Modular Forms and Special Cubic Fourfolds

We study the degree of the special cubic fourfolds in the Hilbert scheme of cubic fourfolds via a computation of the generating series of Heegner divisors of even lattice of signature (2, 20).Comment: 13 page

Integrals on pp-adic upper half planes and Hida families over totally real fields

Bertolini-Darmon and Mok proved a formula of the second derivative of the two-variable $p$-adic $L$-function of a modular elliptic curve over a totally real field along the Hida family in terms of the image of a global point by some $p$-adic logarithm map. The theory of $p$-adic indefinite integrals and $p$-adic multiplicative integrals on $p$-adic upper half planes plays an important role in their work. In this paper, we generalize these integrals for $p$-adic measures which are not necessarily $\mathbf{Z}$-valued, and prove a formula of the second derivative of the two-variable $p$-adic $L$-function of an abelian variety of ${\rm GL}(2)$-type associated to a Hilbert modular form of weight 2.Comment: to appear in Osaka Journal of Mathematic

On the equidistribution of some Hodge loci

We prove the equidistribution of the Hodge locus for certain non-isotrivial, polarized variations of Hodge structure of weight $2$ with $h^{2,0}=1$ over complex, quasi-projective curves. Given some norm condition, we also give an asymptotic on the growth of the Hodge locus. In particular, this implies the equidistribution of elliptic fibrations in quasi-polarized, non-isotrivial families of $K3$ surfaces.Comment: 33 pages. To appear in Journal f\"ur die reine und angewandte Mathematik (Crelles Journal