Fall 2025 number theory day: Friday November 7
Speakers: Edray Goins (Pomona College) and Giada Grossi (IAS/CNRS)
Spring 2025 number theory day: Friday February 7
Speakers: Padmavathi Srinivasan (Boston University) and Preston Wake (Michigan State)
Friday, February 7, 2025
Talks will occur in the Cathedral of Learning on the Pitt campus. Note that the rooms of the two talks are different. The first one is in Cathedral room 332 (third floor) and the second one is in Cathedral room 232 (second floor).
Speakers: Padmavathi Srinivasan (Boston University) and Preston Wake (Michigan State)
11:00am-12:00pm: Preston Wake's talk in Cathedral room 332
The Eisenstein ideal at prime-square level
Abstract: In his famous Eisenstein-ideal paper, Barry Mazur studied congruences modulo p between Eisenstein series and cuspforms of prime level N. Among other things, he proved that such congruences exist only if N is congruent to 1 modulo p. The trickiest part of this result is to prove that such congruences do not exist if N is -1 modulo p. I'll talk about recent work with Jackie Lang in which we prove that, for such primes N, Eisenstein congruences do exist at level N^2, that these congruences can be used to construct non-trivial elements of certain class groups, and that the structure of the congruences is much more rigid than in prime-level case.
12:00pm-2:00pm: Lunch break
2:00pm-3:00pm: Padmavathi Srinivasan's talk in Cathedral room 232
A canonical algebraic cycle associated to a curve in its Jacobian
Abstract: The Ceresa cycle is a canonical homologically trivial algebraic cycle associated to a curve in its Jacobian. In his 1983 thesis, Ceresa showed that this cycle is algebraically nontrivial for the generic curve over genus at least 3. Strategies for proving Fermat curves have infinite order Ceresa cycles due to B. Harris, Bloch, Bertolini-Darmon-Prasanna, Eskandari-Murty use a variety of ideas ranging from computation of explicit iterated period integrals, special values of p-adic L functions and points of infinite order on the Jacobian of Fermat curves. We will survey many recent results around the Ceresa cycle, and present ongoing work with Jordan Ellenberg and Adam Logan where we produce many new explicit examples of curves over number fields with infinite order Ceresa cycles.
Fall 2024 number theory day: Friday November 8
Speakers: Wanlin Li (Washington University) and Ari Shnidman (IAS/Hebrew University)
Friday, November 8, 2024
Talks in room 332, Cathedral of Learning (3rd floor), Pitt campus
Speakers: Wanlin Li (Washington University) and Ari Shnidman (IAS/Hebrew University)
2:00pm-3:00pm: Wanlin Li's talk
Non-vanishing of Ceresa and Gross—Kudla—Schoen cycles
Abstract: The Ceresa cycle and the Gross—Kudla—Schoen modified diagonal cycle are algebraic 1-cycles associated to a smooth algebraic curve. They are algebraically trivial for a hyperelliptic curve and non-trivial for a very general complex curve of genus >2. Given an algebraic curve, it is an interesting question to study whether the Ceresa and GKS cycles associated to it are rationally or algebraically trivial. In this talk, I will discuss some methods and tools to study this problem.
3:00pm-3:30pm: Coffee break
3:30pm-4:30pm: Ari Shnidman's talk
Ceresa cycles and the Northcott property
Abstract: Let C+ be a curve of genus at least 2 embedded in its Jacobian and let C- = {-c : c in C+} be the negative embedding. The Ceresa cycle [C+] - [C-] is the simplest example of an algebraic cycle which is trivial in homology but (generally) non-trivial modulo algebraic equivalence. Hyperelliptic curves have trivial Ceresa class, but only recently examples of non-hyperelliptic curves with torsion Ceresa cycle were found. Gao—Zhang recently proved that the Beilinson—Bloch height of the Ceresa cycle satisfies a Northcott property on a certain mysterious open subset Ug of the moduli space Mg. In work with Jef Laga, we show how to find positive dimensional subvarieties of Mg not contained in Ug, for example the family of Picard curves y3 = x4 + ax2 + bx + c in genus 3. The Northcott property fails on the Picard locus and the orders of the torsion Ceresa cycles are unbounded over the complex numbers. On the other hand, we find that there is a uniform bound on the order of a torsion (Picard curve) Ceresa cycle defined over a fixed number field.
