Plenary talks will be held in Goergen 101
Contributed talks (Parallel session) will be held in Hylan 201 or 202
Speaker: Eran Assaf
Title: Geometric Invariants of Hilbert modular surfaces
Abstract: Hilbert modular surfaces are 2-dimensional analogues of modular curves, in that they parametrize abelian surfaces with endomorphism and level structure. Similar to how curves are stratified by genus, surfaces are organized by their numerical invariants; the Enriques-Kodaira classification organizes smooth surfaces by Kodaira dimension, Hodge numbers, and Chern numbers. In this talk, I will explain how to compute these invariants for a Hilbert modular surface (with *various level structures*, thereby extending results of van der Geer). This is joint work with A. Babei, B. Breen, E. Costa, J. Duque-Rosero, A. Horawa, J. Kieffer, A. Kulkarni, G. Molnar, S. Schiavone and J. Voight. Our implementation can be found at https://github.com/edgarcosta/hilbertmodularforms
Speaker: Bartu Bingol
Title: An attempt to construct obstructed deformation problems
Abstract: Thanks to the not-large-enough space in one of Fermat's notebooks, number theory has seen significantly progressive developments since 1630s. One of those developments were Deformation Theory of Galois Representations, and the theory played a vital role in the proof of Modularity Conjecture, which implied Fermat's Last Theorem. In this talk, we are going to briefly go over the remarkable contribution of the Deformation Theory to the proof of Modularity Conjecture. Then, we are going to describe what a Deformation Problem in this context is. Finally, we are going to introduce our attempt to characterize an important class of Deformation Problems, which is called Obstructed Modular Deformation Problems.
Speaker: Yen-Tsung Chen
Title: Almost holomorphic Drinfeld modular forms
Abstract: In the 1970s, almost holomorphic modular forms and the non-holomorphic operators, the Maass-Shimura operators, were studied extensively by Shimura. Later on, he discovered their connection with the periods of CM elliptic curves. In this talk, we introduce the notion of almost holomorphic Drinfeld modular forms and construct an analogue of the Maass-Shimura operators in this context. Furthermore, we investigate the relation between the periods of CM Drinfeld modules and the special values of almost holomorphic Drinfeld modular forms at CM points. This is joint work with Oguz Gezmis.
Speaker: Alina Cojocaru
Title: Frobenius traces for abelian varieties
Abstract: In the 1970s, Serge Lang and Hale Trotter proposed a conjectural asymptotic formula for the number of primes for which the Frobenius traces of an elliptic curve defined over the rationals equal a given integer. We will discuss results related to generalizations of this conjecture to higher dimensional abelian varieties. This is joint work with Tian Wang (University of Illinois at Chicago).
Speaker: Agniva Dasgupta
Title: Second Moment of Twisted Cusp Forms Along a Coset
Abstract: We prove Lindelöf-on-average upper bound for the second moment of the L-function associated to a level 1 holomorphic cusp form, twisted along a coset of subgroup of the characters modulo q^{2/3} (where q = p^3 for some odd prime p). This result should be seen as a q-aspect analogue of Anton Good’s (1982) result on upper bounds of the second moment of cusp forms in short intervals.
Speaker: Cameron Franc
Title: Modular forms and vertex operator algebras
Abstract: Modular forms are important number theoretic functions that have been at the heart of the theory of vertex operator algebras from the very inception of that theory, starting with monstrous moonshine, up to Zhu's theorem on the characters of strongly regular vertex operator algebras, and beyond. In this talk we will review these results, focusing in particular on the difficult question of classifying vertex operator algebras and what role modular forms play in that problem. We will conclude by discussing some of our recent joint work with Geoff Mason on p-adic properties of vertex operator algebras and their connection with p-adic modular forms.
