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On the critical values of Hecke \(L\)-series. (English) Zbl 0462.14015
Let \(E\) be an elliptic curve over \(\mathbb Q\) with global minimal Weierstrass model \[ y^2+xy=x^3-x^2-2x-1. \] The conductor of \(E\) is \(N=(7^2)\). The curve has complex multiplication over \(K=\mathbb Q(\sqrt{-7})\) by \(\mathfrak O=\mathbb Z[\tfrac12(1+\sqrt{-7})]\). Thus the fundamental real period \(\Omega\) of the Néron differential \(\omega=\frac{dx}{2y+x}\) can be determined explicitly. Moreover, the \(L\)-series of \(E\) is equal to the \(L\)-series of a Hecke character \(\chi\) of \(K\): the conductor of \(\chi\) is the ideal \((\sqrt{-7})\) and for an ideal \(\mathfrak A\) of \(K\) prime to 7 we have \(\chi(\mathfrak A)=\alpha\) where \(\mathfrak A=(\alpha)\) and \(\alpha^3\equiv 1\pmod{\sqrt{-7}}\). The authors have calculated the central critical values of the Hecke \(L\)-series which are associated to odd powers of \(\chi\), i. e. \(L(\chi^{2n-1},s)\) for \(s=n\). They find that for even \(n\) these values vanish; for odd \(n\) \((3\leq n\leq 33)\) they calculate the integers \(a_n\) (also \(a_1=\tfrac12\)) given by \(L(\chi^{2n-1},n)=\Omega^{2n-1}a_n/(n-1)!(2\pi i/\sqrt{-7})^{n-1}\). For a few primes \(p\equiv 1\pmod 4\) similar results are obtained for an elliptic curve \(E_p/\mathbb Q\) which becomes isomorphic to \(E\) over \(\mathbb Q(\sqrt p)\).
Reviewer: R. J. Stroeker

14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11R42 Zeta functions and \(L\)-functions of number fields
11G10 Abelian varieties of dimension \(> 1\)
14K15 Arithmetic ground fields for abelian varieties
Full Text: DOI Numdam EuDML
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