Theta Lifts of Bianchi Modular Forms and Applications to Paramodularity

Abstract

We explain how the work of Johnson-Leung and Roberts on lifting Hilbert modular forms for real quadratic fields to Siegel modular forms can be adapted to imaginary quadratic fields. For this we use archimedean results from Harris, Soudry, Taylor and replace the global arguments of Roberts by the non-vanishing result of Takeda. As an application of our lifting result, we exhibit an abelian surface $B$ defined over $\mathbb{Q}$, which is not restriction of scalars of an elliptic curve and satisfies the Brumer-Kramer Paramodularity Conjecture.

References in corpus (5)

• K3 surfaces and equations for Hilbert modular surfaces
• Examples of abelian surfaces with everywhere good reduction
• The cohomology of lattices in SL(2,C)
• Elliptic Curves over Real Quadratic Fields are Modular
• Invariant representations of GSp(2) under tensor product with a quadratic character

Cited by in corpus (7)

• Genuine Bianchi modular forms of higher level, at varying weight and discriminant
• Certain Abelian varieties bad at only one prime
• Examples of genuine QM abelian surfaces which are modular
• Non-vanishing of fundamental Fourier coefficients of paramodular forms
• On fundamental Fourier coefficients of Siegel cusp forms of degree 2
• On the paramodularity of typical abelian surfaces (and reduction of G-covariant bilinear forms)
• Universal Fourier expansions of Bianchi modular forms