Organizing Committee
Abstract

In the 1980s, Ceresa exhibited one of the first naturally occurring examples of an algebraic cycle, the Ceresa cycle, which is in general homologically trivial but algebraically nontrivial. In the last few years, there has been a renewed interest in the Ceresa cycle, and other cycle classes associated to curves over arithmetically interesting fields, and their interactions with analytic, combinatorial, and arithmetic properties of those curves. We hope to capitalize on this momentum to bring together different communities of arithmetic geometers to fully explore explicit computations around the arithmetic and geometry of cycles when these various approaches are systematically combined.

Image for "The Ceresa Cycle in Arithmetic and Geometry"
Image credit: Daniel Litt

Confirmed Speakers & Participants

Talks will be presented virtually or in-person as indicated in the schedule below.

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

Workshop Schedule

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Monday, May 13, 2024
  • 8:50 - 9:00 am EDT
    Welcome
    11th Floor Lecture Hall
    • Session Chair
    • Brendan Hassett, ICERM/Brown University
  • 9:00 - 9:45 am EDT
    Intro
    11th Floor Lecture Hall
    • Speaker
    • Jordan Ellenberg, University of Wisconsin
    • Session Chair
    • Daniel Litt, University of Toronto
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Heights of Ceresa cycles
    11th Floor Lecture Hall
    • Speaker
    • Wei Zhang, MIT
    • Session Chair
    • Daniel Litt, University of Toronto
    Abstract
    For curves over number fields, heights of Ceresa cycles provide interesting arithmetic invariants. I will survey some previous results and then focus on my joint work with Yuan and S. Zhang in the cases of Shimura curves, where automorphic methods could help relate the heights to L-functions.
  • 11:30 am - 2:00 pm EDT
    Lunch/ Lightning Talks
    Working Lunch - 11th Floor Lecture Hall
  • 2:00 - 2:45 pm EDT
    Ceresa cycles of bielliptic Picard curves
    11th Floor Lecture Hall
    • Speaker
    • Ari Shnidman, Hebrew University of Jerusalem
    • Session Chair
    • Padmavathi Srinivasan, Boston University
    Abstract
    I'll describe recent work with Laga where we relate Ceresa cycles of genus three plane curves with an order 6 automorphism to points on the j-invariant 0 elliptic curve. As an application we deduce the existence of infinitely many plane quartic curves with torsion Ceresa cycle.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EDT
    Jumps in the height of the Ceresa cycle
    11th Floor Lecture Hall
    • Speaker
    • Farbod Shokrieh, University of Washington
    • Session Chair
    • Padmavathi Srinivasan, Boston University
    Abstract
    We give an explicit combinatorial formula for the "height jump" of the Ceresa cycle at a given stable curve in terms of the "slope" of the dual graph. We also characterize those stable curves for which the height jump vanishes. (Based on joint work with Robin de Jong.)
  • 4:30 - 6:00 pm EDT
    Reception
    11th Floor Collaborative Space
Tuesday, May 14, 2024
  • 9:00 - 9:45 am EDT
    TBA
    11th Floor Lecture Hall
    • Speaker
    • Boya Wen, University of Wisconsin - Madison
    • Session Chair
    • Wanlin Li, Washington University in St. Louis
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Normal functions and Ceresa-like cycles
    11th Floor Lecture Hall
    • Speaker
    • Matt Kerr, Washington University in St. Louis
    • Session Chair
    • Wanlin Li, Washington University in St. Louis
  • 11:30 am - 2:00 pm EDT
    Lunch/Free Time
  • 2:00 - 2:45 pm EDT
    Tropical Abel-Jacobi theory
    11th Floor Lecture Hall
    • Speaker
    • Omid Amini, Ecole Polytechnique
    • Session Chair
    • Daniel Corey, University of Nevada, Las Vegas
    Abstract
    I will present joint work with Dan Corey and Leonid Monin in which we define and study an analog of the Abel-Jacobi maps in the tropical setting. I will discuss some applications, in particular to the Ceresa cycle.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EDT
    Open Problems Session
    Problem Session - 11th Floor Lecture Hall
Wednesday, May 15, 2024
  • 9:00 - 9:45 am EDT
    Hyperelliptic curves mapping to abelian surfaces and applications to Beilinson's conjecture for 0-cycles
    11th Floor Lecture Hall
    • Speaker
    • Evangelia Gazaki, University of Virginia
    • Session Chair
