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

##### 2^∞-Selmer groups, 2^∞-class groups, and Goldfeld's conjecture

Take E/Q to be an elliptic curve with full rational 2-torsion (satisfying some extra technical assumptions). In this talk, we will show that 100% of the quadratic twists of E have rank less than two, thus proving that the BSD conjecture implies Goldfeld's conjecture in these families. To do this, we will extend Kane's distributional results on the 2-Selmer groups in these families to 2^k-Selmer groups for any k>1. In addition, using the close analogy between 2^k-Selmer groups and 2^{k+1}-class groups, we will prove that the 2^{k+1}-class groups of the quadratic imaginary fields are distributed as predicted by the Cohen-Lenstra heuristics for all k>1.

##### Cohomology of p-adic Stein spaces

I will discuss a comparison theorem that allows us to recover p-adic (pro-)etale cohomology of p-adic Stein spaces with semistable reduction over local rings of mixed characteristic from complexes of differential forms. To illustrate possible applications, I will show how it allows us to compute cohomology of Drinfeld half-space in any dimension and of its coverings in dimension one. This is a joint work with Pierre Colmez and Gabriel Dospinescu.

##### Kloosterman sums and Siegel zeros

Kloosterman sums arise naturally in the study of the distribution of various arithmetic objects in analytic number theory. The 'vertical' Sato-Tate law of Katz describes their distribution over a fixed field F_p, but the equivalent 'horizontal' distribution as the base field varies over primes remains open. We describe work showing cancellation in the sum over primes if there are exceptional Siegel-Landau zeros. This is joint work with Sary Drappeau, relying on a fun blend of ideas from algebraic geometry, the spectral theory of automorphic forms and sieve theory.

##### Unlikely intersections for algebraic curves in positive characteristic

Please follow this link for the abstract: http://www.math.ias.edu/seminars/abstract?event=131079

##### On residues of Eisenstein series - through a cohomological lens

The cohomology of an arithmetic subgroup of a reductive algebraic group defined over a number field is closely related to the theory of automorphic forms. We discuss in which way residues of Eisenstein series contribute non-trivially to the subspace of square-integrable classes in these cohomology groups.

##### The arithmetic intersection conjecture

The Gan-Gross-Prasad conjecture relates the non-vanishing of a special value of the derivative of an L-function to the non-triviality of a certain functional on the Chow group of a Shimura variety. Beyond the one-dimensional case, there is little hope for proving this conjecture. I will explain a variant of this conjecture (suggested by Wei Zhang) which seems more accessible and report on progress on it. This is joint work with B. Smithling and W. Zhang.

##### Elliptic curves of rank two and generalised Kato classes

The generalised Kato classes of Darmon-Rotger arise as p-adic limits of diagonal cycles on triple products of modular curves, and in some cases, they are predicted to have a bearing on the arithmetic of elliptic curves over Q of rank two. In this talk, we will report on a joint work in progress with Ming-Lun Hsieh concerning a special case of the conjectures of Darmon-Rotger.

##### A converse theorem of Gross-Zagier and Kolyvagin: CM case.

Let E be a CM elliptic curves over rationals and p an odd prime ordinary for E. If the Z_p corank of p^\infty Selmer group for E equals one, then we show that the analytic rank of E also equals one.

This is joint work with Ashay Burungale.

##### Nonlinear descent on moduli of local systems

In 1880, Markoff studied a cubic Diophantine equation in three variables now known as the Markoff equation, and observed that its integral solutions satisfy a form of nonlinear descent.

Generalizing this, we consider families of log Calabi-Yau varieties arising as moduli spaces for local systems on topological surfaces, and prove a structure theorem for their integral points using mapping class group dynamics. The result is reminiscent of the finiteness of class numbers for linear arithmetic group actions on homogeneous varieties, and this Diophantine perspective guides us to obtain new extensions of classical results on hyperbolic surfaces along the way.

##### On the notion of genus for division algebras and algebraic groups.

Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) where D' is a central division K-algebra of degree n having the same isomorphism classes of maximal subfields as D. I will review the results on gen(D) obtained in the last several years, in particular the finiteness theorem for gen(D) when K is finitely generated of characteristic not dividing n. I will then discuss how the notion of genus can be extended to arbitrary absolutely almost simple algebraic K-groups using maximal K-tori in place of maximal subfields, and report on some recent progress in this direction.

(Joint work with V. Chernousov and I. Rapinchuk)

##### Joint Equidistribution of CM Points

A celebrated theorem of Duke states that Picard/Galois orbits of CM points on a complex modular curve equidistribute in the limit when the absolute value of the discriminant goes to infinity.

The equidistribution of Picard and Galois orbits of special points in products of modular curves was conjectured by Michel and Venkatesh and as part of the equidistribution strengthening of the André-Oort conjecture. I will explain the proof of a recent theorem making progress towards this conjecture.

##### Locally symmetric spaces: p-adic aspects

p-adic period spaces have been introduced by Rapoport and Zink as a generaliza- tion of Drinfeld upper half spaces and Lubin-Tate spaces. Those are open subsets of a rigid analytic p-adic flag manifold. An approximation of this open subset is the so called weakly admissible locus obtained by removing a profinite set of closed Schubert varieties. I will explain a recent theorem characterizing when the period space coincides with the weakly admissible locus. The proof consists in a thorough study of modifications of G-bundles on the curve. As an application we can compute the p-adic period space of K3 surfaces with supersingular reduction and other period spaces related to the basic locus in some Shimura varieties. This is joint work with Miaofen Chen and Xu Shen.

##### From counting Markoff triples to Apollonian packings; a path via elliptic K3 surfaces and their ample cones

The number of integer Markoff triples below a given bound has a nice asymptotic formula with an exponent of growth of 2. The exponent of growth for the Markoff-Hurwitz equations, on the other hand, is in generalnot an integer. Certain elliptic K3 surfaces can be thought of as smooth generalizations of the Markoff surface. Like the Markoff surface, their group of automorphisms is discrete and infinite. One can therefore investigate the growth rate of a rational point (or curve) on the surface under the action of the group. The exponent of growth is sometimes an integer, and sometimes not. When it is not, it can be thought* of as the Hausdorff dimension of a fractal associated to the ample cone. For some of them, that fractal is the residual set for the Apollonian packing.

(*Partially known, otherwise conjectured.)