w are two points in C, then there are at most finitely many t in C such that v and w are both preperiodic for the polynomial x^2 + t, unless of course v equals plus or minus w. Here we prove a two-dimensional version of this result, namely that if v, w, and z are distinct complex numbers, then the set of parameters (a,b) such that v,w, and z are all preperiodic under f(x) = x^3 + ax + b cannot be Zariski dense in the affine plane. This represents joint work with Liang-Chung Hsia and Dragos Ghioca.

##### From local class field theory to the curve and vice versa

I will speak about results contained in my article "G-torseurs en théorie de Hodge p-adique" linked to local class field theory. I will in particular explain the computation of the Brauer group of the curve and why its fundamental class is the one from local class field theory.

##### On the Moy-Prasad filtration and supercuspidal representations

Reeder and Yu gave recently a new construction of certain supercuspidal representations of p-adic reductive groups (called epipelagic representations). Their construction relies on the existence of stable vectors in the first Moy-Prasad filtration quotient under the action of a reductive quotient. We will explain these ingredients and present a theorem about the existence of such stable vectors for all primes p. This builds on a result of Reeder and Yu about the existence of stable vectors for large primes and generalizes the paper of the speaker and Romano, which treats the case of an absolutely simple split reductive group. In addition, we will present a general set-up that allows us to compare the Moy-Prasad filtration representations for different primes p.

##### Hasse principle for Kummer varieties

The existence of rational points on the Kummer variety associated to a 2-covering of an abelian variety A over a number field can sometimes be established through the variation of the 2-Selmer group of quadratic twists of A. In the case when the Galois action on A[2] has a large image we prove, under mild additional hypotheses, the Hasse principle for associated Kummer varieties, assuming the finiteness of relevant Shafarevich-Tate groups. This provides further evidence for the conjecture that the Brauer-Manin obstruction controls rational points on K3 surfaces. (Joint work with Yonatan Harpaz)

##### Geometric Deformations of Orthogonal and Symplectic Galois Representations

For a representation of the absolute Galois group of the rationals over a finite field of characteristic p, we would like to know if there exists a lift to characteristic zero with nice properties. In particular, we would like it to be geometric in the sense of the Fontaine-Mazur conjecture: ramified at finitely many primes and potentially semistable at p. For two-dimensional representations, Ramakrishna proved that under technical assumptions, odd representations admit geometric lifts. We generalize this to higher dimensional orthogonal and symplectic representations. The key ingredient is a smooth local deformation condition obtained by analysing unipotent orbits and their centralizers in the relative situation, not just over fields.

##### Several Nonarchimedean Variables, Isolated Periodic Points, and Zhang's Conjecture

**Please note special day, time, and location.** We study dynamical systems in several variables over a complete valued field. If x is a fixed point, we show that in many cases there exist fixed analytic subvarieties through x. These cases include all cases in which x is attracting in some directions and repelling in others, which lets us separate attracting, repelling, and indifferent directions, generalizing results from complex hyperbolic dynamics.

##### Generating series of arithmetic divisors in unitary Shimura varieties

In this talk, I will describe roughly how to define a generating function arithmetic divisors (in Arakelov sense) on a unitary Shimura variety of type (n-1,1). I will then briefly explain why it is modular. If time permits, I will also talk briefly about its application to Gross-Zagier-Zhang type formula and to Colmez conjecture. This is an ongoing joint work with Bruinier, Howard, Kudla and Rapoport.

##### Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits

**Special Number Theory / Ergodic Theory Seminar -- Note special time and place. **

In this talk I will describe some new arithmetic invariants for pairs of torus orbits on inner forms of PGLn and SLn. These invariants allow us to significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank S-arithmetic quotients. An important aspect of our method is that it applies to packets of periodic orbits of maximal tori which are only partially split.

Packets of periodic torus orbits are natural collections of torus orbits coming from a single rational adelic torus and are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer.

##### The first order theory of meromorphic functions

By a result of Julia Robinson, we know that the first order theory of the field of rational numbers is undecidable, and in fact the same holds for any number field. In view of this, it is suggested by analogies studied by Vojta and others that the first order theory of meromorphic functions over a complete algebraically closed field should also be undecidable. Three cases naturally arise: the archimedean case, the non-archimedean case in characteristic zero, and the non-archimedean case of positive characteristic. In this talk I will discuss the main ideas in the proof of undecidability of the third case --the other two remain open. The proof uses Nevanlinna theory, and I will explain what one obtains in the number theoretical side of Vojta's analogy applied to this proof.

##### Modularity and potential modularity theorems in the function field setting

**Please note slight change in time (4:15). ** Let G be a reductive group over a global field of positive characteristic. In a major breakthrough, Vincent Lafforgue has recently shown how to assign a Langlands parameter to a cuspidal automorphic representation of G. The parameter is a homomorphism of the global Galois group into the Langlands L-group $^LG$ of G.

##### Decoupling in harmonic analysis and the Vinogradov mean value theorem

**This is the second Number Theory seminar on this date. Please note the special time.** Based on a new decoupling inequality for curves in R^d, we obtain the essentially optimal form of Vinogradov's mean value theorem in all dimensions (the case d=3 is due to T. Wooley). Various consequences will be mentioned and we will also indicate the main elements in the proof (joint work with C. Demeter and L. Guth).

##### Cycles on the moduli of Shtukas and Taylor coefficients of L-functions

This is joint work with Zhiwei Yun. We prove a generalization of Gross-Zagier formula in the function field setting. Our formula relates self-intersection of certain cycles on the moduli of Shtukas for GL(2) to higher derivatives of L-functions.

##### variance of sums of arithmetic functions over primes in short intervals

**This is a special Analysis/Princeton-IAS Number Theory seminar. **Goldston & Montgomery and Montgomery & Soundararajan have established formulae for the variance of sums of the von Magoldt function over short intervals (i.e. for the variance of the number of primes in these intervals) assuming, respectively, the pair-correlation conjecture and the Hardy-Littlewood conjecture. I will discuss the generalisation of these formulae to other arithmetic functions associated with the Selberg class of L-functions, in the context of both zero statistics and arithmetic correlations. I also hope to discuss the function-field analogues of these generalisations.

##### Statistics of abelian varieties over finite fields

Joint work with Jacob Tsimerman. Let B(g,p) denote the number of isomorphism classes of g-dimensional abelian varieties over the finite field of size p. Let A(g,p) denote the number of isomorphism classes of

principally polarized g dimensional abelian varieties over the finite field of size p. We derive upper bounds for B(g,p) and lower bounds for A(g,p) for p fixed and g increasing. The extremely large gap between the lower

bound for A(g,p) and the upper bound B(g,p) implies some statistically counterintuitive behavior for abelian varieties of large dimension over a fixed finite field.

##### Vanishing Cycles and Bilinear Forms

In joint work with Emmanuel Kowalski and Philippe Michel, we prove two different estimates on sums of coefficients of modular forms - one related to L-functions and another to the level of distribution. A key step in the argument is a careful analysis of vanishing cycles, a tool originally developed by Lefschetz to study the topology of algebraic varieties. We will explain why this is helpful for these problems.