We use C-algebras in the context of modular curves and holomorphic 1-forms. We focus on the “leaf space” X and C-algebra C(X) of the Weil 1-form ωN on the modular curve X0(N). It is proved that K0(C (X)) is isomorphic to a stationary dimension group of rank 4g − 3. An application of this result to Hilbert’s 12th problem is discussed.

We give an overview over recent results concerning rational points on hyperelliptic curves. One result says that ‘most’ hyperelliptic curves of high genus have very few rational points. Another result gives a bound on the number of rational points in terms of the genus and the Mordell-Weil rank, provided the latter is sufficiently small. The first result relies on work by Bhargava and Gross on ...

We study the growth of the Mordell-Weil groups E(Kn) of an elliptic curve E as Kn runs through the intermediate fields of a Zp-extension. We describe those Zp-extensions K∞/K where we expect the ranks to grow to infinity. In the cases where we know or expect the rank to grow, we discuss where we expect to find the submodule of universal norms. 2000 Mathematics Subject Classification: Primary 11...

Given a basis of pseudoidentities for a pseudovariety of ordered semigroups containing the 5-element aperiodic Brandt semigroup B2, under the natural order, it is shown that the same basis, over the most general graph over which it can be read, defines the global. This is used to show that the global of the level 3/2 of the refinement of Straubing-Thérien’s concatenation hierarchy introduced by...

This article deals with a constructive aspect of Hilbert’s seventeenth problem : producing a collection of real polynomials in two variables of degree 8 in one variable which are positive but are not a sum of three squares of rational fractions. To do this we use a reformulation of this problem in terms of hyperelliptic curves due to Huisman and Mahé and we follow a method from Cassels, Ellison...

The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch and Swinnerton-Dyer, Bloch, and Beilinson relate the orders of vanishing of ...

Let S be a surface with genus g and n boundary components and let d(S) = 3g − 3 + n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions CP(S) prove that the Weil-Petersson metric on Teichmüller space Teich(S) is Gromov-hyperbolic if and only if d(S) ≤ 2. When d(S) ≥ 3 the Weil-Petersson metric has higher rank in the sense...

Let E : y = F (x) be an elliptic curve over Q defined by a monic irreducible integral cubic polynomial F (x) with negative and square-free discriminant −D. We determine its 2-Selmer rank in terms of the 2-rank of the class group of the cubic field L = Q[x]/F (x). We then interpret this result as a mod 2 congruence between the Hasse-Weil L-function of E and a degree two Artin L-function associat...