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Organizing Committee
University of Nevada, Las Vegas
University of Wisconsin
Washington University in St. Louis
University of Toronto
MIT
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 credit: Daniel Litt
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Workshop Schedule
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
A graph invariant for the tropical Ceresa cycle
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