We remark the density of the jumping loci of the Picard number of a hyperkähler manifold under small one-dimensional deformation and provide some applications for the Mordell-Weil groups of Jacobian K3 surfaces.

We propose a conjecture combining the Mordell-Lang conjecture with an important special case of the André-Oort conjecture, and explain how existing results imply evidence for it. Mathematics Subject Classification: 14G35, 14K12 (11F32, 11F72, 11G15, 11G18)

We ask questions generalizing uniform versions of conjectures of Mordell and Lang and combining them with the Morton–Silverman conjecture on preperiodic points. We prove a few results relating different versions of such questions.

A rational (resp. integral) distance set is a subset S of the plane R such that for all s, t ∈ S, the distance between s and t is a rational number (resp. is an integer). Huff [4] considered rational distance sets S of the following form: given distinct a, b ∈ Q∗, S contains the four points (0,±a) and (0,±b) on the y-axis, plus points (x, 0) on the x-axis, for some x ∈ Q∗. Such a point (x, 0) m...

This is an extended version of the talk I gave at the summer school in Göttingen in July 2007. We discuss the Mordell-Weil Sieve and some applications.

We prove certain uniform versions of the Mordell Conjecture and of the Shafarevich Conjecture for curves over function fields and their rational points.

We introduce finite and symmetric Mordell–Tornheim type of multiple zeta values give a new approach to the Kaneko–Zagier conjecture stating that satisfy same relations.

Let OK be any domain with field of fractions K . Let F(x, y) ∈ OK [x, y] be a homogeneous polynomial of degree n, coprime to y, and assumed to have unit content (i.e., the coefficients of F generate the unit ideal in OK ). Assume that gcd(n, char(K )) = 1. Let h ∈ OK and assume that the polynomial hzn − F(x, y) is irreducible in K[x, y, z]. We denote by X F,h/K the nonsingular complete model of...

For example, suppose that G ∼= S9 and H ∼= S8, where Sn denotes the symmetric group on n letters. Then ρ is one of the two absolutely irreducible representations of G of dimension 8, and according to our theorem there exists an elliptic curve E over M such that ρ occurs in Q⊗E(K). The other absolutely irreducible representation of G of dimension 8 is ρ ⊗ , where is the “sign” character of G, an...