MSRItemp/ILS

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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.
    If there is time, I will describe work in progress with Charles Li in which we use the simple supercuspidal representations discovered recently by Gross and Reeder to spectrally isolate newforms (holomorphic or Maass) of level N^3, where N is square-free. One consequence is a simple Petersson formula for such newforms.
• 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.
• H. Iwaniec, W. Luo, and, P. Sarnak, Low lying zeros of families of L-functions, Publ. IHES, 2000.

  • Iwaniec-Luo-Sarnak.pdf

• there is also a study guide and reading list by Steven Miller, at Williams: http://www.williams.edu/go/math/sjmiller/public_html/ntandrmt/