NTS ABSTRACTFall2022

Sep 08

Ziquan Yang

The Tate conjecture for h^{2, 0} = 1 varieties over finite fields

The past decade has witnessed a great advancement on the Tate conjecture for varieties with Hodge number h^{2, 0} = 1. Charles, Madapusi-Pera and Maulik completely settled the conjecture for K3 surfaces over finite fields, and Moonen proved the Mumford-Tate (and hence also Tate) conjecture for more or less arbitrary h^{2, 0} = 1 varieties in characteristic 0. In this talk, I will explain that the Tate conjecture is true for mod $p$ reductions of complex projective h^{2, 0} = 1 varieties when p is big enough, under a mild assumption on moduli. By refining this general result, we prove that in characteristic p at least 5 the BSD conjecture holds for a height 1 elliptic curve E over a function field of genus 1, as long as E is subject to the generic condition that all singular fibers in its minimal compacification are irreducible. We also prove the Tate conjecture over finite fields for a class of surfaces of general type and a class of Fano varieties. The overall philosphy is that the connection between the Tate conjecture over finite fields and the Lefschetz (1, 1)-theorem over the complex numbers is very robust for h^{2, 0} = 1 varieties, and works well beyond the hyperkahler world. This is based on joint work with Paul Hamacher and Xiaolei Zhao. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Sep 15

Congling Qiu

Modularity of arithmetic special divisors for unitary Shimura varieties

We construct explicit generating series of arithmetic extensions of Kudla's special divisors on integral models of unitary Shimura varieties over CM fields with arbitrary split levels and prove that they are modular forms valued in the arithmetic Chow groups. This provides a partial solution to Kudla's modularity problem. The main ingredient in our construction is S. Zhang's theory of admissible arithmetic divisors. The main ingredient in the proof is an arithmetic mixed Siegel-Weil formula. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Sep 22

Yousheng Shi

Special cycles on Shimura varieties and theta series

In this talk I will introduce special cycles on Shimura varieties and discuss how to use them to construct geometric and arithmetic theta series. Then I will briefly discuss the connection between these theta series and L functions. In particular I will introduce Kudla-Rapoport conjecture–one key ingredient to make the connection. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Sep 29

Miao (Pam) Gu

A family of period integrals related to triple product L-functions

Let be a number field with ring of adeles . Let be a triple of positive integers and let where the are all cuspidal automorphic representations of . We denote by the corresponding triple product L-function. It is the Langlands L-function defined by the tensor product representation . In this talk I will present a family of Eulerian period integrals, which are holomorphic multiples of the triple product -function in a domain that nontrivially intersects the critical strip. We expect that they satisfy a local multiplicity one statement and a local functional equation. This is joint work with Jayce Getz, Chun-Hsien Hsu and Spencer Leslie. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Oct 13

Raju Krishnamoorthy

Rank 2 local systems and abelian varieties.

Motivated by work of Corlette-Simpson over the complex numbers, we conjecture that all rank 2 \ell-adic local systems with trivial determinant on a smooth variety over a finite field come from families of abelian varieties. We will survey partial results on a p-adic variant of this conjecture. Time permitting, we will provide indications of the proofs, which involve the work of Hironaka and Hartshorne on positivity, the answer to a question of Grothendieck on extending abelian schemes via their p-divisible groups, Drinfeld's first work on the Langlands correspondence for GL_2 over function fields, and the pigeonhole principle with infinitely many pigeons. This is joint with Ambrus Pál. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Oct 20

Tian Wang

Distribution of primes of split reduction of abelian surfaces

Let A defined over the rationals be an absolutely simple abelian surface. We consider the number of primes p less than x, of good reduction for A, such that the reduction of A at p splits (up to isogeny over F_p). It is known that the density of such primes is zero if the endomorphism ring of A is commutative. Under the Generalized Riemann Hypothesis for Dedekind zeta functions, we prove explicit upper bounds on the number of primes such that the reduction of A at p splits. These results improve prior bounds given by J. Achter in 2012 and by D. Zywina in 2018. Under additional conjectures, we get sharper bounds. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Oct 27

Kazuhiro Ito

The Hodge standard conjecture for self-products of K3 surfaces

Grothendieck’s standard conjecture, which is a set of conjectures on algebraic cycles, is wide open. In this talk, I will prove the standard conjecture for the square of a K3 surface in positive characteristic. The new part is the Hodge standard conjecture, which predicts certain positivity of the intersection product. Our main ingredient is the Kuga-Satake period map from the moduli space of K3 surfaces to an orthogonal Shimura variety in mixed characteristic. This is joint work with Tetsushi Ito and Teruhisa Koshikawa. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Nov 03

