This is the website for the University of Arizona Algebra and Number Theory Seminar organized by Serin Hong and myself.
We meet on Tuesdays at 2PM (AZ time), either at ENR2 S395 or online via Zoom, unless specified otherwise.
Upcoming talks:
Previous talks:
Tuesday, September 10 2024, 2-3PM
Speaker: Robert Pollack (University of Arizona)
Title: On the distribution of L-invariants of modular forms
Abstract: The distribution of invariants of modular forms has been studied in many contexts. The Sato-Tate conjecture makes a precise prediction on the distribution of normalized Hecke-eigenvalues for modular forms. Here one fixes a form and varies the eigenvalue. One could also fix the eigenvalue and vary the form and still this invariant has a beautifully predictable distribution.
In this talk, we will discuss p-adic variants of these questions and investigate the distribution of the p-adic size of Hecke-eigenvalues leading to Gouvea's conjecture. Further, we will study a more mysterious p-adic invariant of a modular form, namely the L-invariant. We will give an overview of this invariant and ultimately state a conjecture about its p-adic distribution. This work is joint with John Bergdall.
Tuesday, October 15 2024, 2-3PM
Speaker: Linli Shi, University of Connecticut
Title: On higher regulators of Picard modular surfaces
Abstract: The Birch and Swinnerton-Dyer conjecture relates the leading coefficient of the L-function of an elliptic curve at its central critical point to global arithmetic invariants of the elliptic curve. Beilinson’s conjectures generalize the BSD conjecture to formulas for values of motivic L-functions at non-critical points. In this talk, I will relate motivic cohomology classes, with non-trivial coefficients, of Picard modular surfaces to a non-critical value of the motivic L-function of certain automorphic representations of the group GU(2,1).
Tuesday, October 22 2024, 2-3PM
Speaker: Serin Hong, University of Arizona
Title: Evolution of the Newton stratification
Abstract: The notion of Newton stratification originates from Grothendieck's work on the moduli space of abelian varieties. Since its inception, the notion has significantly evolved to find many surprising applications in arithmetic geometry. In this talk, we provide a friendly overview of this notion and discuss some recent developments, with a particular focus on the question of determining all nonempty strata.
Tuesday, October 29 2024, 2-3PM
Speaker: Anna Medvedovsky, University of Arizona
Title: Mod-p modular forms: explicit Galois representations and applications to densities
Abstract: We tell the story of mod-p modular forms, focusing on the example of p = 2 and p = 3 and level 1: the Hecke algebra acting on the space of forms, the Galois representation carried thereby, and explicit matrix realization thereof. An application is studying the *density* of a mod-p modular form, which captures the distribution of its prime Fourier coefficients. The p = 2 case relies on published and unpublished work of Bellaïche, Nicolas, and Serre.
Tuesday, November 5 2024, 2-3PM
Speaker: Ben Savoie, Rice University
Title: Components of the Moduli Stack of Galois Representations
Abstract: The Emerton-Gee stack for GL_2 serves as a moduli space for 2-dimensional representations of the absolute Galois group of K, where K is a finite, unramified extension of Q_p. This stack is of significant interest because it is expected to play the role of the stack of L-parameters in the conjectural categorical p-adic Langlands correspondence for GL_2(K). In this talk, I will present recent joint work with Kalyani Kansal, where we determine which of the irreducible components of the Emerton-Gee stack are smooth. Among the non-smooth components, we also identify those which are normal or Cohen-Macaulay. This allows us to show that the normalization of every component has fairly mild (resolution-rational) singularities. The talk will begin with a review of Galois representations and modular forms, followed by a discussion of key ideas in the construction of the Emerton-Gee stack. Finally, I will describe how our results update expectations about the categorical p-adic Langlands conjecture.
Wednesday, November 13 2024, 2-3PM at MATH 402 (note the unusual time and location)
Speaker: Florian Sprung, Arizona State University
Title: The logarithm matrix Log(a_p) in terms of p-adic digits
Abstract: For a modular form* f of weight two, one can attach a p-adic L-function, which is `good' if p is an ordinary prime, i.e. the p-th Fourier a_p of f is a p-adic unit. `Good' means `Iwasawa function,' or in even simpler terms `coefficients are bounded.' When p is non-ordinary (i.e. a_p is not a p-adic unit), the p-adic L-functions are `bad' -- they have unbounded growth behavior on their coefficients (and note the plural -- there are now two of them).
However, one can factor out the badness. This was done in R. Pollack's PhD thesis when a_p=0, which gave rise to two functions log+ and log-, the `signed logarithms'. The speaker handled the general non-ordinary case via a 2x2- matrix Log(a_p). This matrix is, when a_p=0, essentially a diagonal matrix in which log+ and log- appear.
But where does Log(a_p) come from? We give a simple description in terms of p-adic digits.
