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ON THE L-FUNCTION OF THE CURVES $\lowercase{y}^2 = \lowercase{x}^5 + A$

Published online by Cambridge University Press:  25 September 2003

MICHAEL STOLL  and
TONGHAI YANG
MICHAEL STOLL
Affiliation:
School of Engineering and Science, International University Bremen, PO Box 75 05 61, 28725 Bremen, Germanym.stoll@iu-bremen.de
TONGHAI YANG
Affiliation:
School of Engineering and Science, International University Bremen, PO Box 75 05 61, 28725 Bremen, Germanym.stoll@iu-bremen.de
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Abstract

Let $C_A/\Q$ be the curve $y^2 = x^5 + A$, and let $L(s, J_A)$ denote the $L$-series of its Jacobian. Under the assumption that the sign in the functional equation for $L(s, J_A)$ is $+1$, the critical value $L(1, J_A)$ is evaluated in terms of the value of a theta series for $\Q(\sqrt{5})$ depending on $A$ at a complex multiplication point coming from $\Q(\zeta_5)$.

Keywords

11G4011G3011G1511F2711F41
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 68 , Issue 2 , October 2003 , pp. 273 - 287
Copyright
The London Mathematical Society 2003

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