On points of Jacobian rank $k$. II

Counting Points on the Jacobian Variety of a

Counting the order of the Jacobian group of a hyperelliptic curve over a nite eld is very important for constructing a hyperelliptic curve cryptosystem (HECC), but known algorithms to compute the order of a Jacobian group over a given large prime eld need very long running times. In this note, we propose a practical polynomial-time algorithm to compute the order of the Jacobian group for a hype...

Nilpotent Symmetric Jacobian Matrices and the Jacobian Conjecture Ii

It is shown that the Jacobian Conjecture holds for all polynomial maps F : k → k of the form F = x + H , such that JH is nilpotent and symmetric, when n ≤ 4. If H is also homogeneous a similar result is proved for all n ≤ 5. Introduction Let F := (F1, . . . , Fn) : C → C be a polynomial map i.e. each Fi is a polynomial in n variables over C. Denote by JF := (i ∂xj )1≤i,j≤n, the Jacobian matrix ...

The Rank of the Jacobian of Modular Curves: Analytic Methods

On Some Recursive Residual Rank Tests for Change~points on Some Recursive Residual Rank Tests for Change-points

A general class of recursive residuals is incorporated in the formulation of suitable (aligned) rank tests for change-points pertaining to some simple linear models. The asymptotic theory of the proposed tests rests on some invariance principles for recursively aligned signed rand statistics, and these are developed. Along with the asymptotic properties of the proposed tests, allied efficiency ...

On the Variation of the Rank of Jacobian Varieties on Unramified Abelian Towers over Number Fields

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.