Related papers: Elliptic K3 surfaces admitting a Shioda-Inose stru…

We study the global structure of the gauge group $G$ of F-theory compactified on an elliptic fibration $Y$. The global properties of $G$ are encoded in the torsion subgroup of the Mordell-Weil group of rational sections of $Y$. Generalising…

We consider K3 surfaces that possess certain automorphisms of prime order p>2 and we present, for these surfaces, a correspondence between the mirror symmetry of Berglund-Huebsch-Chiodo-Ruan and that for lattice polarized K3 surfaces…

We prove that the maximal singular fibres of elliptic K3 surfaces have type I_19 and I_14* unless the characteristic of the ground field is 2. In characteristic 2, the maximal singular fibres are I_18 and I_13*. The paper supplements work…

Let $\mathcal{W}\subset\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$ be a surface given by the vanishing of a $(2,2,2)$-form. These surfaces admit three involutions coming from the three projections…

This paper is concerned with the construction of extremal elliptic K3 surfaces. It gives a complete treatment of those fibrations which can be derived from rational elliptic surfaces by easy manipulations of their Weierstrass equations. In…

A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve $E/\mathbb{Q}$, a positive proportion of its quadratic twists $E^{(d)}$ have rank 1. Using tools from Galois cohomology, we give criteria on E and d which…

We describe the possible Mordell-Weil groups for degree 1 elliptic threefold with rational base and constant $j$-invariant. Moreover, we classify all such elliptic threefolds if the $j$-invariant is nonzero. We can use this classification…

We study the injectivity of the Kudla-Millson lift of genus 2 Siegel cusp forms, vector-valued with respect to the Weil representation associated to an even lattice L. We prove that if L splits off two hyperbolic planes and is of…

Let $K$ be a number field, $\bar{K}$ an algebraic closure of $K$ and $E/K$ an elliptic curve defined over $K$. In this paper, we prove that if $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for each $\sigma\in…

Let $S$ be a K3 surface with primitive curve class $\beta$. We solve the relative Gromov-Witten theory of $S \times \mathbb{P}^1$ in classes $(\beta,1)$ and $(\beta,2)$. The generating series are quasi-Jacobi forms and equal to a…

In this paper we complete the classification of the elliptic fibrations on K3 surfaces which admit a non-symplectic involution acting trivially on the N\'eron--Severi group. We use the geometric method introduced by Oguiso and moreover we…

This article primarily aims at classifying, on certain K3 surfaces, the elliptic fibrations induced by conic bundles on smooth del Pezzo surfaces. The key geometric tool employed is the Alexeev-Nikulin correspondence between del Pezzo…

We study the structure of complex points on real surfaces, embedded into complex Elliptic surfaces. We show, for example, that any compact surface has a totally real embedding into a blow-up of a K3 surface. We also exhibit smooth disc…

We use rational parametrizations of certain cubic surfaces and an explicit formula for descent via 3-isogeny to construct the first examples of elliptic curves E_k: x^3 + y^3 = k of ranks 8, 9, 10, and 11 over Q. As a corollary we produce…

Let $K$ be a finitely generated field over $\mathbb{Q}$. Let $\mathcal{X}\to \mathcal{B}$ be a family of elliptic surfaces over $K$ such that each elliptic fibration has the same configuration of singular fibers. Let $r$ be the minimum of…

We compute the Donaldson-Thomas invariants of a local elliptic surface with section. We introduce a new computational technique which is a mixture of motivic and toric methods. This allows us to write the partition function for the…

Let S be a nonsingular projective K3 surface. Motivated by the study of the Gromov-Witten theory of the Hilbert scheme of points of S, we conjecture a formula for the Gromov-Witten theory (in all curve classes) of the Calabi-Yau 3-fold S x…

We classify pairs $(S, \gamma)$, consisting of a rational elliptic surface $S$ and a Galois cover $\gamma$ of the base, which satisfy a condition we call $\mathcal{L}$-stability. We explain how to use the theory of Mordell-Weil lattices to…

K3 surfaces have been studied from many points of view, but the positivity of the cotangent bundle is not well understood. In this paper we explore the surprisingly rich geometry of the projectivised cotangent bundle of a very general…

This paper concerns complex algebraic K3 surfaces with an automorphism which acts trivially on the Neron-Severi group. Complementing a result by Vorontsov and Kondo, we determine those K3 surfaces where the order of the automorphism is a…