Pursuing ideas in [6], we determine the isometry classes of unimodular lattices of rank $28$, as well as the isometry classes of unimodular lattices of rank $29$ without nonzero vectors of norm $\leq 2$. We also provide some invariant that allows to distinguish these lattices and to independently check with a computer that our lists are complete.

We construct the $p$-adic standard $L$-functions for ordinary families of Hecke eigensystems of the symplectic group $\operatorname{Sp}(2n)_{/\mathbb{Q}}$ using the doubling method. We explain a clear and simple strategy of choosing the local sections for the Siegel Eisenstein series on the doubling group $\operatorname{Sp}(4n)_{/\mathbb{Q}}$, which guarantees the nonvanishing of local zeta integrals and allows us to $p$-adically interpolate the restrictions of the Siegel Eisenstein series to $\operatorname{Sp}(2n)_{/\mathbb{Q}}\times \operatorname{Sp}(2n)_{/\mathbb{Q}}$.

We define weight changing operators for automorphic forms on Grassmannians, that is, on orthogonal groups, and investigate their basic properties. We then evaluate their action on theta kernels, and prove that theta lifts of modular forms, in which the theta kernel involves polynomials of a special type, have some interesting differential properties.

We prove vanishing of the μ-invariant of the p-adic Katz L-function in N. M. Katz [p-adic L-functions for CM fields, Invent. Math. 49 (1978), 199–297].

We explain three methods for showing that the $p$-adic monodromy of a modular family of abelian varieties is ‘as large as possible', and illustrate them in the case of the ordinary locus of the moduli space of $g$-dimensional principally polarized abelian varieties over a field of characteristic $p$. The first method originated from Ribet's proof of the irreducibility of the Igusa tower for Hilbert modular varieties. The second and third methods both exploit Hecke correspondences near a hypersymmetric point, but in slightly different ways. The third method was inspired by work of Hida, plus a group theoretic argument for the maximality of $\ell$-adic monodromy with $\ell\neq p$.

It has been known that mixed automorphic forms arise naturally as holomorphic forms on elliptic varieties and that they include classical automorphic forms as a special case. In this paper, we show how to construct mixed automorphic forms of type (k, l) from elliptic modular forms to give nontrivial examples of mixed automorphic forms.