Speaker: Sheng-Yang Kevin Ho
Title: On the Rational Cuspidal Divisor Class Groups of Drinfeld Modular Curves X_0(\mathfrak{p}^r)
Abstract: We study the structure of the rational cuspidal divisor class group C(\mathfrak{p}^r) of the Drinfeld modular curve X_0(\mathfrak{p}^r) for a prime power level \mathfrak{p}^r \in \mathbb{F}_q[T]. We relate the rational cuspidal divisors in C(\mathfrak{p}^r) with \Delta-quotients, where \Delta is the Drinfeld discriminant function. As a result, we are able to determine explicitly the structure of C(\mathfrak{p}^r) for arbitrary prime \mathfrak{p}\in \mathbb{F}_q[T] and r \geq 2.
Speaker: Xiaoyu Huang
Title: On the Universal Deformation Ring of A Residual Galois Representation with Three Jordan Holder Factors
Abstract: In this paper, we study Fontaine-Laffaille, essentially self-dual deformations of a mod p non-semisimple Galois representation of dimension n with its Jordan-Holder factors being three mutually non-isomorphic absolutely irreducible representations. We show that under some conditions regarding the orders of certain Selmer groups, the universal deformation ring is a discrete valuation ring. Given enough information on the Hecke side, we also prove an R=T theorem. We then apply our results to abelian surfaces with cyclic rational isogenies and 6-dimensional representations arising from automorphic forms congruent to Ikeda lifts. Particularly, assuming Bloch-Kato conjecture, our result identifies the special L values conditions for unique abelian surface isogeny class and an R=T theorem.
Speaker: Caleb Ji
Title: The Shafarevich conjecture for hypersurfaces in tori
Abstract: A Shafarevich conjecture is used to describe the statement that in various scenarios, there are only finitely many smooth varieties over a number field with good reduction outside some finite set of places. Faltings proved the Shafarevich conjecture for curves and abelian varieties as an essential step in his proof of the Mordell conjecture. Since then, many authors have made progress towards the Shafarevich conjecture in other cases. In this talk we will report on work in progress towards establishing the Shafarevich conjecture for hypersurfaces in tori. The method involves establishing large monodromy statements through computing a Tannakian monodromy group and combining this with the Lawrence-Venkatesh strategy in their alternative proof of Faltings's theorem.
Speaker: Jeffrey Katen
Title: Isomorphism Classes of Drinfeld Modules over Finite Fields
Abstract: We study isogeny classes of Drinfeld A-modules over finite fields k with commutative endomorphism algebra D, in order to describe the isomorphism classes in a fixed isogeny class. We study when the minimal order A[π] of D occurs as an endomorphism ring by proving when it is locally maximal at π, and show that this happens if and only if the isogeny class is ordinary or k is the prime field. We then describe how the monoid of fractional ideals of the endomorphism ring E of a Drinfeld module ϕ up to D-linear equivalence acts on the isomorphism classes in the isogeny class of ϕ, in the spirit of Hayes. We show that the action is free when restricted to kernel ideals, of which we give three equivalent definitions, and determine when the action is transitive. In particular, the action is free and transitive on the isomorphism classes in an isogeny class which is either ordinary or defined over the prime field, yielding a complete and explicit description in these cases.
Speaker: Harun Kir (Queen’s University)
Title: The Refined Humbert Invariant for Abelian Product Surfaces A = E1 × E2 with Complex Multiplication and its Applications
Abstract: In this talk, I will briefly mention the classification of the refined Humbert invariants associated with a product abelian surface A ≃ E1 ×E2, for isogenous CM elliptic curves E1 and E2. In this case, the refined Humbert invariant can be seen as a ternary quadratic form. Using this classification, I will illustrate some results about genus 2 curves. For example, I will discuss what automorphism group of the genus 2 curve C is, and whether it has an elliptic subcover of degree n, for a given n, and how this classification can be used to find CM points on Shimura curves.
