NTS ABSTRACTFall2021

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Sep 9

Anwesh Ray

Arithmetic statistics and diophantine stability for elliptic curves

In 2017, B. Mazur and K. Rubin introduced the notion of diophantine stability for a variety defined over a number field. Given an elliptic curve E defined over the rationals and a prime number p, E is said to be diophantine stable at p if there are abundantly many p-cyclic extensions $L/\mathbb{Q}$ such that $E(L)=E(\mathbb{Q})$. In particular, this means that given any integer $n>0$, there are infinitely many cyclic extensions with Galois group $\mathbb{Z}/p^n\mathbb{Z}$, such that $E(L)=E(\mathbb{Q})$. It follows from more general results of Mazur-Rubin that $E$ is diophantine stable at a positive density set of primes p. In this talk, I will discuss diophantine stability of average for pairs $(E,p)$, where $E$ is a non-CM elliptic curve and $p\geq 11$ is a prime number at which $E$ has good ordinary reduction. First, I will fix the elliptic curve and vary the prime. In this context, diophantine stability is a consequence of certain properties of Selmer groups studied in Iwasawa theory. Statistics for Iwasawa invariants were studied recently in joint work with Debanjana Kundu. As an application, one shows that if the Mordell Weil rank of E is zero, then, $E$ is diophantine stable at $100\%$ of primes $p$. One also shows that standard conjectures (like rank distribution) imply that for any prime $p\geq 11$, a positive density set of elliptic curves (ordered by height) is diophantine stable at $p$. I will also talk about related results for stability and growth of the p-primary part of the Tate-Shafarevich group in cyclic p-extensions.

Sep 16

Qiao He

Kudla-Rapoport conjecture at a ramified prime

Kudla-Rapoport conjecture predicts that there is an identity between the intersection number of special cycles on unitary Rapoport-Zink space and the derivative of local density of certain Hermitian form. However, the original conjecture was only formulated at an unramified prime. In this talk, I will motivate the original conjecture and discuss how to modify it at a ramified prime. Finally, I will sketch how to verify the modified conjecture for n=3. This is a joint work with Yousheng Shi and Tonghai Yang.

Sep 23

Boya Wen

A Gross-Zagier Formula for CM cycles over Shimura Curves

In this talk I will introduce my thesis work to prove a Gross-Zagier formula for CM cycles over Shimura curves. The formula connects the global height pairing of special cycles in Kuga varieties over Shimura curves with the derivatives of the L-functions associated to weight-2k modular forms. As a key original ingredient of the proof, I will introduce some harmonic analysis on local systems over graphs, including an explicit construction of Green's function, which we apply to compute some local intersection numbers.

Sep 30

Boya Wen

Arithmetic local systems

In this talk, I will discuss several problems in arithmetic geometry that are inspired by classical questions in number theory. Each one of them is about studying a certain local system arising from a family of curves. Specifically, these include: Grothendieck's section conjecture on the boundary of \bar{M}_g where we show the non-existence of relative rational points for families of curves with certain degeneration types; Chowla's conjecture over function fields in which we study quadratic characters over F_q(t) whose L-functions vanish at the central point using knowledge of the cohomology of a twisted Hurwitz space; a generalization to Elkies's theorem where we consider the Galois representation of an arithmetic family over Spec Z.

Nov 11

Gunther Cornelissen

Is there a Prime Number Theorem in algebraic groups?

How does the number of primes below a bound, or the number of irreducible polynomials over a finite field of given degree grow with the bound? The Prime Number Theorem, and the Prime Polynomial Theorem, provide answers. We study such questions for finite orbits of endomorphisms of algebraic groups in positive characteristic; this encompasses counting fixed points of such endomorphisms, starting from Steinberg’s work on the cardinality of finite groups of Lie type, and leads to dichotomies for dynamical zeta functions as they occur for topological groups. (Joint work with Jakub Byszewski & Marc Houben.)