arXiv subjects
Publications and source records attributed to Matthias Schuett.
We study the elliptic modular surface attached to the commutator subgroup of the modular group. This has an elliptic curve as base and only one singular fibre. We employ an algebraic approach and then consider some arithmetic questions.
This paper gives upper and lower bounds for the degree of the field of definition of a singular K3 surface, generalising a recent result by Shimada. We use work of Shioda-Mitani and Shioda-Inose and classical theory of complex multiplication.
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 particular, this approach enables us to find explicit equations for 38 semi-stable extremal elliptic K3 fibrations, 32 of which are indeed defined over the rationals. They are realized as pull-back of non-semi-stable extremal rational elliptic surfaces via base change. This is related to work of J. Top and N. Yui which exhibited the same procedure for the semi-stable extremal rational elliptic 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 of Shioda, published in 2003 and 2005.
In this paper we discuss recent progress on the modularity of Calabi-Yau varieties. We focus mostly on the case of surfaces and threefolds. We will also discuss some progress on the structure of the L-function in connection with mirror symmetry. Finally, we address some questions and open problems.
This paper studies the arithmetic of the extremal elliptic K3 surface with configuration of singular fibres [19,1,1,1,1,1]. We give a model over Q such that the Neron Severi group is generated by divisors over Q, and we describe the local Hasse-Weil zeta-functions in terms of a modular form of weight 3. Furthermore we verify the Tate conjecture for the reduction at 3 and comment on a conjecture of T. Shioda concerning the similarity of the lattice of transcendental cycles and a lattice resulting from supersingular reduction.
We investigate the modularity of three non-rigid Calabi-Yau threefolds with bad reduction at 11 which arise as fibre products of rational elliptic surfaces. For this purpose, we apply a method by Serre to compare two-dimensional 2-adic Galois representations with uneven trace. Hereby, we associate two of the threefolds (or, more precisely, a two-dimensional piece of their middle cohomology group) to newforms of weight 4 and level 22 and 55, respectively. This enables us to compute the zeta-functions of these varieties up to the Euler factors of the bad primes.
This paper presents five new examples of modular rigid Calabi-Yau threefolds arising from the modular elliptic surface of level 6. Explicit correspondences to newforms of weight 4 and level 10, 17, 21, and 73 are given.