In this paper we study the half-linear differential equation ( r(t)Φp(x ′) ) ′ + c(t)Φp(x) = 0, where Φp(x) = |x|p−2x, p > 1. Using modified Riccati technique and suitable local estimates for terms in modified Riccati equation we derive new characterization of principal solution and new nonoscillation criteria.

It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base, and in particular identifies the moduli space of cubic surfaces with an open substack of a certain moduli space of abelian varieties. This answers, away from the prime 2, an old question ...

Let X be a cubic surface over a local number field k. Given an Azumaya algebra on X, we describe the local evaluation map X(k)→ Q/Z in two cases, showing a sharp dependence on the geometry of the reduction of X. This extends results of Colliot-Thélène, Kanevsky and Sansuc.

We show that many of Ramanujan’s modular equations of degree 3 can be interpreted in terms of integral ternary quadratic forms. This way we establish that for any n ∈ N

For a smooth plane cubic B, we count curves C of degree d such that the normalizations of C\B are isomorphic to A, for d ≤ 7 (for d = 7 under some assumption). We also count plane rational quartic curves intersecting B at only one

We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power savings in the error term. We also obtain a large-sieve type result for order three (and six) Dirichlet characters.