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.
![Image for "The Ceresa Cycle in Arithmetic and Geometry"](http://app.icerm.brown.edu/img/488_image.png)
Confirmed Speakers & Participants
Talks will be presented virtually or in-person 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
-
Samuel 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 L-functions.
-
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 Ceresa-Zharkov invariant of Corey and Li.
-
11:50 am - 12:00 pm EDTZero-cycles 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 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 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 EDTDedekind-Rademacher 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 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 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 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 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 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 EDTCoffee Break11th Floor Collaborative Space
-
10:30 - 11:15 am EDTNormal functions and Ceresa-like 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 Abel-Jacobi 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 Abel-Jacobi 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 0-cycles11th 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 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 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 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 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 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 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 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 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 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 p-adic 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 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 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 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 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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