Explicit methods in automorphic forms

Location: University College London (UCL)

Date: 19-20 June 2025 

Venue: Room 505, UCL mathematics building

Registration:  Please fill out this Google form.

This workshop emphasizes two related themes: development of new versions of the trace formula for use in applications to analytic number theory, and numerical computations using the trace formula.

Speakers 

1. Andrew Booker  (University of Bristol)

2. Jack Buttcane (University of Maine)

3. Andrew Knightly (University of Maine)

4. Min Lee (University of Bristol)

5. Didier Lesesvre (Université de Lille)

6. Asbjørn Nordentoft  (University of Copenhagen)

7. Nicole Raulf (Université de Lille)

Schedule (Thursday 19 June)

• 1:00-1:30 --- Registration

• 1:30-2:30 --- Andrew Booker

Title: L-functions from nothing

Abstract: Long before the proof of the modularity theorem, evidence for it was amassed in the 1972 "Antwerp IV" tables comparing elliptic curves and modular forms of small conductor; in particular, D. J. Tingley's calculation of spaces of modular forms was essential for filling some gaps in the elliptic curve table, and gave a de facto complete list of all elliptic curves of conductor up to 200. In the decades that followed this list was extended by John Cremona and is now a major part of the L-functions and Modular Forms Database (LMFDB).

Elliptic curves are abelian varieties of dimension 1. In recent decades similar efforts have been undertaken in dimension 2, including a 2015 tabulation of genus 2 curves corresponding to abelian surfaces which can be found in the LMFDB. But we lack an analogue of Tingley's table of modular forms, and with it any proof of completeness or evidence of gaps. Directly computing the relevant spaces of modular forms is prohibitively difficult, even for small conductors.

In the talk I will describe joint work with Andrew Sutherland extending a method of Farmer, Koutsoliotas, and Lemurell that makes it possible to compute these spaces indirectly. The result is a provably complete tabulation of L-functions of modular forms of conductor up to 1368, and partial results for some conductors as large as 2401. For instance, we obtain a complete classification of modular abelian surfaces with good reduction away from 7.

• 2:30-3:00 --- Coffee break (Room 502)

• 3:00-4:00 --- Nicole Raulf

Title: Automorphic forms and quantum unique ergodicity

Abstract: In this talk I will discuss various results related to quantum unique ergodicity and its refinements with a focus on dimensions 2 and 3. This is joint work with D. Chatzakos, R. Frot and Y. Petridis, M. Risager.

• 4:00-5:00 --- Andrew Knightly

Title: Counting newforms with ramified supercuspidal local type

Abstract: Using a specialized test function in the adelic trace formula for GL(2), we obtain a general formula for the number of cuspidal newforms whose local representation type at a fixed prime p is a fixed supercuspidal $\pi_p$.  However, the relevant local elliptic orbital integral for a supercuspidal of large conductor exponent is complicated and hasn't been computed explicitly. Kimball Martin and I found a way to use the general framework together with his recent formula for traces of local Atkin--Lehner operators to give explicit dimension formulas when $\pi_p$ has conductor $p^{2r+1}$ for any r>0, without having to compute those complicated local orbital integrals.  These dimensions depend only on the root number and the ramified quadratic extension of $\mathbb{Q}_p$ attached to $\pi_p$.

Schedule (Friday 20 June)

• 9:30-10:30 --- Asbjørn Nordentoft

Title: A mixing version of QUE

Abstract: In this talk I will describe a refinement of the Quantum Unique Ergodicity (QUE) conjecture in the level aspect given by pushing forward to the product of two modular curves via the Hecke correspondence. This gives rise to a two-variable equidistribution problem resembling in many ways the mixing conjecture of Michel--Venkatesh. I will explain how to resolve the conjecture in the cocompact case under GRH. This is joint work in progress with Radu Toma.

• 10:30-11:00 --- Coffee break (Room 502)

• 11:00-12:00 --- Min Lee

Title: Joint equidistribution of rational points on expanding closed horocycles related by different powers 

Abstract: We study joint equidistribution of rational points on expanding closed horocycles related by different powers using methods from analytic number theory. By applying the spectral theory of automorphic forms, we reformulate this problem in L-functions of automorphic forms twisted by characters. Our approach makes explicit the error bound in a recent work of Aka, Einsiedler, Luethi, Michel, and Wieser. This talk is based on joint work in progress with Bingrong Huang and Andreas Strömbergsson. 

• 12:00-2:00 --- Lunch break (Room 502)

• 2:00-3:00 --- Didier Lesesvre

Title: Kuznetsov trace formula for GSp(4) and applications

Abstract: Trace formulas relate statistics on automorphic forms, which often remain mysterious yet central in number theory, with statistics on geometric or arithmetic quantities, which one hopes to be more explicit and better understood. We will discuss how to establish such a Kuznetsov-type trace formula in the case of the symplectic group GSp(4), and will study the precise analytic behaviour of both the spectral and the arithmetic transforms arising in the formula. These fundamental properties can be used to establish various results on the family of Maaß automorphic forms on GSp(4) in the spectral aspect: the Weyl law, a density result on the non-tempered spectrum, large sieve inequalities, bounds on the second moment of the spinor and standard L-functions, as well as a statement on the distribution of the low-lying zeros of these L-functions.

• 3:00-3:30 --- Coffee break (Room 502)

• 3:30-4:30 --- Jack Buttcane

Title: The spectral Kuznetsov formula on GL(n)

Abstract: The past few years, I've been on a quest to find integral representations for the Bessel functions occurring in the GL(n) Kuznetsov formula.  We have these for GL(2), GL(3) and now GL(4), and I think I know how to construct them for GL(n).  This talk will be about the spectral families on GL(n) and their corresponding Kuznetsov formulas.  I'll talk about the integral representations we want for the Bessel functions and how (I hope) they lead to applications like Weyl laws, moments and subconvexity of L-functions.

Organizers: Ian Petrow and Chung-Hang Kwan