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Note on a meromorphic Jacobi form. (Note sur une forme de Jacobi méromorphe.) (French. Abridged English version) Zbl 0885.11035

Summary: Let \(L\) be a complex lattice. Our object of study is a function \(D_L(z;\varphi)\), periodic with period lattice \(L\) in the second variable, and analytic in the first variable with normalization condition \(\lim_{z\to 0} zD_L(z;\varphi)= 1\); up to an exponential factor, this function is related to the form \(F_\tau(u, v)= \theta'(0)\theta(u+ v)/(\theta(u)\theta(v))\) [see §3 of D. Zagier, Invent. Math. 104, 449-465 (1991; Zbl 0742.11029)], analytic in \(\tau\in{\mathcal H}\) (the upper half plane) and in \(u,v\in\mathbb{C}\), with \((u,v)\) proportional to \((z,\varphi)\) and \(\theta\) the Jacobi triple product.
Our main result is that \(D_L\) also satisfies a simple additive distribution relation. Indeed, if \(\Lambda\) is a lattice such that \(L\subset\Lambda\) and \([\Lambda:L]= l\), we have: \[ \sum_t D_L(lz; \varphi+ t)= D_\Lambda(z;\varphi), \] where \(t\) runs over a representative system of \(\Lambda/L\). When \(\varphi\) is a torsion point of \(\mathbb{C}/L\), we recover known results.

MSC:

11F50 Jacobi forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations

Citations:

Zbl 0742.11029
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