The analytic rank of \(J_0(q)\) and zeros of automorphic \(L\)-functions.
type: Item
instance of: scholarly article
title: The analytic rank of \(J_0(q)\) and zeros of automorphic \(L\)-functions.
zbMATH Open document ID: 1161.11359
DOI: 10.1215/S0012-7094-99-10017-2
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: For this important paper we cite the introduction: This paper is motivated by the conjecture of Birch and Swinnerton-Dyer relating the rank of the Mordell-Weil group of an abelian variety defined over a number field with (in its crudest form) the order of vanishing of its Hasse-Weil \(L\)-function at the central critical point. \textit{J.-F. Mestre} [Compos. Math. 58, 209--232 (1986; Zbl 0607.14012)] began the study of the implications of this conjecture towards providing upper bounds for the rank. He used ``explicit formulae'' similar to that of Riemann-Weil and assumed the analytic continuation and (perhaps more significantly) the Riemann hypothesis for those \(L\)-functions. \textit{A. Brumer} [Astérisque 228, 41--68 (1995; Zbl 0851.11035)] first studied the special case of the Jacobian variety \(J_0(q)\) of the modular curve \(X_0(q)\). This is an abelian variety defined over \(Q\) of dimension about \(q/12\). Here analytic continuation is known, by the work of Eichler and \textit{G. Shimura} [Introduction to the arithmetic theory of automorphic functions (1971; Zbl 0221.10029), reprint (1994; Zbl 0872.11023)]. Assuming only the Riemann hypothesis for the \(L\)-functions of automorphic forms (of weight 2 and level \(q\)), Brumer proved \[ \text{rank}_a J_0(q)\leq\left(\frac 32 +o(1)\right) \dim J_0(q) \] and conjectured that \[ \text{rank} J_0(q) = \text{rank}_a J_0(q) \text{sym}\frac 12\dim J_0(q) \] (based on the fact that the sign of the functional equation for the automorphic \(L\)-functions of weight 2 and level \(q\) is approximately half the time \(+1\) and half the time \(-1\)). Other authors, notably \textit{M. Ram Murty} [CMS Conf. Proc. 15, 263--277 (1995; Zbl 0851.11036)] (who first applied the Petersson formula in this context), considered the same problem. Most recently, \textit{W. Luo, H. Iwaniec}, and \textit{P. Sarnak} [Low lying zeros for families of \(L\)-functions, preprint, 1998], using the same assumptions, proved an estimate \[ \text{rank}…
reviewed by: Olaf Ninnemann
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11G10
zbMATH DE Number: 1425264
Wikidata QID: Q57268267
author: E. Kowalski
author: Philippe Michel
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1425264