*normalized eigenform
Tuesday, November 19 2024, 2-3PM
Speaker: Shaunak Deo, Indian Institute of Science
Title: The Eisenstein ideal of weight $k$ and ranks of Hecke algebras
Abstract: Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We will give a necessary and sufficient condition for the $Z_p$-rank of the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight $k$ and level $\Gamma_0(\ell)$ at the maximal Eisenstein ideal containing $p$ to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We will begin with a brief review of modular forms.
Tuesday, November 26 2024, 2-3PM
Speaker: Chengyang Bao, UCLA
Title: Computing crystalline deformation rings via the Taylor-Wiles-Kisin patching
Abstract: Crystalline deformation rings play an important role in Kisin's proof of the Fontaine-Mazur conjecture for GL2 in most cases. One crucial step in the proof is to prove the Breuil-Mezard conjecture on the Hilbert-Samuel multiplicity of the special fiber of the crystalline deformation ring. In pursuit of formulating a horizontal version of the Breuil-Mezard conjecture, we develop an algorithm to compute arbitrarily close approximations of crystalline deformation rings. Our approach, based on reverse-engineering the Taylor-Wiles-Kisin patching method, aims to provide detailed insights into these rings and their structural properties, at least conjecturally.
Tuesday, December 10 2024, 2-3PM
Speaker: Kimball Martin, University of Oklahoma
Title: Distributions of root numbers and Fourier coefficients of modular forms
Abstract: Root numbers are signs that determine symmetries of L-functions. While asymptotically root numbers are +1 half the time and -1 half the time, there is in fact a bias towards sign +1. Moreover, an unexpected correlation between root numbers and Fourier coefficients of modular forms, termed murmurations, was recently discovered. I will explain these phenomena, as well as analogues for local root numbers.
Tuesday, January 21 2025, 2-3PM
Speaker: Deewang Bhamidipati, UC Santa Cruz
Title: Strata intersections in some unitary Shimura varieties
Abstract: Unitary Shimura varieties are moduli spaces of abelian varieties with a certain extra structure, including a signature condition. An effective way to understand these spaces in positive characteristic is by stratifying them, of which two are of interest: the Ekedah-Oort (EO) stratification, defined using the p-torsion group scheme structure up to isomorphism, and the Newton stratification, defined using the p-divisible group structure up to isogeny. We will see in several concrete examples that these two stratifications are very different, reflecting that these two invariants capture very different attributes of the abelian varieties. In joint work with E. Anne, M. Fox, H. Goodson, S. Groen, and S. Nair, we take a specific stratum in the Newton stratification - the supersingular stratum - and study its intersection with the EO stratification in some low signature cases.
Tuesday, January 28 2025, 2-3PM
Speaker: Kirti Joshi, University of Arizona
Title: Arithmetic Teichmüller Theory of a Number Field I
Abstract: The goal of these three talks is to provide an introduction to the notion of Teichmuller type deformations of a fixed number field. I will begin a brief introduction to Classical Teichmüller Theory (of Riemann surfaces). That such Teichmuller type deformations of a number field exist was suggested by Shinchi Mochizuki in his Inter-Universal Teichmüller Theory and underlies his work on the abc-conjecture. The three talks will focus mainly on my paper (available on the arxiv) Constructions of Arithmetic Teichmüller Spaces II(1/2): Deformations of Number Fields.
Tuesday, February 4 2025, 2-3PM
Speaker: Kirti Joshi, University of Arizona
Title: Arithmetic Teichmüller Theory of a Number Field II
Abstract: The goal of these three talks is to provide an introduction to the notion of Teichmuller type deformations of a fixed number field. I will begin a brief introduction to Classical Teichmüller Theory (of Riemann surfaces). That such Teichmuller type deformations of a number field exist was suggested by Shinchi Mochizuki in his Inter-Universal Teichmüller Theory and underlies his work on the abc-conjecture. The three talks will focus mainly on my paper (available on the arxiv) Constructions of Arithmetic Teichmüller Spaces II(1/2): Deformations of Number Fields.
Tuesday, February 11 2025, 2-3PM
Speaker: Kirti Joshi, University of Arizona
Title: Arithmetic Teichmüller Theory of a Number Field III
Abstract: The goal of these three talks is to provide an introduction to the notion of Teichmuller type deformations of a fixed number field. I will begin a brief introduction to Classical Teichmüller Theory (of Riemann surfaces). That such Teichmuller type deformations of a number field exist was suggested by Shinchi Mochizuki in his Inter-Universal Teichmüller Theory and underlies his work on the abc-conjecture. The three talks will focus mainly on my paper (available on the arxiv) Constructions of Arithmetic Teichmüller Spaces II(1/2): Deformations of Number Fields.
