On the low-lying zeros of Hasse-Weil L-functions for elliptic curves

Stephan Baier, Liangyi Zhao

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse–Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. This upper bound for the average rank enables us to deduce that, under the same assumption, a positive proportion of elliptic curves have algebraic ranks equaling their analytic ranks and finite Tate–Shafarevich group. Statements of this flavor were known previously [M.P. Young, Low-lying zeros of families of elliptic curves, J. Amer. Math. Soc. 19 (1) (2005) 205–250] under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper.
Original languageEnglish
Pages (from-to)952-985
Number of pages34
JournalAdvances in Mathematics
Issue number3
Publication statusPublished - 20 Oct 2008

Cite this