    • Congling Qiu, MIT
    Abstract
    The Chow group of zero-cycles is a generalization to higher dimensions of the Picard group of a smooth projective curve. When X is a curve over an algebraically closed field k its Picard group can be fully understood by the Abel-Jacobi map, which gives an isomorphism between the degree zero elements of the Picard group and the k-points of the Jacobian variety of X. In higher dimensions however the situation is much more chaotic, as the Abel-Jacobi map in general has a kernel, which over large fields like the complex numbers can be enormous. On the other extreme, a famous conjecture of Beilinson predicts that if X is a smooth projective variety over the algebraic closure of the rational numbers, then this kernel is zero. For a variety X with positive geometric genus this conjecture is very hard to establish. In fact, there are hardly any examples in the literature. In this talk I will discuss joint work with Jonathan Love where we make substantial progress on this conjecture for an abelian surface A. First, we will describe a very large collection of relations in the kernel of the Abel-Jacobi arising from hyperelliptic curves mapping to A. Second, we will show that at least in the special case when A is isogenous to a product of two elliptic curves, such hyperelliptic curves are plentiful. Namely, we will describe a construction that produces for infinitely many values of g countably many hyperelliptic curves of genus g mapping birationally into A.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Archimedean heights of generalized Heegner cycles
    11th Floor Lecture Hall
    • Speaker
    • David Lilienfeldt, Leiden University
    • Session Chair
    • Congling Qiu, MIT
    Abstract
    In the 1980s, Gross and Zagier famously proved a formula equating, on the one hand, the central value of the first derivative of the Rankin-Selberg L-function of a weight 2 eigenform with the theta series of a class group character of an imaginary quadratic field, and on the other hand, the height of a Heegner point on the corresponding modular curve. Two important generalizations present themselves: to allow eigenforms of higher weight, and to further allow Hecke characters of infinite order. The former generalization is due to S. Zhang. The latter one is the subject of this talk and requires the calculation of the archimedean heights of generalized Heegner cycles. These cycles were first introduced by Bertolini, Darmon, and Prasanna, and are relevant to the study of Chow groups of Jacobians with CM. This is joint work with Ari Shnidman.
  • 11:30 am - 12:15 pm EDT
    Ceresa cycles of Fermat curves
    11th Floor Lecture Hall
    • Speaker
    • Payman Eskandari, The University of Winnepeg
    • Session Chair
    • Congling Qiu, MIT
    Abstract
    The study of Ceresa cycles of Fermat curves has a rich history, going back to Bruno Harris’ fundamental work in the early 80s, where he showed via a Hodge-theoretic argument that the Ceresa cycle of the Fermat curve F(4) of degree 4 is algebraically nontrivial, thereby giving the first explicit example of an algebraic cycle that is homologically trivial but algebraically nontrivial. Soon after, Bloch used an l-adic argument to show that the Ceresa cycle of F(4) is, in fact, of infinite order modulo algebraic equivalence. Since then, Harris’ and Bloch’s approaches have been adapted to other Fermat curves (in particular, by Otsubo, Tadokoro, and Kimura), giving rise to many interesting results. However, despite these efforts, until very recently the nontriviality of Ceresa cycles of Fermat curves modulo rational equivalence (let alone, algebraic equivalence) was not known unconditionally for most prime degrees. The goal of this talk is to discuss some recent developments in this direction. The talk is based on a joint work with Kumar Murty.
  • 12:30 - 3:00 pm EDT
    Work/Free Time
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 5:00 pm EDT
    Work/Free Time
Thursday, May 16, 2024
  • 10:00 - 10:45 am EDT
    The rank of the normal function of the Ceresa cycle
    11th Floor Lecture Hall
    • Virtual Speaker
    • Richard Hain, Duke University
    • Session Chair
    • Jordan Ellenberg, University of Wisconsin
    Abstract