Zhiyu Zhang

Arithmetic transfer identities on unitary Rapoport—Zink spaces

In this talk, I will formulate and prove some arithmetic transfer identities generalizing the arithmetic fundamental lemma. These identities relate local arithmetic intersection numbers to local twisted orbital integrals. They could be used in the proof of (p-adic) global arithmetic GGP conjecture for unitary groups with mild levels. I will focus on geometric part of the story, in particular how to define suitable derived fixed point locus when the total space is singular. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Nov 10

Qiao He

A proof of the Kudla-Rapoport conjecture for the Kramer model

In this talk, I will talk about a proof of the Kudla-Rapoport conjecture for the Kramer model, which is a precise identity between certain derived intersection number on Rapoport-Zink space and derived local density. After an introduction to the global motivation, I will recall the conjecture and describe the proof strategy. On the geometric side, we completely avoided Tate conjecture and explicit calculation. On the analytic side, we establish a surprisingly simple formula for the primitive derived local density. Combining these two novel ingredients, applying partial Fourier transform proves the conjecture. This is a joint work with Chao Li, Yousheng Shi and Tonghai Yang. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Nov 17

Tristan Phillips

Counting Points on Modular Curves over Global Fields

Let E be an elliptic curve over a number field K. The Mordell-Weil Theorem states that the set of rational points E(K) of E forms a finitely generated abelian group. In particular, we may write E(K)≅ E(K)tors⊕ ℤr, where E(K)tors is a finite torsion group, called the torsion subgroup of E, and r is a non-negative integer, called the rank of E. In this talk I will discuss some results regarding how frequently elliptic curves with a prescribed torsion subgroup occur, and how one can bound the average analytic rank of elliptic curves over number fields. One of the main ideas behind these results is to use methods from Diophantine geometry to count points of bounded height on modular curves. If time permits, I will discuss some ongoing work on a function field analog of some of these results, which has applications to counting Drinfeld modules with prescribed level structures. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Dec 01

Sachi Hashimoto

p-Adic Gross--Zagier and rational points on modular curves

Faltings' theorem states that there are finitely many rational points on a nice projective curve defined over the rationals of genus at least 2. The quadratic Chabauty method makes explicit some cases of Faltings' theorem. Quadratic Chabauty has recent notable success in determining the rational points of some modular curves. In this talk, I will explain how we can leverage information from p-adic Gross--Zagier formulas to give a new quadratic Chabauty method for certain modular curves. Gross--Zagier formulas relate analytic quantities (special values of p-adic L-functions) to invariants of algebraic cycles (the p-adic height and logarithm of Heegner points). By using p-adic Gross--Zagier formulas, this new quadratic Chabauty method makes essential use of modular forms to determine rational points. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)

Dec 08

Padmavathi Srinivasan

A canonical algebraic cycle associated to a curve in its Jacobian

We will talk about the Ceresa class, which is the image under a cycle class map of a canonical homologically trivial algebraic cycle associated to a curve in its Jacobian. In his 1983 thesis, Ceresa showed that the generic curve of genus at least 3 has nonvanishing Ceresa cycle modulo algebraic equivalence. Strategies for proving Fermat curves have infinite order Ceresa cycles due to B. Harris, Bloch, Bertolini-Darmon-Prasanna, Eskandari-Murty use a variety of ideas ranging from computation of explicit iterated period integrals, special values of p-adic L functions and points of infinite order on the Jacobian of Fermat curves. In fact, Bloch's results about the Ceresa cycle of Fermat quartics provided the first concrete evidence for the generalization of the BSD conjecture to the Bloch-Beilinson conjectures. We will survey several recent results about the Ceresa cycle and the Ceresa class. The Ceresa class vanishes for all hyperelliptic curves and was expected to be nonvanishing for non-hyperelliptic curves. We will present joint work with Dean Bisogno, Wanlin Li and Daniel Litt, where we construct a non-hyperelliptic genus 3 quotient of the Fricke--Macbeath curve with torsion Ceresa class, using the character theory of the automorphism group of the curve, namely, PSL2(F8). Time permitting, we will also outline joint work in progress with Jordan Ellenberg, Adam Logan and Akshay Venkatesh for an algorithm for certifying non-triviality of Ceresa classes. Zoom ID: 93014934562 Password: The order of A9 (the alternating group of 9 elements)