Given a family of Abelian varieties over a positive-dimensional base, we prove that for a sufficiently general curve in the base, every rational section of the family over the curve is contained in a unique rational section over the entire base. 1. Main results sec-intro The starting point for this article is the following theorem. Theorem 1.1. [GHMS05, Theorem 6.2] Let B be a smooth, quasi-pro...

Introduction Recall that for proper smooth and connected curves of genus g ≥ 2 over an algebraically closed eld of characteristic 0 the structure of the étale fundamental group π g is well known and depends only on the genus g. Namely it is the pro-nite completion of the topological fundamental group of a compact orientable topological surface of genus g. In contrast to this, the structure of t...

Early Days of the Department Education at University Buckingham: A Personal View

Corollary 3 of [4] is not known unconditionally, as cohomological automorphic forms on GL2 over an imaginary quadratic field are not known to satisfy the Ramanujan conjecture. We shall briefly describe the reason for this and discuss what information Theorem 1 of [4] does give in the case of imaginary quadratic fields. Let K be an imaginary quadratic field with nontrivial automorphism c, and le...

We present the formalization of Dirichlet’s theorem on the infinitude of primes in arithmetic progressions, and Selberg’s elementary proof of the prime number theorem, which asserts that the number π(x) of primes less than x is asymptotic to x/ log x, within the proof system Metamath.

Let X be a connected normal complex space of dimension n ≥ 2 which is (n − 1)-complete, and let π : M → X be a resolution of singularities. By use of Takegoshi’s generalization of the Grauert-Riemenschneider vanishing theorem, we deduce H cpt(M,O) = 0, which in turn implies Hartogs’ extension theorem on X by the ∂-technique of Ehrenpreis.

Let X be a connected normal complex space of dimension n ≥ 2 which is (n − 1)-complete, and let π : M → X be a resolution of singularities. By use of Takegoshi’s generalization of the Grauert-Riemenschneider vanishing theorem, we deduce H cpt(M,O) = 0, which in turn implies Hartogs’ extension theorem on X by the ∂-technique of Ehrenpreis.