Speaker: Joe Kramer-Miller (Lehigh)
Title: Zeros of equicharacteristic zeta functions
Abstract: The Riemann hypothesis is perhaps the most significant open problem in number theory. It asserts that the zeros of the Riemann zeta function have real part 1/2. In the early 2000s, David Goss wrote several papers speculating about an equicharacteristic Riemann hypothesis for function fields. In this talk we will give an overview of these conjectures and the progress made on these conjectures. We will then describe joint work with James Upton, which proves a corrected version of Goss's conjecture for ordinary curves. In particular, we prove the conjecture for `almost all' function fields. We also explain analogous results for the finite place interpolation of such zeta functions.
Speaker: Sarah Lamoureux
Title: ADOs on Compact DVRs and the Completions of their Maximal Unramified Extensions
Abstract: Let R be a compact DVR with uniformizer \pi and residue field \mathbb{F}_q. Denote by S = \hat{R}^{\textup{ur}} the completion of the maximal unramified extension of R, and let \phi: S\to S be the lift of the q-th-power map S / \pi S \to S / \pi S. We define \delta: S -> S by \delta(x)=\frac{\phi(x)-x^q}{\pi}. A map f: S^d \to S is called an arithmetic differential operator (ADO) when it can be represented by a restricted power series in \delta and its iterates; a similar definition is made for maps f: R^d \to R. This talk explores the relationship between ADOs f:R^d \to R and ADOs g: S^d \to S.
Speaker: Daniel Li-Huerta
Title: Local-global compatibility over function fields
Abstract: We prove that V. Lafforgue's global Langlands correspondence is compatible with Fargues–Scholze's semisimplified local Langlands correspondence. As a result, we canonically lift Fargues–Scholze’s construction to a non-semisimplified local Langlands correspondence for fields of characteristic p ≥ 5. We also deduce that Fargues–Scholze’s construction agrees with that of Genestier–Lafforgue, which answers a question of Fargues–Scholze, Hansen, Harris, and Kaletha.
Speaker: Steve Miller (Williams College)
Title: The Katz-Sarnak Density Conjecture and Bounding Central Point Vanishing of L-Functions
Video: https://youtu.be/Be3zNr4kKVw
Abstract: Spacings between zeros of L-functions occur throughout modern number theory, such as in Chebyshev's bias and the class number problem. Montgomery and Dyson discovered in the 1970's that random matrix theory (RMT) seems to model these spacings away from the central point s = 1/2. While we have an incomplete understanding as to why a correlation exists between RMT and number theory, this interplay has proved useful for conjecturing answers to classical problems. These RMT models are insensitive to finitely many zeros, and thus miss the behavior near the central point. This is the most arithmetically interesting place; for example, the Birch and Swinnerton Dyer conjecture states that the rank of the Mordell-Weil group equals the order of vanishing of the associated L-function there.
To investigate the zeros near the central point, Katz and Sarnak developed a new statistic, the $n$-level density; one application is to bound the average order of vanishing at the central point for a given family of L-functions by an integral of a weight against some test function \phi. It is therefore of interest to choose \phi optimally to minimize the integral and obtain the best bound possible. While the 1-level density has been studied in prior work, larger n yield better bounds, but new technical problems emerge in the higher level densities. We discuss recent work on extending the class of test functions for which we can do the analysis. We see agreement with the RMT predictions, and show how larger n and good choices of test functions lead to significant improvements in bounding how often cuspidal newforms vanish to a given order at the central point. To highlight the key ideas of the calculations, we often concentrate on easier cases, such as Dirichlet L-functions.
This talk is joint with numerous REU students.
Speaker: Ajith Nair
Title: Gauss composition and Higher Composition Laws
Abstract: In this talk, I will describe Bhargava's remarkable generalization of Gauss composition on binary quadratic forms to higher degree forms such as binary cubic forms, senary alternating 3-forms, pairs of binary quadratic forms and pairs of quaternary alternating 2-forms. I will explain Bhargava's methodology and how these spaces of forms parametrize arithmetic objects associated to quadratic number fields. Lastly, I will give an outline of an ongoing work with my advisor Gautam Chinta (as part of my PhD) on formulating the higher composition laws in a manner similar to Gauss's formulation in the binary quadratic forms case.
