## Low-lying zeros seminar

- chair: Kaneenika Sinha
- times: Wednesdays 1.30 - 2.30 pm. Please note: The first meeting on 19 Jan will be from 2-3 pm in the Baker boardroom.
Wed. 19 Jan in the Common Room: Henryk Iwaniec,

*Introduction to low lying zeros of L-functions*- Abstract: The content of the paper "Low lying zeros of families of L-functions" by Iwaniec, Luo and Sarnak will be described in general terms.

- Wed. 26 Jan in Baker boardroom: Henryk Iwaniec, "The paper of Iwaniec, Luo and Sarnak"
- Abstract: We will continue to describe the contents of the above mentioned paper.

- Wed. 9 Feb in Baker boardroom. H. Iwaniec will continue to present the contents of the above paper.
- 16 Feb in Baker boardroom:
- Speaker: David Farmer Title: What you need to know about random matrix theory

- 23 Feb in Baker boardroom, Andrew Knightly, Title: Petersson's fomula and related topics,
- Abstract: A crucial ingredient in the ILS paper is an estimate for a sum of Hecke eigenvalues. This is handled using a variant of the trace formula due to Petersson, but there are some obstacles, especially the fact that Petersson's formula includes the contribution of old forms. I will briefly outline the ILS strategy for isolating the newform contribution when N is square-free. I will then propose an alternate strategy coming from representation theory. Along the way, I will give a very basic introduction to the trace formula, and explain how one can use it to derive various identities of interest in classical analytic number theory. These include:
- The trace of a Hecke operator; The Petersson and Kuznetsov formulas; Formulas for averages/moments of L-functions; Vertical Sato-Tate laws for Hecke eigenvalues.

- Abstract: A crucial ingredient in the ILS paper is an estimate for a sum of Hecke eigenvalues. This is handled using a variant of the trace formula due to Petersson, but there are some obstacles, especially the fact that Petersson's formula includes the contribution of old forms. I will briefly outline the ILS strategy for isolating the newform contribution when N is square-free. I will then propose an alternate strategy coming from representation theory. Along the way, I will give a very basic introduction to the trace formula, and explain how one can use it to derive various identities of interest in classical analytic number theory. These include:
- 2nd March: Duc Khiem Huynh in Simons Auditorium, Title: Lower order terms for the one-level density of elliptic curve L-functions
- Abstract: The L-function ratios conjectures - motivated by analogous random matrix theorems - give precise predictions for averages over families

of ratios of products of shifted L-functions. We present an application by deriving lower order terms for the one-level density of elliptic curve L-functions. We discuss how these terms together with other arithmetical information can model elliptic curve L-functions in terms of random matrices.

9th March: Kaneenika Sinha in Baker boardroom, Title: Analytic rank of J_0(q)

Abstract: We will survey different methods of finding explicit upper bounds for the analytic rank of the Jacobian of the modular curve X_0(N). The papers of Brumer, Murty, Iwaniec-Luo-Sarnak and Michel-Kowalski address this important question. To begin with, we will focus on the papers of Brumer and Murty which assume the Riemann hypothesis for L-functions of newforms.

- 16th March: Henryk Iwaniec, Title: Critical zeros of L-functions
- 23rd March: Rob Rhoades, Title: Alternatives to Dirichlet Series Amplifiers
- Abstract: I'll discuss an approach to subconvexity that uses a "geometric" mollifier. I'll give some concrete examples to illustrate the idea. If time permits I'll try to discuss why mollification does not have a straightforward analog in this setting and an approach toward mollification that uses relative trace formula. The amplification ideas are due to Venkatesh and Venkatesh-Michel.

- 4th April (Monday 1 pm at Simons Auditorium): Brooke Feigon, Title: Averages of central L-values
- Abstract: I will discuss computing averages of central values of certain families of L-functions via the relative trace formula and explicit period formulas. The two examples I will focus on are central values of certain twisted quadratic base change L-functions averaged over Hilbert modular forms and triple product L-functions averaged over weight two newforms of fixed level. This is joint work with David Whitehouse.

- H. Iwaniec, W. Luo, and, P. Sarnak, Low lying zeros of families of L-functions, Publ. IHES, 2000.
there is also a study guide and reading list by Steven Miller, at Williams: http://www.williams.edu/go/math/sjmiller/public_html/ntandrmt/