We use Hodge theory to prove a new upper bound on the ranks of Mordell-Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension a...

The variety DA of finite monoids has a huge number of different characterizations, ranging from two-variable first-order logic FO to unambiguous polynomials. In order to study the structure of the subvarieties of DA, Trotter and Weil considered the intersection of varieties of finite monoids with bands, i.e., with idempotent monoids. The varieties of idempotent monoids are very well understood ...

Néron showed that an elliptic surface with rank 8, and with base B = P1Q, and geometric genus =0, may be obtained by blowing up 9 points in the plane. In this paper, we obtain parameterizations of the coefficients of the Weierstrass equations of such elliptic surfaces, in terms of the 9 points. Manin also describes bases of the Mordell-Weil groups of these elliptic surfaces, in terms of the 9 p...

Consider a smooth, geometrically irreducible, projective curve of genus $g\ge 2$ defined over number field degree $d \ge 1$. It has at most finitely many rational points by the Mordell Conjecture, theorem Faltings. We show that is bounded only in terms $g$, $d$ and Mordell–Weil rank curve's Jacobian, thereby answering affirmative question Mazur. In addition we obtain uniform bounds, $g$ $d$, fo...

Given a genus 2 curve C with rational Weierstrass point defined over number field, we construct family of 5 curves that realize descent by maximal unramified abelian two-covers C, and describe explicit models the isogeny classes their Jacobians as restrictions scalars elliptic curves. All constructions this paper are accompanied formulas implemented in Magma and/or SageMath. We apply these algo...

In this article we would like to address the following situation. Let f : X → C be an elliptic surface with section S0. We further assume that X is relatively minimal and smooth, and that all singular fibers are semistable, that is, are cycles of P’s (type “Im” in Kodaira’s notation, see [K]). Given a section S of f , it will meet one and only one component in every singular fiber. The fundamen...

The moduli space M of semi-stable rank 2 bundles with trivial determinant over a complex curve Σ carries involutions naturally associated to 2-torsion points on the Jacobian of the curve. For every lift of a 2-torsion point to a 4-torsion point, we define a lift of the involution to the determinant line bundle L. We obtain an explicit presentation of the group generated by these lifts in terms ...

We give a proof that the Riemann hypothesis for hypersurfaces over finite fields implies the result for all smooth proper varieties, by a deformation argument which does not use the theory of Lefschetz pencils or the `-adic Fourier transform.

Arguments against scientific misconduct one finds in the literature generally fail to support current policies on research fraud: they may not prove wrong what is typically considered research misconduct and they tend to make wrong things that are not usually seen as scientific fraud, in particular honest errors. I argue that society cannot set a rule enjoining scientists to be honest, so any s...

In this paper, we introduce and study two new types of non-abelian zeta functions for curves over finite fields, which are defined by using (moduli spaces of) semi-stable vector bundles and non-stable bundles. A Riemann-Weil type hypothesis is formulated for zeta functions associated to semi-stable bundles, which we think is more canonical than the other one. All this is motivated by (and hence...