We describe an algebra of meromorphic functions on the Siegel domain genus two which contains modular forms for arithmetic index six subgroup symplectic group, is closed under three canonical derivations domain. The main ingredients our study are moduli enhanced curves, Gauss-Manin connection and vector fields living such spaces.

Carl Ludwig Siegel showed in [Siegel 1969] (English translation, [Siegel 1980]) that the constant terms of certain level one negative-weight modular forms Th are non-vanishing (“ Satz 2 ”), and that this implies an upper bound on the least positive exponent of a non-zero Fourier coefficient for any level one entire modular form of weight h with a non-zero constant term. Level one theta function...

We consider a special theta lift $\theta(f)$ from cuspidal Siegel modular forms $f$ on $\mathrm{Sp}_4$ to "modular forms" $\mathrm{SO}(4,4)$. This can be considered an analogue of the Saito-Kurokawa lift, where now image is representations $\mathrm{SO}(4,4)$ that are quaternionic at infinity. relate Fourier coefficients those $f$, and in particular prove nonzero has algebraic if does. Restricti...

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 and 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 s...