Nonvanishing of \(L\)-series associated to cubic twists of elliptic curves. (English) Zbl 0817.11029

Lieman, Daniel B.

Ann. Math. (2) 140, No. 1, 81-108 (1994).

The purpose of this paper is to prove that in any arithmetic progression \(D: D\equiv c\pmod p\) (\(p\neq 3\), a prime and \((c,p)=1\)) there are infinitely many cube-free elements \(D\) so that the \(L\)-function of the elliptic curve \(X^ 3+ Y^ 3=d\) does not vanish at 1. In view of the theorem of Coates-Wiles it follows that the curve has only finitely many rational points. The proof is based on an observation of D. Bump and J. Hoffstein [Invent. Math. 84, 481-505 (1986; Zbl 0591.10018)], that the \(L\)-functions in question essentially arise as the Fourier coefficients of a cubic metaplectic form, namely an Eisenstein series induced from a cubic metaplectic form on GL(2). The author computes these coefficients carefully and makes use of the Rankin-Selberg convolution with a metaplectic Eisenstein series on GL(2) to obtain the necessary analytic continuation to deduce the theorem above.

Reviewer: S.J.Patterson (Göttingen)

MSC:

11F67	Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G05	Elliptic curves over global fields

Keywords:

finitely many rational points; Fourier coefficients; cubic metaplectic form; Eisenstein series; Rankin-Selberg convolution

Citations:

Zbl 0591.10018