Lower bounds for relative class numbers of imaginary abelian number fields and CM-fields

Louboutin, Stéphane

doi:10.4064/aa121-3-1

Cited by 5 publications

(2 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here h K and h + K denote the class numbers of K and its real quadratic subfield K + . In [18,Corollaries 29 and 32], we find the Brauer-Siegel type result that the logarithm of h − K is asymptotic to 1 2…”

Section: Genus-2 Curves and Jacobiansmentioning

confidence: 88%

Genus-2 curves and Jacobians with a given number of points

Bröker¹,

Howe²,

Lauter³

et al. 2015

LMS J. Comput. Math.

View full text Add to dashboard Cite

We study the problem of efficiently constructing a curve C of genus 2 over a finite field F for which either the curve C itself or its Jacobian has a prescribed number N of F-rational points. In the case of the Jacobian, we show that any `CM-construction' to produce the required genus-2 curves necessarily takes time exponential in the size of its input. On the other hand, we provide an algorithm for producing a genus-2 curve with a given number of points that, heuristically, takes polynomial time for most input values. We illustrate the practical applicability of this algorithm by constructing a genus-2 curve having exactly 10^2014 + 9703 (prime) points, and two genus-2 curves each having exactly 10^2013 points. In an appendix we provide a complete parametrization, over an arbitrary base field k of characteristic neither 2 nor 3, of the family of genus-2 curves over k that have k-rational degree-3 maps to elliptic curves, including formulas for the genus-2 curves, the associated elliptic curves, and the degree-3 maps.Comment: Made a number of clarifications and corrected some typographical error

show abstract

Section: Genus-2 Curves and Jacobiansmentioning

confidence: 88%

Genus-2 curves and Jacobians with a given number of points

Bröker¹,

Howe²,

Lauter³

et al. 2015

LMS J. Comput. Math.

View full text Add to dashboard Cite

show abstract

“…For (5), see [15,19]. For (6), see [23 Theorem 7;28], in which a uniform proof of (5) and (6) is given. However, we may take…”

Section: Bounds On Residues Of Zeta Functionsmentioning

confidence: 99%

The Brauer–siegel Theorem

Louboutin

2005

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

Explicit bounds are given for the residues at s = 1 of the Dedekind zeta functions of number fields. As a consequence, a simple proof of the Brauer-Siegel theorem and explicit lower bounds for class numbers of number fields are obtained. Compared with Stark's original approach, the paper is explicit and more satisfactory for number fields containing quadratic subfields. Examples are given of fully explicit lower bounds for class numbers of various types of number fields, for example normal and non-normal number fields of odd degree, with an emphasis on cubic fields, real cyclic quartic number fields, and non-normal quartic number fields containing an imaginary quadratic subfield.

show abstract

Note on Bernoulli numbers associated to some Dirichlet character of prime conductor

Ichimura

2016

Arch. Math.

View full text Add to dashboard Cite

Lower bounds for relative class numbers of imaginary abelian number fields and CM-fields

Cited by 5 publications

References 25 publications

Genus-2 curves and Jacobians with a given number of points

Genus-2 curves and Jacobians with a given number of points

The Brauer–siegel Theorem

Note on Bernoulli numbers associated to some Dirichlet character of prime conductor

Contact Info

Product

Resources

About