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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
Page: Show Scanned Page ##### MSC:
 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
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