Speaker: Evangelos Nastas
Title: Exploring Reciprocity in Some Sawtooth function Product Sums & Semigroups
Abstract: Sawtooth function Product Sums, especially Dedekind sums play a key role in number theory and have links with a broad range of topics, such as modular forms, quadratic forms and elliptic curves. This talk explores the affinity between Dedekind sums and formulas for numerical semigroups, which are sets of integers closed under addition and having a finite complement in the positive integers. How reciprocity theorems can be utilized to understand more the structure of variations of Dedekind sums and their close sums in numerical semigroups will be particularly emphasized. Reciprocity theorems are instrumental in accelerating the calculation of these sums by allowing the use of the Euclidean algorithm. The aim here is to provide insight into the interplay between these two important areas of number theory and highlight some recent developments in this field.
Speaker: Arghya Sadhukhan
Title: Dimension of some (union of) affine Deligne-Lusztig varieties via the quantum Bruhat graph
Abstract: First introduced by Rapoport, the affine Deligne-Lusztig varieties (ADLVs) play an important role in the study of mod-p reduction of Shimura varieties and related geometric structures, thus finding applications in arithmetic geometry and the Langlands program; for instance, precise information about the connected components of ADLVs has proved to be useful in Kisin's proof of the Langlands-Rapoport conjecture. In this talk, we will discuss a dimension formula for a certain union of ADLVs (in the affine flag variety) associated with a quasi-split group and some partial description of the dimension and top-dimensional irreducible components in the non-quasi-split cases.
Speaker: Anurag Sahay
Title: Moments of the Hurwitz zeta function on the critical line
Abstract: The Hurwitz zeta function is a shifted integer analogue of the Riemann zeta function. We will consider the integral moments of the Hurwitz zeta function on the critical for rational shift parameters. In this case, the Hurwitz zeta function decomposes as a linear combination of Dirichlet L-functions, which leads us into investigating moments of products of L-functions. Using heuristics from random matrix theory, we conjecture an asymptotic of the same form as the moments of the Riemann zeta function.
Speaker: Ciaran Schembri
Title: Torsion points on abelian surfaces with many endomorphisms
Abstract: In a celebrated work Mazur classified which torsion subgroups can occur for elliptic curves defined over the rationals. A natural analogue is to consider surfaces with geometric endomorphisms by a quaternion order, since the associated moduli space is 1-dimensional. In this talk I will discuss progress towards classifying which torsion subgroups are possible for these surfaces. This is joint work (in progress) with Jef Laga, Ari Shnidman and John Voight.
Speaker: Gabrielle Scullard
Title: Nonisomorphic Group Structures of Elliptic Curves in Finite Field Extensions
Abstract: Let E and E' be rationally 2-isogenous elliptic curves defined over Q. We call a (good) odd prime p anomalous if for the reductions of E and E' mod p, the group structures E(\mathb{F}_p) and E'(\mathbb{F}_p) are isomorphic, but the group structures E(\mathbb{F}_{p^2}) and E'(\mathbb{F}_{p^2}) are not isomorphic. (Cullinan has shown that being isomorphic over \mathbb{F}_p and \mathbb{F}_{p^2} in fact guarantees isomorphisms over all finite extensions.) Recently, Cullinan and Kaplan computed that the proportion of anomalous primes in the "generic" case (for which the elliptic curves have maximal 2-adic image with respect to the constraints of the problem) is 1/30. This project addresses the more general case: We give a formula for the proportion of anomalous primes, given the 2-adic images of the elliptic curves, and we address the case of CM curves. This is joint work (through the Rethinking Number Theory workshop) with John Cullinan, Shanna Dobson, Jorge de Mello, Asimina Hamakiotes, and Roberto Hernandez.
