Let X be a curve over a number field K with genus g > 2, p a prime of OK over an unramified rational prime p > 2r, J the Jacobian of X, r = rank J(K), and X a regular proper model of X at p. Suppose r < g. We prove that #X(K) 6 #X (Fp) + 2r, extending the refined version of the Chabauty–Coleman bound to the case of bad reduction. The new technical insight is to isolate variants of the classical...

Let C be a smooth projective curve defined over a number field k, X/k(C) a smooth projective curve of positive genus, JX the Jacobian variety of X and (τ, B) the k(C)/k-trace of JX . We estimate how the rank of JX(k(C))/τB(k) varies when we take an unramified abelian cover π : C ′ → C defined over k.

A simplified method of descent via isogeny is given for Jacobians of curves of genus 2. This method is then used to give applications of a theorem of Coleman for computing all the rational points on certain curves of genus 2. 0 Introduction The following classical result of Chabauty [3] is a curiosity of the literature in that there has been a 50 year period during which applications have been ...

Abstract We consider the problem of identifying a parallel Wiener-Hammerstein structure from Volterra kernels. Methods based on kernels typically resort to coupled tensor decompositions However, in case systems, such methods require nontrivial constraints factors decompositions. In this paper, we propose an entirely different approach: by using special sampling (operating) points for Jacobian n...

A nonlinear least squares problem with nonlinear constraints may be ill posed or even rank-deficient in two ways. Considering the problem formulated as minx 1/2‖f2(x)‖2 subject to the constraints f1(x) = 0, the Jacobian J1 = ∂f1/∂x and/or the Jacobian J = ∂f/∂x, f = [f1; f2], may be ill conditioned at the solution. We analyze the important special case when J1 and/or J do not have full rank at ...

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...

The existence problem for vector bundles on a smooth compact complex surface consists in determining which topological complex vector bundles admit holomorphic structures. For projective surfaces, Schwarzenberger proved that a topological complex vector bundle admits a holomorphic (algebraic) structure if and only if its first Chern class belongs to the Neron-Severi group of the surface. In con...

The existence problem for vector bundles on a smooth compact complex surface consists in determining which topological complex vector bundles admit holomorphic structures. For projective surfaces, Schwarzenberger proved that a topological complex vector bundle admits a holomorphic (algebraic) structure if and only if its first Chern class belongs to the Neron-Severi group of the surface. In con...

A nonlinear least squares problem with nonlinear constraints may be ill posed or even rank-deecient in two ways. Considering the problem formulated as min x 1=2kf 2 (x)k 2 2 subject to the constraints f 1 (x) = 0, the Jacobian J 1 = @f 1 =@x and/or the Jacobian J = @f=@x, f = f 1 ; f 2 ], may be ill conditioned at the solution. We analyze the important special case when J 1 and/or J do not have...

We show that for all odd primes p, there exist ordinary elliptic curves over Fp(x) with arbitrarily high rank and constant j-invariant. This shows in particular that there are elliptic curves with arbitrarily high rank over these fields for which the corresponding elliptic surface is not supersingular. The result follows from a theorem which states that for all odd prime numbers p and l, there ...