    The goal of this talk is to explain what the rank of a normal function is and to sketch a proof that the rank of the normal function of the genus g Ceresa cycle is 3g-3 provided g > 2. I will review the basics of normal functions and then sketch a proof of the result. The motivation comes from work of Ziyang Gao and Shou-Wu Zhang on the Arakelov theory of moduli spaces of curves. I understand that Gao has given an independent proof of the rank result using Ax-Schanuel.
  • 11:00 - 11:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 11:30 am - 12:15 pm EDT
    Tropical iterated integrals and a unipotent Torelli theorem
    11th Floor Lecture Hall
    • Speaker
    • Eric Katz, The Ohio State University
    • Session Chair
    • Jordan Ellenberg, University of Wisconsin
    Abstract
    The cycle pairing on graphs takes a pair of cycles to their oriented intersection. While purely combinatorial, it arises in Picard-Lefschetz theory as a way of studying monodromy of families of algebraic curves, variations of Hodge structures, and asymptotics of period integrals. The cycle pairing, once properly packaged, determines a graph up to two moves by the graph Torelli theorem of Caporaso and Viviani. In this talk, we discuss tropical iterated integrals, a mildly non-Abelian extension of the cycle pairing. We relate them to asymptotics of iterated integrals and monodromy on the fundamental group. We discuss the obstructions to a more precise unipotent Torelli theorem. This is joint work with Raymond Cheng.
  • 12:30 - 2:00 pm EDT
    Strategies for collaborating across disciplines in pure math
    Working Lunch - 11th Floor Collaborative Space
  • 2:00 - 2:45 pm EDT
    Geometric cycles in locally symmetric manifolds
    11th Floor Lecture Hall
    • Speaker
    • Bena Tshishiku, Brown University
    • Session Chair
    • Wanlin Li, Washington University in St. Louis
    Abstract
    Geometric cycles are totally geodesic immersed submanifolds in a locally symmetric manifold. We discuss what is known about the contribution of these cycles to homology.
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 4:15 pm EDT
    Brainstorming Stategies
    Open Discussion - 11th Floor Lecture Hall
Friday, May 17, 2024
  • 9:00 - 9:45 am EDT
    Variation of p-adic Ceresa classes
    11th Floor Lecture Hall
    • Speaker
    • Alexander Betts, Harvard University
    • Session Chair
    • Daniel Corey, University of Nevada, Las Vegas
    Abstract
    If X is a curve over the p-adic rationals, then the l-adic etale Ceresa class of X is always trivial when l is different from p, for weight reasons. The p-adic Ceresa class, by contrast, contains much more information about X, and might reasonably be expected to be non-trivial for a suitably generic X. In this talk, I will describe some recently initiated work with Wanlin Li, where we show such a generic non-triviality result.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    C IS NOT EQUIVALENT TO -C IN ITS JACOBIAN: A TROPICAL POINT OF VIEW
    11th Floor Lecture Hall
    • Speaker
    • Ilia Zharkov, Kansas State University
    • Session Chair
    • Daniel Corey, University of Nevada, Las Vegas
    Abstract
    We show that the Abel-Jacobi image of a tropical curve C in its Jacobian J(C) is not algebraically equivalent to its reflection by using a simple calculation in tropical homology.
  • 11:30 am - 12:15 pm EDT
    Ceresa cycles of genus 3 curves with automorphisms
    11th Floor Lecture Hall
    • Speaker
    • Jef Laga, University of Cambridge
    • Session Chair
    • Daniel Corey, University of Nevada, Las Vegas
    Abstract
    Consider the locus of the moduli space of genus 3 curves where the Ceresa cycle is torsion (modulo rational or algebraic equivalence). This locus is a countable union of proper closed algebraic subvarieties and contains the hyperelliptic locus, but little is known beyond this. I will report on joint work with Ari Shnidman, where we study this locus for curves with extra automorphisms, focusing on Picard curves.
  • 12:30 - 2:00 pm EDT
    Lunch/Free Time
  • 2:00 - 3:00 pm EDT
    Refining open problems generated, and Identifying which are ready to be addressed
    Closing Remarks - 11th Floor Lecture Hall
    • Daniel Corey, University of Nevada, Las Vegas
    • Jordan Ellenberg, University of Wisconsin
    • Wanlin Li, Washington University in St. Louis
    • Daniel Litt, University of Toronto
    • Congling Qiu, MIT
    • Padmavathi Srinivasan, Boston University
  • 3:00 - 3:30 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 3:30 - 5:00 pm EDT
    Open Discussion
    - 11th Floor Collaborative Space

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