Explicit formula for the $(\text{GL}_2, \text{GL}_2)$ theta lift via Bruhat decomposition

Peter Xu

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arXiv · 2409.06940

Explicit formula for the $(\text{GL}_2, \text{GL}_2)$ theta lift via Bruhat decomposition

Peter Xu

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Abstract

Using combinations of weight-1 and weight-2 of Kronecker-Eisenstein series to construct currents in the distributional de Rham complex of a squared elliptic curve, we find a simple explicit formula for the type II $(\text{GL}_2, \text{GL}_2)$ theta lift without smoothing, analogous to the classical formula of Siegel for periods of Eisenstein series. For $K$ a CM field, the same technique applies without change to obtain an analogous formula for the $(\text{GL}_2(K),K^\times)$ theta correspondence.

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Peter Xu. 2024-09-19. Explicit formula for the $(\text{GL}_2, \text{GL}_2)$ theta lift via Bruhat decomposition. https://arxiv.org/abs/2409.06940

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