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Some remarks on signs in functional equations. (English) Zbl 1057.11052

The Artin root number, \(W(\chi)\), occurs in the functional equation of the extended Artin \(L\)-function \(\widetilde L(s,\chi)\) and it has profound arithmetical consequences, for example for real valued \(\chi\) the sign of \(W (\chi)\) has an interpretation in terms of the existence of normal integral bases, a remarkable connection that led to the theory of Galois module structure [see A. Fröhlich, Galois module structure of algebraic integers. Springer-Verlag, Berlin (1983; Zbl 0501.12012)].
This paper, which very appropriately is dedicated to the memory of Robert Rankin, surveys the problem of determining the sign in functional equations and outlines questions of absorbing interest, referring the reader to authoritative sources for background.
Let \(k\) be a number field and let \(M\) be a pure motive of weight \(n\) over \(k\), and suppose that there exists a non-degenerate pairing \(M\times M\to\mathbb{Q}(-n)\). Such a pairing arises in the case when \(M\) is given by the cohomology in degree \(n\) of a projective, smooth geometrically connected variety, \(X\), of dimension \(n\) over \(k\) in which case the pairing is given by the cup product. The conjectured functional equation for the complete \(L\)-function of \(M\) (including local factors at infinity) has the form \[ L(M,s)= \varepsilon(M,s) L(M,n+1-s), \] where \(\varepsilon(M,s)= \varepsilon(M) f(M)^{(n+1)/2-s}\), where \(| f(M)|\geq 1\) is an integer and \(\varepsilon(M)= \pm 1\), the ‘sign’ of the title of the paper.
The case where \(L(M,s)\) possesses an analytical interpretation is similar to the one mentioned above and many very interesting formulae may be obtained by computing \(\varepsilon(M)\) in different ways. For example the calculation of the sign of the quadratic Gauss sum and Dirichlet’s interpretation of quadratic reciprocity in terms of the \(L\)-function, \(L(s,\chi)\), in the case \(k=\mathbb{Q}\) and \(\chi\) a quadratic character. The paper also contains references to such fundamental results; for example to Serre’s proof of Hecke’s theorem on the class of the absolute different, which occupies coronidis loco in A. Weil’s [Basic number theory. Springer-Verlag, Berlin (1967; Zbl 0176.33601)].
The author uses ideas derived from a paper by Fröhlich and the reviewer to obtain the corresponding result for the strict ideal class group.
This brief but fascinating survey ends with some remarks on signs in functional equations for Abelian varieties. Let \(A\) be an Abelian variety of dimension \(g\) over \(k\) and let \(M_A=H_1 (A)\), a motive of weight \(-1\) and rank \(2g\). The Weil pairing gives a non-degenerate, alternating bilinear form \[ M_A\times M_A\to\mathbb{Q} (1). \] Now denote by \(M_\rho\) an orthogonal motive of weight 0 and rank 2 over \(k\). Then one is led to consider the motive \(M=M_A\otimes M_\rho\) and it is plausible that the conductor \(f_A\) of \(A\) is prime to the conductor of \(M_\rho\) and \[ \varepsilon(M)= \chi(f_A)(-1)^{g\#T}, \] where \(\#T\) denote the number of real places in \(k\) that become complex in the quadratic étale extension corresponding to \(M_\rho\).
A method due to Rankin gives a proof of the functional equation for \(L(M, s)\) when \(k\) is totally real and \(A\) appears in connection with the Jacobian of a Shimura curve. Such considerations led the author to the starting point for his work with Zagier on the Birch and Swinnerton-Dyer conjecture.

MSC:

11R42 Zeta functions and \(L\)-functions of number fields
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