Arithmetic of a singular K3 surface

This paper is concerned with the arithmetic of the elliptic K3 surface with configuration [1,1,1,12,3*]. We determine the newforms and zeta-functions associated to X and its twists. We verify conjectures of Tate and Shioda for the reductions of X at 2 and 3.Comment: 12 pages, 2 figures; final version: two typos corrected, references update

Dynamics on supersingular K3 surfaces

For any odd characteristic p=2 mod 3, we exhibit an explicit automorphism on the supersingular K3 surface of Artin invariant one which does not lift to any characteristic zero model. Our construction builds on elliptic fibrations to produce a closed formula for the automorphism's characteristic polynomial on second cohomology, which turns out to be an irreducible Salem polynomial of degree 22 with coefficients varying with p.Comment: 12 pages, 3 figures; v2: main result improved to Salem degree 2

CM newforms with rational coefficients

We classify newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. Under the extended Riemann hypothesis for odd real Dirichlet characters, these newforms are finite in number. We produce tables for weights 3 and 4, where finiteness holds unconditionally.Comment: 17 pages, 3 tables; final version: Rem 7.2, Thm 9.2 added, references updated, minor change

K3 surfaces with non-symplectic automorphisms of 2-power order

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 2-power and equals the rank of the transcendental lattice. We also study the arithmetic of these K3 surfaces and comment on mirror symmetryComment: 19 pages, 1 figure; v3: exposition improved thanks to referee's comment

Two lectures on the arithmetic of K3 surfaces

In these lecture notes we review different aspects of the arithmetic of K3 surfaces. Topics include rational points, Picard number and Tate conjecture, zeta functions and modularity.Comment: 26 pages; v4: typos corrected, references update

New examples of modular rigid Calabi-Yau threefolds

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.Comment: 9 pages; journal-ref. added; minor mistakes correcte