The Ceresa Cycle in Arithmetic and Geometry

Institute for Computational and Experimental Research in Mathematics (ICERM)

https://icerm.brown.edu/topical_workshops/tw-24-ccag
May 13, 2024 - May 17, 2024
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 - 11:40 am EDT
    Canonical curves and Griffiths infinitesimal invariant
    Lightning Talks - 11th Floor Lecture Hall
    • Speaker
    • Haohua Deng, Duke University
    • Session Chair
    • Daniel Litt, University of Toronto
    Abstract
    I will explain the relation between the infinitesimal invariant coming from the Ceresa cycle normal function and the canonical embedding of a curve. This include both known results for genus less than 4 and some new results for higher genus.
  • 11:40 - 11:50 am EDT
    Zero-cycles on K3 surfaces over local fields
    Lightning Talks - 11th Floor Lecture Hall
    • Speaker
    • Caelan Ritter, University of Washington
    • Session Chair
    • Daniel Litt, University of Toronto
    Abstract
    Zharkov defined the tropical Ceresa cycle and proved an algebraic nontriviality result for tropical curves overlying the complete graph on four vertices. Building on his methods, we generalize this result by defining a graph invariant that provides information about the "universal" Ceresa cycle in a family of tropical Jacobians; we show that our invariant being trivial has a forbidden minor characterization, suggesting a close relationship to the Ceresa-Zharkov invariant of Corey and Li.
  • 11:50 am - 12:00 pm EDT
    Zero-cycles on K3 surfaces over local fields
    Lightning Talks - 11th Floor Lecture Hall
    • Speaker
    • Jonathan Love, McGill University
    • Session Chair
    • Daniel Litt, University of Toronto
    Abstract
    For a certain class of K3 surfaces over a finite extension $k$ of $\mathbb{Q}_p$, we show that if $k/\mathbb{Q}_p$ is unramified, then the Chow group of zero-cycles of degree $0$ is a divisible group. On the other hand, we give examples to demonstrate that for ramified extensions $k/\mathbb{Q}_p$, the quotient of the Chow group by its maximal divisible subgroup can be an arbitrarily large finite group. This is joint work with Evangelia Gazaki.
  • 12:00 - 12:10 pm EDT
    Clarifying Yan Zhou's example of a cluster variety with a disconnected mutation graph
    Lightning Talks - 11th Floor Lecture Hall
    • Speaker
    • Ricardo Acuna, Washington University in St. Louis
    • Session Chair
    • Daniel Litt, University of Toronto
    Abstract
    Yan Zhou exhibited an example of a cluster variety in dimension 6 with two inequivalent cluster structures. Her proof of this fact is written in the language of scattering diagrams and broken lines used by Gross, Hacking, Keel and Kontsevitch. At the behest of Alessio Corti, I've rewritten the example using only quiver mutations, recovered the mutations she writes down that give the example. We've checked the mutation is volume preserving. But we haven't yet verified it cannot be factored into a product of standard mutations, that would completely reprove her assertion. However, recovering her map explicitly was an important step, as it was unclear from her work how she had computed the formulas on her paper. The problem is interesting because for cluster surfaces the mutation graph is always connected, and her example is the first example of a disconnected mutation graph in dimension > 2.
  • 12:10 - 12:20 pm EDT
    Dedekind-Rademacher Cocycle and Explicit Class Field Theory
    Lightning Talks - 11th Floor Lecture Hall
    • Speaker
    • Jae Hyung Sim, Boston University
    • Session Chair
    • Daniel Litt, University of Toronto
    Abstract
    Darmon and Vonk's theory of rigid cocycles is a p-adic analogue of CM theory which has been computationally demonstrated to generate Brumer-Stark units and Stark-Heegner points over real quadratic fields. In particular, recent work of Darmon, Pozzi, and Vonk proved that the Dedekind-Rademacher (DR) cocycle in particular generates Gross-Stark units at real quadratic points in the p-adic upper-half plane. In this talk, we will review the construction of the DR cocycle by using modular units in an adelic point of view which will show a direct relationship with the partial modular symbols of Darmon and Dasgupta and provide some unexplained observations from Dasgupta, Kakde, Liu, and Fleischer.
  • 12:30 - 2:00 pm EDT
    Lightning Talk Discussions
    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
    Ceresa Cycles on Modular Curves and Quadratic Chabauty
    11th Floor Lecture Hall
    • Speaker
    • Boya Wen, University of Wisconsin - Madison
    • Session Chair
    • Wanlin Li, Washington University in St. Louis
    Abstract
    Given a prime number p, the quadratic Chabauty function on the collection of Q_p points on a curve X is defined to be the difference between the global p-adic height and the local p-adic height at the prime p. It is an essential ingredient in finding the complete set of rational points on certain curves, including some modular curves. In joint work in progress with Jordan Ellenberg and Sachi Hashimoto, we rewrite the quadratic Chabauty function near a cusp of the modular curve as a series expansion in terms of the Tate parameter q. We also explore the relationship between the coefficient of the q^1 term in this series and the Ceresa cycle on modular curves, which is largely in progress but I’ll share our speculations.
  • 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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