Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-14T07:26:20.276Z Has data issue: false hasContentIssue false
  • English
  • Français

Semistable models of elliptic curves over residue characteristic 2

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  29 April 2020

Jeffrey Yelton
Jeffrey Yelton*
Affiliation:
Emory University, Mathematics & Science Center, Suite E431, Atlanta, Georgia30322, USA
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Given an elliptic curve E in Legendre form $y^2 = x(x - 1)(x - \lambda )$ over the fraction field of a Henselian ring R of mixed characteristic $(0, 2)$, we present an algorithm for determining a semistable model of E over R that depends only on the valuation of $\lambda $. We provide several examples along with an easy corollary concerning $2$-torsion.

MSC classification

Primary: 11G07: Elliptic curves over local fields
Secondary: 11G20: Curves over finite and local fields
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 154 - 162
Check for updates
Copyright
© Canadian Mathematical Society 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bouw, I. I. and Wewers, S., Semistable reduction of curves and computation of bad Euler factors of L-functions. ICERM course notes, 2015.Google Scholar
Coleman, R., Computing stable reductions. In: Séminaire de théorie des nombres, Paris 1985–86, Prog. Math., 71, Springer, 1987, pp. 1–18. https://doi.org/10.1007/9781475742671Google Scholar
Lehr, C. and Matignon, M., Wild monodromy and automorphisms of curves. Duke Math. J. 135(2006), 569–586. https://doi.org/10.1215/S0012-7094-06-13535-4Google Scholar
Serre, J.-P.. Abelian ℓ-adic representations and elliptic curves. 2nd ed., Addison-Wesley, Advanced Book Program, Redwood City, CA, 1989. https://doi.org/10.1201/9781439863862Google Scholar
Silverman, J. H., The arithmetic of elliptic curves. 2nd ed., Graduate Texts in Mathematics, 106, Springer, Dordrecht, 2009. https://doi.org/10.1007/978-0-387-09494-6CrossRefGoogle Scholar