TORA XII - Abstracts

Venue: All talks will take place in PHSC 108. Tea will be served in PHSC 424.

 Friday October 13

• 3:30   Colloquium: Ellen Eischen, Creating Clarity from Confusion: Three Lessons from Number Theory
  Abstract: Regardless of your research area or career stage, you almost certainly have stories about "mistakes" or "failures" in your research. This colloquium talk will introduce three crucial developments in number theory, alongside some of the mistakes, false starts, and otherwise suboptimal approaches that led to them. These three strands--one from each of the 18th, 19th, and 20th centuries--eventually merged spectacularly to enable progress in the 21st century, including in the speaker's work. Intertwined with these specific mathematical triumphs are valuable lessons about grappling with -- and sometimes adding to -- confusion while trying to solve hard problems. The focus of this talk will be on progression of key ideas, rather than on technical details. Graduate students and faculty from all areas of math are encouraged to attend.

 Saturday October 14

• 9:00   John Bergdall, p-adic slope distributions for modular forms
  Abstract: This short talk will survey non-Archimedean distribution results and questions for modular eigenforms. We will describe the p-adic slope problem, which aims to determine the p-adic norms of Hecke eigenvalues for modular forms of level co-prime to p. It is analogous to the Archimedean question, which results in the Sato-Tate conjecture and its "vertical" analogue. Until the past two years, little was truly known about the slope problem, so we will definitely describe a recent breakthrough in this area due to Liu, Truong, Xiao, and Zhao. Some of the research presented will be my own, in which case it is part of a long-term collaboration with Robert Pollack.
• 9:30   Raghuram, Why do cusp forms exist?
  Abstract: I will begin this talk by reviewing the Eichler-Shimura isomorphism between the space of cusp forms of weight k and level N, and a certain cohomology group. This is based on Chapter 8 of Shimura's book "Introduction to the Arithmetic Theory of Automorphic Functions." Shimura called this cohomology group as parabolic cohomology. In the context of automorphic forms on higher groups, such a cohomology group takes the form of cuspidal cohomology. The existence of cusp forms of prescribed weight then may be reformulated into the nonvanishing of cuspidal cohomology with prescribed coefficients. I will review the relevant definitions to be able to state the nonvanishing problem for cuspidal cohomology for GL(n) over a number field. Then I will briefly survey the known results. Finally, I will discuss recent results obtained in joint work with my student Darshan Nasit, making some modest progress towards this problem.
• 11:00   Debanjana Kundu, λ-invariant stability in Families of Modular Galois Representations
  Abstract: Consider a family of modular forms, all of whose residual (mod p) Galois representations are isomorphic. It is well-known that their corresponding Iwasawa λ-invariants may vary. We will discuss this variation from a quantitative perspective, providing lower bounds on the frequency with which these λ-invariants grow or remain stable. This is joint work with Jeff Hatley.
• 11:30   Speed TORA I
• 2:00   Rahul Dalal, Computing Statistics of Automorphic Representations
  Abstract: I will motivate why we might want to compute statistics of families of automorphic representations---generalizations of questions like: how does the number of weight-k, level-N modular forms grow as either N or k go to infinity? What fraction of them asymptotically have Hecke eigenvalue at 17 within a certain interval? In the particular case of discrete-at-infinity automorphic representations (the analogues of holomorphic modular forms as opposed to Maass forms) there is a "simple" version of the Arthur-Selberg trace formula that lets us answer such questions reasonably precisely on general groups. While this produces many interesting results, the ways it can restrict the representations being counted are too coarse for many applications---for example, it cannot distinguish whether a Siegel modular form is holomorphic or not.
  The theory of endoscopy and endoscopic classifications allows us to further refine our counts and even extend them slightly past the case of discrete-at-infinity. Specifically, it allows "cancelling out" unwanted contributions with appropriate counts transferred from smaller groups. I will end by discussing recent work with Mathilde Gerbelli-Gauthier using this idea to understand statistics of certain extremely sparse families on unitary groups violating the Ramanujan conjecture.
• 3:00   Pan Yan, Product of Rankin-Selberg convolutions and a new proof of Jacquet's local converse conjecture
  Abstract: We construct a family of integrals which represent the product of Rankin-Selberg L-functions of GL(l) x GL(m) and of GL(l) x GL(n) when m+n<l. When n=0, these integrals are those defined by Jacquet--Piatetski-Shapiro--Shalika up to a shift. We discuss basic properties of these integrals. In particular, we define local gamma factors using this new family of integrals. As an application, we obtain a new proof of Jacquet's local converse conjecture using these new integrals and Cogdell-Shahidi-Tsai's theory of partial Bessel functions. This is joint work with Qing Zhang.
• 4:00   Speed TORA II

 Sunday October 15

• 9:00   Jonathan Cohen, Invariant vectors in depth-zero supercuspidal representations of GSp(4)
  Abstract: We will discuss an approach to compute dimensions of certain spaces of classical Siegel modular forms. This leads to the question of dimensions of fixed vectors under certain sequences of congruence subgroups in representations of p-adic GSp(4). For general representations, these dimensions remain unknown. We will illustrate the situation in the special case of depth-zero supercuspidal representations; these include the first examples of (generic) representations for which these fixed-vector dimensions have been computed.
• 9:30   Ellen Eischen, Algebraic and p-adic aspects of L-functions, with a view toward Spin L-functions for GSp_6
  Abstract: I will discuss recent developments and ongoing work for algebraic and p-adic aspects of L-functions. Interest in p-adic properties of values of L-functions originated with Kummer’s study of congruences between values of the Riemann zeta function at negative odd integers, as part of his attempt to understand class numbers of cyclotomic extensions. After presenting an approach to studying analogous congruences for more general classes of L-functions, I will conclude by introducing ongoing joint work of G. Rosso, S. Shah, and myself (concerning Spin L-functions for GSp 6). I will explain how this work fit into the context of earlier developments, while also indicating where new technical challenges arise. All conference participants who are curious about this topic are welcome at this talk, even without prior experience with p-adic L-functions or Spin L-functions.
• 11:00   Yangbo Ye, Bounds toward Hypothesis S for cusp forms
  Abstract: Hypothesis S as conjectured by Iwaniec, Luo, and Sarnak predicts cancellation in a sum of exponential functions of fractional powers. When weighted by Fourier coefficients of a fixed cusp form, such a sum exhibits a resonance phenomenon and hence prevents further cancellation. In this talk the cusp form is allowed to change with its weight tending to infinity. The hope is that this way it might become possible to break the resonance barrier. Non-trivial bound is proved as the first step toward Hypothesis S with moving cusp forms.
• 11:30   Mike Hanson, Ramanujan congruences for overpartitions with restricted odd differences
  Abstract: In this talk, we investigate Ramanujan congruences for a modified partition function which counts overpartitions with restricted odd differences. Our method involves using the theory of modular forms to prove a more general theorem which bounds the number of primes possible for Ramanujan congruences in a certain class of modular forms.