We compute all K3 surfaces with Picard rank 20 over Q. Our proof uses modularity, the Artin-Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell-Weil rank 18 over Q is impossible for an elliptic K3 surface. We also apply our methods to general singular K3 surfaces, i.e. with geometric Picard rank 20, but not necessaril...

We study the orbits of a polynomial f ∈ C[X], namely the sets {α, f(α), f(f(α)), . . . } with α ∈ C. We prove that if two nonlinear complex polynomials f, g have orbits with infinite intersection, then f and g have a common iterate. More generally, we describe the intersection of any line in C with a d-tuple of orbits of nonlinear polynomials, and we formulate a question which generalizes both ...

Inspired by work of McMullen, we show that any orbit of the diagonal group in the space of lattices accumulates on the set of stable lattices. As consequences, we settle a conjecture of Ramharter concerning the asymptotic behavior of the Mordell constant, and reduce Minkowski’s conjecture on products of linear forms to a geometric question, yielding two new proofs of the conjecture in dimension...

We give a structure theorem for the $m$-torsion of Jacobian general superelliptic curve $y^m=F(x)$. study existence torsion on curves form $y^q=x^p-x+a$ over finite fields characteristic $p$. apply those results to bound from below Mordell-Weil ranks Jacobians certain $\mathbb Q$.

We study the growth of the Mordell-Weil groups E(Kn) of an elliptic curve E as Kn runs through the intermediate fields of a Zp-extension. We describe those Zp-extensions K∞/K where we expect the ranks to grow to infinity. In the cases where we know or expect the rank to grow, we discuss where we expect to find the submodule of universal norms. 2000 Mathematics Subject Classification: Primary 11...