The Chicago Number Theory Day 2020 was a virtual number theory conference held on Saturday, June 20, 2020 via Zoom.
All times are in Central Daylight Time (Chicago)
09:50 — 10:00 Welcome
10:00 — 10:50 David Zureick-Brown [slides]
11:00 — 12:30 Lightning talks
12:30 — 13:00 Break
13:00 — 13:50 Jennifer Balakrishnan
14:00 — 14:50 John Voight [slides]
David Zureick-Brown — Sporadic points on modular curves [slides]
I'll survey various results about "sporadic" (or "unexpected") points on modular curves, and then focus on recent joint work with Derickx, Etropolski, van Hoeij, and Morrow about torsion on elliptic curves over cubic number fields.
Jennifer Balakrishnan — Quadratic Chabauty over number fields
We describe the extent to which p-adic height pairings can allow us to determine the set of integral or rational points on curves, in the spirit of Kim's nonabelian Chabauty program. In particular, we discuss what aspects of the quadratic Chabauty method can be made practical for certain hyperelliptic curves over number fields. This is joint work with Amnon Besser, Francesca Bianchi, and Steffen Mueller.
John Voight — Archimedean aspects of the Cohen-Lenstra heuristics [slides]
Like rational points on elliptic curves, units in number rings are gems of arithmetic. Refined questions about units remain difficult to answer, often embedded within difficult questions about class groups. For example: how often in a number ring is it that all totally positive units are squares?
Absent theorems, we may still try to predict the answer to these questions. In this talk, we present heuristics (and some theorems!) for signatures of unit groups, inspired by the Cohen-Lenstra heuristics, formulating precise conjectures and providing evidence for them. A key role is played by a lustrous structure of number rings we call the 2-Selmer signature map. This structure clarifies the provenance of reflection theorems, like those due to Leopoldt, Armitage-Frohlich, and Gras.
This is joint work with David S. Dummit and Richard Foote and with Ben Breen, Noam Elkies, and Ila Varma.
Jack Petok — Composite level Galois representations and low genus modular curves [slides]
Nicholas Triantafillou — Restriction of scalars, Chabauty's method, and the S-unit equation [slides]
Shiva Chidambaram — Abelian surfaces with fixed 3-torsion [slides]
Jef Laga — Statistics of 2-Selmer groups of nonhyperelliptic genus 3 curves
Arda H. Demirhan — Counting rational points of toric varieties
Seoyoung Kim — From the Birch and Swinnerton-Dyer conjecture to Nagao's conjecture [slides]
Neelima Borade — Curious squares via elliptic curves [slides]
Vlad A. Matei — Counting curves in projective space with a prescribed number of F_q intersection points
Eric Stubley — Partial Weight One Hilbert Modular Forms [slides]
Kelly Isham — A lower bound for the number of subrings in Z^n [slides]
Sarthak Chimni — Some results on the cotype of subrings of Z[t]/(t^3) and Z[t]/(t^4)
Please don't hesitate to contact the organizers if you have any questions.
Nathan Jones (University of Illinois at Chicago)
Jacob Mayle (University of Illinois at Chicago)