Organizing Committee
 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
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.
Confirmed Speakers & Participants
Talks will be presented virtually or inperson as indicated in the schedule below.
 Speaker
 Poster Presenter
 Attendee
 Virtual Attendee

Jeff Achter
Colorado State University

Ricardo Acuna
Washington University in St. Louis

Ivan Aidun
University of Wisconsin–Madison

Omid Amini
Ecole Polytechnique

Jennifer Balakrishnan
Boston University

Alexander Betts
Harvard University

Tejasi Bhatnagar
University of Wisconsin Madison

Jerson Caro
Boston University

Shiva Chidambaram
Massachusetts Institute of Technology

Daniel Corey
University of Nevada, Las Vegas

Alejandro De Las Penas Castano
University of Virginia

Haohua Deng
Duke University

Anna Dietrich
Brown University

Jordan Ellenberg
University of Wisconsin

Payman Eskandari
The University of Winnepeg

Sam Freedman
Brown University

Yu Fu
Caltech

Evangelia Gazaki
University of Virginia

Asvin Gothandaraman
Hebrew University of Mathematics, Jerusalem

Richard Hain
Duke University

Sachi Hashimoto
Brown University

Amanda Hernandez
Brown University

Liqiang Huang
Boston University

Thomas Jaklitsch
University of Virginia

Eric Katz
The Ohio State University

Enis Kaya
KU Leuven

Matt Kerr
Washington University in St. Louis

Jef Laga
University of Cambridge

Aaron Landesman
Harvard University

Joshua Lehman
University of Notre Dame

Wanlin Li
Washington University in St. Louis

David Lilienfeldt
Leiden University

Jonathan Love
McGill University

Kaiwen Lu
Brown University

Hao Peng
MIT

Bjorn Poonen
MIT

Rachel Pries
Colorado State University

Congling Qiu
MIT

Caelan Ritter
University of Washington

Nick Salter
University of Notre Dame

Soumya Sankar
Utrecht University

Chad Schoen
Duke University

Ari Shnidman
Hebrew University of Jerusalem

Farbod Shokrieh
University of Washington

Joseph Silverman
Brown University

Jae Hyung Sim
Boston University

Padmavathi Srinivasan
Boston University

Vijay Srinivasan
MIT

Bena Tshishiku
Brown University

Isabel Vogt
Brown University

Boya Wen
University of Wisconsin  Madison

Michael Wills
University of Virginia

Chenxi Wu
university of wisconsin at madison

Ziquan Yang
University of Wisconsin Madison

Mohao Yi
Washington University in St. Louis

Yuri Zarhin
Pennsylvania State University

Robin Zhang
Massachusetts Institute of Technology

Wei Zhang
MIT

Ilia Zharkov
Kansas State University

Xinyu Zhou
Boston University

Eric Zhu
Brown University
Workshop Schedule
Monday, May 13, 2024

8:50  9:00 am EDTWelcome11th Floor Lecture Hall
 Session Chair
 Brendan Hassett, ICERM/Brown University

9:00  9:45 am EDTIntro11th Floor Lecture Hall
 Speaker
 Jordan Ellenberg, University of Wisconsin
 Session Chair
 Daniel Litt, University of Toronto

10:00  10:30 am EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTHeights of Ceresa cycles11th 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 Lfunctions.

11:30  11:40 am EDTCanonical curves and Griffiths infinitesimal invariantLightning 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 EDTA graph invariant for the tropical Ceresa cycleLightning 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 CeresaZharkov invariant of Corey and Li.

11:50 am  12:00 pm EDTZerocycles on K3 surfaces over local fieldsLightning 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 zerocycles 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 EDTClarifying Yan Zhou's example of a cluster variety with a disconnected mutation graphLightning 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 EDTDedekindRademacher Cocycle and Explicit Class Field TheoryLightning 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 padic analogue of CM theory which has been computationally demonstrated to generate BrumerStark units and StarkHeegner points over real quadratic fields. In particular, recent work of Darmon, Pozzi, and Vonk proved that the DedekindRademacher (DR) cocycle in particular generates GrossStark units at real quadratic points in the padic upperhalf 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 EDTLightning Talk DiscussionsWorking Lunch  11th Floor Lecture Hall

2:00  2:45 pm EDTCeresa cycles of bielliptic Picard curves11th 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 jinvariant 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 EDTCoffee Break11th Floor Collaborative Space

