## 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

Zbl 0742.11029
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