Tag - Algebraic varieties

Bhargav Bhatt: p-adic motives II

7th April, 2023

In the 1960s, Grothendieck dreamt that algebraic varieties can be linearized in a universal way, leading to his philosophy of motives. Subsequent ideas of many mathematicians (especially Beilinson and Deligne) led to a beautiful conjectural framework surrounding the notion of a motive. In the last decade, thanks to the discovery of perfectoid geometry and subsequent developments, some aspects of this framework have also been realized unconditionally in the context of p-adic motives on p-adic varieties. In these lectures, I will survey some of this landscape, with an emphasis on the concrete applications that have guided the theoretical developments.

Bhargav Bhatt: p-adic motives I

7th April, 2023

In the 1960s, Grothendieck dreamt that algebraic varieties can be linearized in a universal way, leading to his philosophy of motives. Subsequent ideas of many mathematicians (especially Beilinson and Deligne) led to a beautiful conjectural framework surrounding the notion of a motive. In the last decade, thanks to the discovery of perfectoid geometry and subsequent developments, some aspects of this framework have also been realized unconditionally in the context of p-adic motives on p-adic varieties. In these lectures, I will survey some of this landscape, with an emphasis on the concrete applications that have guided the theoretical developments.

Wanlin Li: Ordinary and Basic Reductions of Abelian Varieties

16th February, 2023

Given an abelian variety A defined over a number field, a conjecture attributed to Serre states that the set of primes at which A admits ordinary reduction is of positive density. This conjecture had been proved for elliptic curves (Serre, 1977), abelian surfaces (Katz 1982, Sawin 2016) and certain higher dimensional abelian varieties (Pink 1983, Fite 2021, etc). In this talk, we will discuss ideas behind these results and recent progress for abelian varieties with non-trivial endomorphisms, including some cases of A with almost complex multiplication by an abelian CM field, based on joint work with Cantoral-Farfan, Mantovan, Pries, and Tang. Apart from ordinary reduction, we will also discuss the set of primes at which an abelian variety admits basic reduction, generalizing a result of Elkies on the infinitude of supersingular primes for elliptic curves.

Paolo Stellari: Stability conditions in the trivial canonical bundle case: Hilbert schemes of points

15th March, 2022

The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem, especially when the canonical bundle is trivial. In this talk, I will review some results and techniques related to the latter setting. I will specifically concentrate on the case of Hilbert scheme of points on K3 surfaces and (as a work in progress) on generic abelian varieties of any dimension. This is joint work in progress with C. Li, E. Macri' and X. Zhao. 

Aleksander Horawa: Motivic action on coherent cohomology of Hilbert modular varieties

3rd February, 2022

A surprising property of the cohomology of locally symmetric spaces is that Hecke operators can act on multiple cohomological degrees with the same eigenvalues. We will discuss this phenomenon for the coherent cohomology of line bundles on modular curves and, more generally, Hilbert modular varieties. We propose an arithmetic explanation: a hidden degree-shifting action of a certain motivic cohomology group (the Stark unit group). This extends the conjectures of Venkatesh, Prasanna, and Harris to Hilbert modular varieties.

Jacob Tsimerman: Abelian varieties not isogenous to Jacobians

1st December, 2021

Katz and Oort raised the following question: Given an algebraically closed field k, and a positive integer g>3, does there exist an abelian variety over k not isogenous to a Jacobian over k? There has been much progress on this question, with several proofs now existing over ℚ. We discuss recent work with Ananth Shankar, answering this question in the affirmative over 𝔽q(T). Our method introduces new types of local obstructions, and can be used to give another proof over ℚ.

Will Sawin: The Shafarevich Conjecture for Hypersurfaces in Abelian Varieties

18th March, 2021

Faltings proved the statement, previously conjectured by Shafarevich, that there are finitely many abelian varieties of dimension n, defined over a fixed number field, with good reduction outside a fixed finite set of primes, up to isomorphism. In joint work with Brian Lawrence, we prove an analogous finiteness statement for hypersurfaces in a fixed abelian variety with good reduction outside a finite set of primes. I will give a broad introduction to some of the ideas in the proof, which builds on p-adic Hodge theory techniques from work of Lawrence and Venkatesh as well as sheaf convolution in algebraic geometry.

Alexander Smith: Selmer groups and a Cassels-Tate pairing for finite Galois modules

25th February, 2021

I will discuss some new results on the structure of Selmer groups of finite Galois modules over global fields. Tate's definition of the Cassels-Tate pairing can be extended to a pairing on such Selmer groups with little adjustment, and many of the fundamental properties of the Cassels-Tate pairing can be reproved with new methods in this setting. I will also give a general definition of the theta/Mumford group and relate it to the structure of the Cassels-Tate pairing, generalizing work of Poonen and Stoll.

Salim Tayou: Exceptional Jumps of Picard Rank of K3 Surfaces over Number Fields

18th February, 2021

Given a K3 surface X over a number field K, we prove that the set of primes of K where the geometric Picard rank jumps is infinite, assuming that X has everywhere potentially good reduction. This result is formulated in the general framework of GSpin Shimura varieties and I will explain other applications to abelian surfaces. I will also discuss applications to the existence of rational curves on K3 surfaces.

The results in this talk are joint work with Ananth Shankar, Arul Shankar and Yunqing Tang.

Joel Nagloo: Ax-Lindemann-Weierstrass Theorem for Fuchsian automorphic functions

21st January, 2021

Over the last decades, following works around the Pila-Wilkie counting theorem in the context of o-minimality, there has been a surge in interest around functional transcendence results, in part due to their connection with special points conjectures. A prime example is Pila's modular Ax-Lindemann-Weierstrass (ALW) Theorem and its role in his proof of the André-Oort conjecture.

In this talk we will discuss how an entirely new approach, using the model theory of differential fields, can be used to prove the ALW Theorem with derivatives for Fuchsian automorphic functions - a direct generalization of Pila’s ALW theorem. We will also explain how new cases of the André-Pink conjecture can be obtained using this new approach.

This is joint work with G. Casale and J. Freitag.