3:30  4:15 pm EDTJumps in the height of the Ceresa cycle11th 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 EDTReception11th Floor Collaborative Space
Tuesday, May 14, 2024

9:00  9:45 am EDTCeresa Cycles on Modular Curves and Quadratic Chabauty11th 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 padic height and the local padic 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 EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTNormal functions and Ceresalike cycles11th Floor Lecture Hall
 Speaker
 Matt Kerr, Washington University in St. Louis
 Session Chair
 Wanlin Li, Washington University in St. Louis
Abstract
I’ll discuss limits and differential invariants of normal functions: how they relate to the Ceresa cycle on M_3; and how they predict families of cycles (some known, some unknown) over other moduli spaces. The main computation is a classification of infinitesimal normal functions over locally symmetric varieties (begun with R. Keast, and recently extended with X. Cheng and W. Li). I will discuss its implications for cycles on abelian varieties and for special subvarieties in the Torelli locus.

11:25  11:30 am EDTGroup Photo (Immediately After Talk)11th Floor Lecture Hall

11:30 am  2:00 pm EDTLunch/Free Time

2:00  2:45 pm EDTTropical AbelJacobi theory11th Floor Lecture Hall
 Virtual 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 AbelJacobi maps in the tropical setting. I will discuss some applications, in particular to the Ceresa cycle.

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space

3:30  4:15 pm EDTOpen Problems SessionProblem Session  11th Floor Lecture Hall
Wednesday, May 15, 2024

9:00  9:45 am EDTHyperelliptic curves mapping to abelian surfaces and applications to Beilinson's conjecture for 0cycles11th Floor Lecture Hall
 Speaker
 Evangelia Gazaki, University of Virginia
 Session Chair
 Congling Qiu, MIT
Abstract
The Chow group of zerocycles 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 AbelJacobi map, which gives an isomorphism between the degree zero elements of the Picard group and the kpoints of the Jacobian variety of X. In higher dimensions however the situation is much more chaotic, as the AbelJacobi 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 AbelJacobi 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 EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTArchimedean heights of generalized Heegner cycles11th 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 RankinSelberg Lfunction 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 EDTCeresa cycles of Fermat curves11th 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 Hodgetheoretic 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 ladic 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 EDTWork/Free Time

3:00  3:30 pm EDTCoffee Break11th Floor Collaborative Space

3:30  5:00 pm EDTWork/Free Time
Thursday, May 16, 2024

10:00  10:45 am EDTThe rank of the normal function of the Ceresa cycle11th 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 3g3 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 ShouWu Zhang on the Arakelov theory of moduli spaces of curves. I understand that Gao has given an independent proof of the rank result using AxSchanuel.

11:00  11:30 am EDTCoffee Break11th Floor Collaborative Space

11:30 am  12:15 pm EDTTropical iterated integrals and a unipotent Torelli theorem11th 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 PicardLefschetz 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 nonAbelian 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 EDTStrategies for collaborating across disciplines in pure mathWorking Lunch  11th Floor Collaborative Space

2:00  2:45 pm EDTGeometric cycles in locally symmetric manifolds11th 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 EDTCoffee Break11th Floor Collaborative Space

3:30  4:15 pm EDTBrainstorming StategiesOpen Discussion  11th Floor Lecture Hall
Friday, May 17, 2024

9:00  9:45 am EDTVariation of padic Ceresa classes11th 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 padic rationals, then the ladic etale Ceresa class of X is always trivial when l is different from p, for weight reasons. The padic Ceresa class, by contrast, contains much more information about X, and might reasonably be expected to be nontrivial for a suitably generic X. In this talk, I will describe some recently initiated work with Wanlin Li, where we show such a generic nontriviality result.

10:00  10:30 am EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTC IS NOT EQUIVALENT TO C IN ITS JACOBIAN: A TROPICAL POINT OF VIEW11th Floor Lecture Hall
 Speaker
 Ilia Zharkov, Kansas State University
 Session Chair
 Daniel Corey, University of Nevada, Las Vegas
Abstract
We show that the AbelJacobi 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 EDTCeresa cycles of genus 3 curves with automorphisms11th 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 EDTLunch/Free Time

2:00  3:00 pm EDTRefining open problems generated, and Identifying which are ready to be addressedClosing 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 EDTCoffee Break11th Floor Collaborative Space

3:30  5:00 pm EDTOpen Discussion 11th Floor Collaborative Space
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