Speaker: Vaishavi Sharma
Title: p-adic valuations of sequences
Abstract: Given a prime p and any positive integer n, the p-adic valuation of n, denoted by \nu_p(n), is the highest power of p that divides $n$. This notion is extended to \mathbb{Q} by \nu_p(\frac{a}{b})=\nu_p(a)-\nu_p(b) and by setting \nu_p(0) = \infty$. For any sequence {a_n} and a fixed prime p, the sequence of valuations \nu_p(a_n) often presents interesting challenges and we try to obtain a closed form for the valuations. In this talk, I will discuss p-adic valuations of some common integer sequences.
Speaker: Owen Sweeney
Title: On the second case of Fermat's last theorem over cyclotomic fields
Abstract: In the nineteenth century, Kummer proved Fermat's last theorem for regular prime exponents, not just over the rational numbers but over certain cyclotomic fields. Whether this is true for all prime exponents is still an open problem. We discuss the development of Kummer's cyclotomic approach and more recent criteria applicable in the more general setting when the prime exponent does not satisfy the regularity hypothesis. Time permitting, we say a few words about an application of the uniform abc conjecture to this problem.
Speaker: Dinesh Thakur (UR)
Title: Multizeta values for function fields and structures underlying their arithmetic
Abstract: We will introduce these values, some variants and related structures, and describe various results, conjectures and some recent developments in the subject.
Speaker: Fang-Ting Tu (LSU)
Title: Hypergeometric Functions over Finite Fields
Abstract: Hypergeometric functions over finite fields are expressed in terms of certain character sums and determine the corresponding hypergeometric Galois representations.
In this talk, I will give a brief introduction to hypergeometric character sums, Galois representations introduced by Katz and Beukers-Cohen-Mellit, and my recent joint works. In a joint project with Wen-Ching Winnie Li and Ling Long, we consider the hypergeometric Galois representations corresponding to a formula due to Whipple which relates certain hypergeometric values $_7F_6(1)$ and $_4F_3(1)$. We explain a special structure of the representations behind Whipple's formula leading to a decomposition that can be described by the Fourier coefficients of Hecke eigenforms. Based on this observation and the works of Frechette-Ono-Papanikolas and Scholl, in a joint work with Jerome William Hoffman, Wen-Ching Winnie Li, and Ling Long, we obtain trace formulas of Hecke operators of certain spaces of modular forms in terms of character sums.
Speaker: Peter Vang Uttenthal
Title: Density of Selmer ranks in families of even Galois representations
Abstract: This talk concerns an even, reducible residual Galois representation in even characteristic. By thickening the image with cohomology classes, all lifts of the representation are ensured to be irreducible. Smooth quotients of the local deformation rings at the primes where the representation is ramified are found, and the reciprocity law of Galois cohomology is applied to deform the representation globally. By using the generic smoothness of the local deformation rings at trivial primes and the Wiles-Greenberg formula, a balanced global setting is created, in the sense that the Selmer group and the dual Selmer group have the same rank. Finally, a family of even representations is obtained by allowing ramification at auxiliary primes, and the distribution over primes of the ranks of their Selmer groups is studied. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192.
Speaker: Benjamin York
Title: On the adelic image of Galois representations attached to elliptic curves with CM
Abstract: Let E be an elliptic curve defined over a number field K, and let \rho_E be the adelic Galois representation attached to E/K. The goal of the so-called Mazur's "Program B" is to classify the possibilities for the image of \rho_E as a subgroup of GL(2, \widehat{Z}), up to conjugation. In this talk, we will discuss a method for computing adelic images of elliptic curves with complex multiplication. Joint work with \'Alvaro Lozano-Robledo.
The Eleventh annual Upstate New York Number Theory Conference is funded by the National Science Foundation Award #1902052 and the Journal of Number theory.