Tuesday, February 18 2025, 2-3PM
Speaker: Jinyue Luo, University of Chicago
Title: Criteria for pseudorepresentations to arise from genuine representations
Abstract: In application to modularity, one often seeks for an R=T theorem, that is, the deformation ring R is isomorphic to a (localized) Hecke algebra T. However, sometimes only the framed deformation ring exists. With the framing variables, it is obviously larger than the Hecke algebra. Pseudorepresentations, which is a generalization of the notion of traces of representations, was invented to get around this issue. We will introduce the notion of pseudorepresentations and discuss the criteria for pseudo representations to arise from genuine representations. Next, we will introduce the algorithm used to explicitly compute usual deformation rings and pseudodeformation rings for finitely presented groups, which leads to the discovery of a counterexample.
Tuesday, March 4 2025, 2-3PM
Speaker: Jaclyn Lang, Temple University
Title: Eisenstein congruences in prime-square level
Abstract: In his celebrated Eisenstein ideal paper, Mazur studied congruences modulo a prime p between Eisenstein series and cusp forms in prime level N. If p is at least 5, he showed that such congruences exist if and only if N is congruent to 1 modulo p. I will discuss recent work with Preston Wake in which we investigate Eisenstein-cuspidal congruences when the level is N^2, where N is a prime congruent to -1 modulo p. We show that such congruences exist in this case, and that they are remarkably uniform compared with Mazur’s setting. Moreover, one can use a mild extension of Ribet’s method to produce from our congruences nontrivial elements in the class group of Q(N^{1/p}).
Tuesday, March 11 2025
No seminar due to Arizona Winter School: Representation theory of p-adic groups
Tuesday, March 25 2025, 2-3PM
Speaker: Gaurish Korpal, University of Arizona
Title: Gross lattices of supersingular elliptic curves
Abstract: Chevyrev and Galbraith (2013) and Goren and Love (2023) show that the successive minima of the Gross lattice of a supersingular elliptic curve can be used to characterize the endomorphism ring of that curve. We show that the third successive minimum D3 of the Gross lattice gives necessary and sufficient conditions for the curve to be defined over the field Fp or over the field Fp2. In the case where the curve E is defined over Fp, the value of D3 can even yield finer information about the endomorphism ring of E. This talk is based on my joint work with Chenfeng He, Ha Tran, and Christelle Vincent.
Tuesday, April 1 2025, 2-3PM
Speaker: Pan Yan, University of Arizona
Title: L-functions for Sp(2n)xGL(k) via non-unique models
Abstract: In the usual paradigm of the Rankin-Selberg method, the Eulerian factorization of a global integral relies on the uniqueness of a model such as the Whittaker model, or the uniqueness of an invariant bilinear form between an irreducible representation and its contragredient. Examples of Rankin-Selberg integrals which unfold to non-unique models are very rare because standard tools for local unramified computation such as the Casselman-Shalika formula are not applicable. In this talk we derive new global integrals for Sp(2n)xGL(k) where n is even, from the generalized doubling method of Cai, Friedberg, Ginzburg and Kaplan, following a strategy and extending a previous result of Ginzburg and Soudry on the case n=k=2. We show that these new global integrals unfold to non-unique models on Sp(2n). Using the New Way method of Piatetski-Shapiro and Rallis, we show that these new global integrals represent the L-functions for Sp(2n)xGL(k), generalizing a previous work of Piatetski-Shapiro and Rallis on Sp(2n)xGL(1). This is joint work with Yubo Jin.
Tuesday, April 8 2025, 2-3PM
Speaker: Katharine Woo, Princeton University
Title: Manin's conjecture for Châtelet surfaces
Abstract: We resolve Manin's conjecture for all Châtelet surfaces over Q (surfaces given by equations of the form x^2 + ay^2 = f(z)) -- we establish asymptotics for the number of rational points of increasing height. The key analytic ingredient is estimating sums of Fourier coefficients of modular forms along polynomial values.
Tuesday, April 22 2025, 2-3PM
Speaker: Serin Hong, University of Arizona
Title: An explicit construction of vector-valued automorphic forms on unitary groups
Abstract: In order to study automorphic forms on general groups, one must work with not only scalar-valued forms but also vector-valued forms. One of major challenges for studying vector-valued forms is lack of explicit examples to work with. In this talk, we describe a method for constructing vector-valued automorphic forms on unitary groups from scalar-valued ones, inspired by the work of Clery and van der Geer for Siegel modular forms. This is joint work with T. Browning, P. Coupek, E. Eischen, C. Freschette, S. Y. Lee, and D. Marcil, which began as a project from the 2022 Arizona Winter School.
Tuesday, May 6 2025, 2-3PM
Speaker: Chapman Howard, University of Arizona
Title: Epsilon Dichotomy and a Conjecture of Prasad and Takloo-Bighash
Abstract: The philosophy of using L-functions (analytic invariants) to study algebraic objects has produced many important results in the last century. We’ll discuss how this philosophy can be refined to a more interesting invariant (the epsilon factor or local root number), which in our case completely determines the representations we will study. With the importance of the local root number well understood, we’ll move toward the conjecture, explain previously proven cases, what work we have done so far, and how one might prove our case.