A new conjectured Jacquet-Langlands-type correspondence between degree-two Hermitian cusp forms and SO(6) algebraic modular forms, with matching L-functions, backed by dimension and eigenform computations.
The basis problem for modular forms for the Weil representation
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abstract
The vector valued theta series of a positive-definite even lattice is a modular form for the Weil representation of $\mathrm{SL}_2(\mathbb{Z})$. We show that the space of cusp forms for the Weil representation is generated by such functions. This gives a positive answer to Eichler's basis problem in this case. As applications we derive Waldspurger's result on the basis problem for scalar valued modular forms and give a new proof of the surjectivity of the Borcherds lift based on the analysis of local Picard groups.
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Hermitian modular forms and algebraic modular forms on $SO(6)$
A new conjectured Jacquet-Langlands-type correspondence between degree-two Hermitian cusp forms and SO(6) algebraic modular forms, with matching L-functions, backed by dimension and eigenform computations.