We determine all complex K3 surfaces with Picard rank 20 over Q. Here the NéronSeveri group has rank 20 and is generated by divisors which are defined 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 a...

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch and Swinnerton-Dyer, Bloch, and Beilinson relate the orders of vanishing of ...

Let C : Y 2 = anX n + · · · + a 0 be a hyperelliptic curve with the a i rational integers, n ≥ 5, and the polynomial on the right irreducible. Let J be its Jacobian. We give a completely explicit upper bound for the integral points on the model C, provided we know at least one rational point on C and a Mordell–Weil basis for J(Q). We also explain a powerful refinement of the Mordell–Weil sieve ...

We construct explicit examples of E8 lattices occurring in arithmetic for which the natural Galois action is equal to the full group of automorphisms of the lattice, i.e., the Weyl group of E8. In particular, we give explicit elliptic curves over Q(t) whose Mordell-Weil lattices are isomorphic to E8 and have maximal Galois action. Our main objects of study are del Pezzo surfaces of degree 1 ove...

CONTENTS We analyze the 128-dimensional Mordell-Weil lattice of a cer1 . Introduction j -a jn elliptic curve over the rational function field k(t)f where k is 2. Statement of Results a finite field of 2 elements. By proving that the elliptic curve 3. Proof of Rank, Discriminant and Tate-Safarevic Group has trivial Tate-Safarevic group and nonzero rational points of 4. Proof of Minimal Norm, Den...

Let $p$ be an odd prime. Associated to a pair $(E, \mathcal{F}_\infty)$ consisting of rational elliptic curve $E$ and $p$-adic Lie extension $\mathcal{F}_\infty$ $\mathbb{Q}$, is the $p$-primary Selmer group $Sel_{p^\infty}(E/\mathcal{F}_\infty)$ over $\mathcal{F}_\infty$. In this paper, we study arithmetic statistics for algebraic structure group. The results provide insights into asymptotics ...