Joint IAS/PU Number Theory

Joint equidistribution of CM points

Ilya Khayutin
Princeton University; Veblen Research Instructor, School of Mathematics
November 21, 2017

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.

On the notion of genus for division algebras and algebraic groups

Andrei Rapinchuk
University of Virginia
November 2, 2017
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 $n$.

Nonlinear descent on moduli of local systems

Junho Peter Whang
Princeton University
October 31, 2017
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.

Elliptic curves of rank two and generalised Kato classes

Francesc Castella
Princeton University
October 24, 2017
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.

\(2^\infty\)-Selmer groups, \(2^\infty\)-class groups, and Goldfeld's conjecture

Alex Smith
Harvard University
September 14, 2017
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 \(k > 1\).

The $p$-curvature conjecture and monodromy about simple closed loops

Ananth Shankar
Harvard University
May 11, 2017
The Grothendieck-Katz $p$-curvature conjecture is an analogue of the Hasse Principle for differential equations. It states that a set of arithmetic differential equations on a variety has finite monodromy if its $p$-curvature vanishes modulo $p$, for almost all primes $p$. We prove that if the variety is a generic curve, then every simple closed loop has finite monodromy.

Congruences between motives and congruences between values of $L$-functions

Olivier Fouquet
Université Paris-Sud
April 13, 2017
If two motives are congruent, is it the case that the special values of their respective $L$-functions are congruent? More precisely, can the formula predicting special values of motivic $L$-functions be interpolated in $p$-adic families of motives? I will explain how the formalism of the Weight-Monodromy filtration for $p$-adic families of Galois representations sheds light on this question (and suggests a perhaps surprising answer).

Basic loci of Shimura varieties

Xuhua He
University of Maryland; von Neumann Fellow, School of Mathematics
April 6, 2017

In mod-$p$ reductions of modular curves, there is a finite set of supersingular points and its open complement corresponding to ordinary elliptic curves. In the study of mod-$p$ reductions of more general Shimura varieties, there is a "Newton stratification" decomposing the reduction into finitely many locally closed subsets, of which exactly one is closed. This closed set is called the basic locus; it recovers the supersingular locus in the classical case of modular curves.

Galois Representations for the general symplectic group

Arno Kret
University of Amsterdam
March 30, 2017

In a recent preprint with Sug Woo Shin (https://arxiv.org/abs/1609.04223) I construct Galois representations corresponding for cohomological cuspidal automorphic representations of general symplectic groups over totally real number fields under the local hypothesis that there is a Steinberg component. In this talk I will explain some parts of this construction that involve the eigenvariety.