An exact control theorem is proven for nearly ordinary $p$-adic automorphic forms on symplectic and unitary groups over totally real fields if the algebraic group is split at $p$. In particular, a given nearly ordinary holomorphic Hecke eigenform can be lifted to a family of holomorphic Hecke eigenforms indexed by weights of the standard maximal split torus of the group. Their $q$-expansion coefficients are Iwasawa functions on the Iwasawa algebra of ${\mathbb{Z}}_p$-points of the split torus. The method is applicable to any reductive algebraic groups yielding Shimura varieties of PEL type under mild assumptions on the existence of integral toroidal compactification of the variety. Even in the Hilbert modular case, the result contains something new: freeness of the universal nearly ordinary Hecke algebra over the Iwasawa algebra, which eluded my scrutiny when I studied general theory of the $p$-adic Hecke algebra in the 1980s.

AMS 2000 Mathematics subject classification: Primary 11F03; 11F30; 11F33; 11F41; 11F60; 11G15; 11G18

In this paper, we construct explicitly endoscopic representations of $\widetilde{\mathrm{Sp}}_{2n}$, the metaplectic cover of a symplectic group of rank $n$. We do this in the automorphic case, and also in the local case, over a $p$-adic field.

AMS 2000 Mathematics subject classification: Primary 11F70

We show that a continuous, odd, regular (non-exceptional), ordinary, irreducible, two-dimensional, $l$-adic representation of the absolute Galois group of the rational numbers is modular over some totally real field. We deduce that it occurs in the $l$-adic cohomology of some variety over the rationals and that its $L$-function has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

AMS 2000 Mathematics subject classification: Primary 11F80. Secondary 11G40

We construct embedded periodic minimal surfaces with planar ends. We use the Weierstrass representation and solve the period problem using the implicit function theorem at a singular point.

AMS 2000 Mathematics subject classification: Primary 53A10