Spring 2024 number theory day: Thursday, April 18, 2024
Speakers: Katherine Stange (Colorado-Boulder) and Frank Thorne (South Carolina)
Thursday, April 18, 2024
Talks and coffee in room 7218, Wean Hall, Carnegie-Mellon University campus
Speakers: Katherine Stange (Colorado-Boulder) and Frank Thorne (South Carolina)
10:00am-11:00am: Katherine Stange's talk
Title: The local-global conjecture for Apollonian circle packings is false
Abstract: Primitive integral Apollonian circle packings are fractal arrangements of tangent circles with integer curvatures. The curvatures form an orbit of a 'thin group,' a subgroup of an algebraic group having infinite index in its Zariski closure. The curvatures that appear must fall into one of six or eight residue classes modulo 24. The twenty-year old local-global conjecture states that every sufficiently large integer in one of these residue classes will appear as a curvature in the packing. We prove that this conjecture is false for many packings, by proving that certain quadratic and quartic families are missed. The new obstructions are a property of the thin Apollonian group (and not its Zariski closure), and are a result of quadratic and quartic reciprocity, reminiscent of a Brauer-Manin obstruction. Based on computational evidence, we formulate a new conjecture. This is joint work with Summer Haag, Clyde Kertzer, and James Rickards. Time permitting, I will discuss some new results, joint with Rickards, that extend these phenomena to certain settings in the study of continued fractions.
11:00am-11:30am: Coffee break
11:30am-12:30pm: Frank Thorne's talk
Title: Fourier analysis in arithmetic statistics
Abstract: "Arithmetic statistics" is all about counting arithmetic objects, and it's been the subject of a great deal of recent and ongoing work.
I will give an overview of some of what's been happening in this field recently. Most talks on the subject emphasize the underlying algebraic constructions or the use of the "geometry of numbers", and I will approach the topic from a different angle – and explain how Fourier analysis can be incredibly helpful.
Fall 2023 number theory day: Thursday, September 14, 2023
Speakers: John Bergdall (University of Arkansas) and Ila Varma (University of Toronto)
Thursday, September 14, 2023
Talks and coffee in room 703 (7th floor), Thackeray Hall, University of Pittsburgh campus
1:30pm-2:30pm: Ila Varma's talk
Title: Counting number fields and predicting asymptotics
Abstract: A guiding question in number theory, specifically in arithmetic statistics, is: Fix a degree n and a Galois group G in Sn. How does the count of number fields of degree n whose normal closure has Galois group G grow as their discriminants tend to infinity? In this talk, we will discuss the history of this question and take a closer look at the story in the case that n = 4, i.e. the counts of quartic fields.
2:30pm-3:00pm: Coffee break
3:00pm-4:00pm: John Bergdall's talk
Title: P-adic slope distributions for modular forms
Abstract: The Sato--Tate conjecture predicts statistical properties of point counts on elliptic curves. Its proof over the rationals relies on elliptic curves being modular. In the modular world, there is a wider class of distributions to investigate. For instance, Serre and Conrey--Duke--Farmer established (c. 2000) the distribution of fixed Hecke eigenvalues in modular weights growing to infinity.
This seminar will survey non-Archimedean analogues of these distributions. We first describe the p-adic slope problem for modular eigenforms. Then, we survey the past 20-30 years of research on slopes. One aim is Liu, Truong, Xiao, and Zhao's recent results on the "ghost conjecture" of myself and Pollack. Among its by-products, we learn about p-adic distributions in weight aspects, as previously predicted by Gouvêa.
Organized by Shabnam Akhtari (Penn State), Theresa Anderson (CMU), and Carl Wang-Erickson (Pitt)