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Algebraic theta functions and the \(p\)-adic interpolation of Eisenstein-Kronecker numbers. (English) Zbl 1205.11076
Let \(\Gamma\) be a lattice in \(\mathbb C\), and let \(A\) denote the area of its fundamental domain divided by \(\pi\). Then for integers \(a\) and \(b > a+2\), and for complex numbers \(z\) and \(w\), the Eisenstein-Kronecker numbers (see in particular, A. Weil’s book [Elliptic functions according to Eisenstein and Kronecker. Berlin: Springer (1999; Zbl 0955.11001)]) are defined by \[ e_{a,b}^*(z,w) = \sum_{\gamma \in \Gamma \setminus \{-z\}} \frac{(\overline{z} + \overline{\gamma})^a}{(z+\gamma)^b}\, \langle \gamma, w \rangle_\Gamma, \] where \[ \langle z, w \rangle_\Gamma = \exp\Big(\frac{z\overline{w} - w \overline{z}}A\Big). \] The sum converges for all \(b > a-2\), but can be given a meaning for any \(a \geq 0\) and \(b > 0\) by analytic continuation. The Eisenstein-Kronecker numbers can be used to express special values of Hecke \(L\)-functions associated to the field of complex multiplication of a lattice \(\Gamma\), just as generalized Bernoulli numbers can be used to express special values of Dirichlet \(L\)-functions.
In Section 1, the authors introduce theta functions attached to the Poincaré bundle of elliptic curves, and show that the Kronecker theta function is a generating function for the Eisenstein-Kronecker numbers. Section 2 covers algebraic and \(p\)-adic properties of reduced theta functions on abelian varieties with complex multiplication, and in Section 3 these are applied to the construction of a \(p\)-adic measure and R. I. Yager’s [Ann. Math. (2) 115, 411–449 (1982; Zbl 0496.12010)] \(p\)-adic \(L\)-function. Section 4 is devoted to the calculation of the Laurent expansion of the Kronecker theta function. In a subsequent article, the authors intend to study the \(p\)-divisibility of critical values of Hecke \(L\)-functions associated to Hecke characters of complex quadratic number fields.

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14K25 Theta functions and abelian varieties
14K22 Complex multiplication and abelian varieties
14K05 Algebraic theory of abelian varieties
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