Exponential decay in the frequency of analytic ranks of automorphic \(L\)-functions
type: Item
instance of: scholarly article
title: Exponential decay in the frequency of analytic ranks of automorphic \(L\)-functions
zbMATH Open document ID: 1166.11326
DOI: 10.1215/S0012-7094-00-10235-9
author: D. R. Heath-Brown
author: Philippe Michel
published in: Duke Mathematical Journal
publication date: 2003-05-19T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.dmj/1092749339
review text: From the text: This note should be seen as an addendum to the work of \textit{E. Kowalski} and the second author [Duke Math. J. 100, 503--542 (1999; Zbl 1161.11359)] and deals with the problem of bounding, unconditionally, the order of vanishing at the critical point in a family of \(L\)-functions. This problem is illustrated in the particular case of \(L\)-functions of weight-2 primitive modular forms of prime level. Recall the notation from that paper: For q a prime number, let \(S_2(q)^*\) be the set of primitive forms of weight 2 and level \(q\), normalized so that their first Fourier coefficient is 1; for \(f (z) =\sum_{n\geq 1} \lambda_f(n)n^{1/2}e(nz) \in S_2(q)^*\), \(\lambda_f (1) = 1\), let \(L(f, s) :=\sum_{n\geq 1}\lambda_f (n)n^{-s}\), the associated (normalized) \(L\)-function; it admits analytic continuation to \(\mathbb C\) with a functional equation relating \(L(f, s)\) to \(L(f,1-s),\) and we call \(r_f := \text{ord}_{s=1/2} L(f, s)\) the analytic rank of \(f\) . In [Kowalski-Michel (loc. cit.)] the following was proved: There exists an absolute constant \(C_1 \geq 0\) such that, for all \(q\) prime, \(\sum_{f \in S_2(q)^*} r_f \leq C_1\,| S_2(q)^*|.\) After that, much progress has been made concerning this question. In particular, in [\textit{E.~Kowalski, P. Michel} and \textit{J. M. VanderKam}, J. Reine Angew. Math. 526, 1--34 (2000; Zbl 1020.11033)], a sharp explicit value was given for the constant \(C_1\) (\(C_1 = 1.1891\) for \(q\) large enough). In the course of the proof, a uniform bound for the square of the ranks was obtained: \(\sum_{f \in S_2(q)^*}r_f ^2 \leq C_2\,| S_2(q)^*|.\) However, the latter improvement used only a slight variant of the methods of Kowalski-Michel (loc. cit.). In fact, it is possible to pursue this idea further and it turns out that much more is true; this is the subject of the present note. Consider a finite probability space \((\Omega,\mu)\), where \(\mu(\omega) > 0\) for every \(\omega\in \Omega\). For each \(\…
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 1455498
Wikidata QID: Q57268252
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1455498