A generalization of the Jacobian variety

Author(s):  
A. N. Paršin
Keyword(s):  
1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.


1982 ◽  
Vol 88 ◽  
pp. 197-212 ◽  
Author(s):  
Tsutomu Sekiguchi

Throughout the paper, a scheme means a noetherian scheme. By a curve C over a scheme S of genus g, we mean a proper and smooth S-scheme with irreducible curves of genus g as geometric fibres. In the previous paper [15], the author showed that the field of moduli for a non-hyperelliptic curve over a field coincides with the one for its canonically polarized jacobian variety, and in [16], he gave a partial result on the coincidence of the fields of rationality for a hyperelliptic curve and for its canonically polarized jacobian variety. In the present paper, we will discuss the isomorphy of the isomorphism schemes of two curves over a scheme and of their canonically polarized jacobian schemes, by using Oort-Steenbrink’s result [12].


2016 ◽  
Vol 12 (08) ◽  
pp. 2241-2264
Author(s):  
Alan Hertgen

Let [Formula: see text] be a complete discrete valuation field. Let [Formula: see text] be its ring of integers. Let [Formula: see text] be its residue field which we assume to be algebraically closed of characteristic exponent [Formula: see text]. Let [Formula: see text] be a semi-abelian variety. Let [Formula: see text] be its Néron model. The special fiber [Formula: see text] is an extension of the identity component [Formula: see text] by the group of components [Formula: see text]. We say that [Formula: see text] has split reduction if this extension is split. Whereas [Formula: see text] has always split reduction if [Formula: see text] we prove that it is no longer the case if [Formula: see text] even if [Formula: see text] is tamely ramified. If [Formula: see text] is the Jacobian variety of a smooth proper and geometrically connected curve [Formula: see text] of genus [Formula: see text], we prove that for any tamely ramified extension [Formula: see text] of degree greater than a constant, depending on [Formula: see text] only, [Formula: see text] has split reduction. This answers some questions of Liu and Lorenzini.


Author(s):  
Jean-Marc Couveignes

This chapter addresses the problem of computing in the group of lsuperscript k-torsion rational points in the Jacobian variety of algebraic curves over finite fields, with an application to computing modular representations. An algorithm in this chapter usually means a probabilistic Las Vegas algorithm. In some places it gives deterministic or probabilistic Monte Carlo algorithms, but this will be stated explicitly. The main reason for using probabilistic Turing machines is that there is a need to construct generating sets for the Picard group of curves over finite fields. Solving such a problem in the deterministic world is out of reach at this time. The unique goal is to prove, as quickly as possible, that the problems studied in this chapter can be solved in probabilistic polynomial time.


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