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

Wanlin Li

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.

Oct 7

Ziquan Yang

Isogenies between K3 surfaces and the Hecke orbit conjecture

It is well known that quasi-polarized K3 surfaces can be parametrized by orthogonal Shimura varieties, so that we can define Hecke actions on the moduli of K3's. However, what does these Hecke actions mean geometrically for the K3's? In this talk, I will describe an isogeny theory for K3 surfaces which answer this question. Then I will explain that, the Hecke orbit conjecture for orthogonal Shimura varieties implies a Neron-Ogg-Shafarevich criterion for K3's. Finally, I describe the ongoing work on the Hecke orbit conjecture, which is joint with Ananth, Ruofan, and Yunqing.

Oct 14

Asvin Gothandaraman

The variation of the characteristic polynomial in l-adic towers

We study how the characteristic polynomial varies in towers of curves such as . When this tower corresponds to the Fermat curves, we show that the characteristic polynomials are all determined by a finite amount of data and follow a very regular pattern while in the general case, we show that the characteristic polynomials satisfy a l-adic congruence as . We prove this result by proving a more general result about non commutative Iwasawa modules that might be of independent interest.

Oct 21

Chi-Yun Hsu

Partial classicality of Hilbert modular forms

Modular forms are global sections of certain line bundles on the modular curve, while p-adic overconvergent modular forms are defined only over a strict neighborhood of the ordinary locus. The philosophy of classicality theorems is that when the p-adic valuation of Up-eigenvalue is small compared to the weight (called a small slope condition), an overconvergent Up eigenform is automatically classical, namely it can be extended to the whole modular curve. In the case of Hilbert modular forms, there are the partially classical forms which are defined over a strict neighborhood of a “partially ordinary locus”. Modifying Kassaei’s method of analytic continuation, we show that under a weaker small slope condition, an overconvergent form is automatically partially classical.

Oct 28

Yotam Hendel

A number theoretic characterization of (FRS) morphisms: uniform estimates over finite rings of the form $\mathrm{Z}/p^k\mathrm{Z}$

Let $f:X\rightarrow Y$ be a morphism between smooth algebraic varieties defined over the integers. We show its fibers satisfy an extension of the Lang-Weil bounds with respect to finite rings of the form $\mathrm{Z}/p^k\mathrm{Z}$ uniformly in $p$, $k$ and in the base point $y$ if and only if $f$ is flat and its fibers have rational singularities, a property abbreviated as (FRS). This characterization of (FRS) morphisms serves as a joint refinement of two results of Aizenbud and Avni; namely a similar characterization in the case of a single variety, and a characterization of (FRS) morphisms which is non-uniform in the prime $p$. Aizenbud and Avni's argument in the case of a variety breaks in the relative case due to bad behaviour of resolution of singularities in families with respect to taking points over $\mathrm{Z}$ and $\mathrm{Z}/p^k\mathrm{Z}$. To bypass this, we prove a key model theoretic statement on a certain nice class of positive functions (formally non-negative motivic functions), which allows us to efficiently approximate their suprema. Based on arXiv:2103.00282, joint with Raf Cluckers and Itay Glazer.

Nov 4

Junho Peter Whang

Decidability of integral points on some moduli spaces

In this talk, we discuss the Diophantine study of integral points on certain moduli spaces that arise in geometry. First, building on a prior work of ours, we establish the decidability and effective finite generation of integral points on the moduli spaces of SL2-local systems on surfaces with prescribed boundary monodromy. The second family we discuss are the moduli spaces of Stokes matrices (upper triangular unipotent matrices) with prescribed Coxeter invariants. For these, we present recent and ongoing work on the structure of the integral points under a nonlinear braid group action (partly joint with Yu-Wei Fan), and highlight an exceptional connection to the first moduli spaces.

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.)

Nov 18

Yunqing Tang

The unbounded denominators conjecture

The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL_2(Z) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. In this talk, we will give a sketch of the proof of this conjecture based on a new arithmetic algebraization theorem. This is joint work with Frank Calegari and Vesselin Dimitrov.

Dec 2

Jingwei Xiao

A relative trace formula approach to Unitary Friedberg-Jacquet periods

Let G be a reductive group over a number field F and H a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on $G$ over $H(F)\backslash H(A_F)$. They are often related to special values of certain L functions. In this talk, I will explain my work in progress with Wei Zhang that study $(G,H)=(U(2n), U(n)\times U(n))$ and its inner twists using a relative trace formula comparison. We prove the required fundamental lemma using a limit of the Jacquet-Rallis fundamental lemma.

Dec 9

Yingkun Li

Algebraicity of higher Green functions at CM points

By the classical theory of complex multiplication, the modular j-function takes algebraic values at CM points. It is an interesting question to ask about the algebraic nature of other types of automorphic functions at CM points. For the automorphic Green function at integral parameters, Gross and Zagier conjectured in the 1980s that their values at CM points are essentially logarithms of algebraic numbers. In this talk, we will discuss recent progress toward this conjecture and its generalization to the setting of orthogonal Shimura varieties. This is partly joint with Jan Bruinier and Tonghai Yang.

Dec 16

Ananth Shankar

Canonical heights on Shimura varieties and the Andre-Oort conjecture

Let S be a Shimura variety. The Andre-Oort conjecture posits that the Zariski closure of special points must be a sub Shimura subvariety of S. The Andre-Oort conjecture for A_g (the moduli space of principally polarized Abelian varieties) — and therefore its sub Shimura varieties — was proved by Jacob Tsimerman. However, this conjecture was unknown for Shimura varieties without a moduli interpretation. I will describe joint work with Jonathan Pila and Jacob Tsimerman (with an appendix by Esnault-Groechenig) where we prove the Andre Oort conjecture in full generality.