Rocky Mountain Journal of Mathematics

On quadratic twists of hyperelliptic curves

Mohammad Sadek

Full-text: Open access

Abstract

Let $C$ be a hyperelliptic curve of good reduction defined over a discrete valuation field $K$ with algebraically closed residue field $k$. Assume moreover that $\text{char\,} k\ne2$. Given $d\in K^*\setminus K^{*2}$, we introduce an explicit description of the minimal regular model of the quadratic twist of $C$ by $d$. As an application, we show that if $C/\q$ is a nonsingular hyperelliptic curve given by $y^2=f(x)$ with $f$ an irreducible polynomial, there exists a positive density family of prime quadratic twists of $C$ which are not everywhere locally soluble.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 3 (2014), 1015-1026.

Dates
First available in Project Euclid: 28 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1411945677

Digital Object Identifier
doi:10.1216/RMJ-2014-44-3-1015

Mathematical Reviews number (MathSciNet)
MR3264495

Zentralblatt MATH identifier
1304.14041

Citation

Sadek, Mohammad. On quadratic twists of hyperelliptic curves. Rocky Mountain J. Math. 44 (2014), no. 3, 1015--1026. doi:10.1216/RMJ-2014-44-3-1015. https://projecteuclid.org/euclid.rmjm/1411945677


Export citation

References

  • L. Halle, Stable reduction of curves and tame ramification, Math. Z. \bf265 (2009), 529-550.
  • Q. Liu, Modèles entiers des courbes hyperelliptiques sur un corps de valuations discrète, Trans. Amer. Math. Soc. \bf348 (1996), 4577-4610.
  • –––, Algebraic geometry and arithmetic curves, Oxford Grad. Texts Math. \bf6, Oxford University Press, Oxford, 2002.
  • D. Lorenzini, Models of curves and wild ramification, Pure Appl. Math. Quart., to appear.
  • –––, Wild quotient singularities of surfaces, preprint.
  • –––, Dual graphs of degenerating curves, Math. Ann. \bf287 (1990), 135-150.
  • J.S. Milne, Algebraic number theory (v3.01), 2008, available at www.jmilne.org/math/.
  • M.R. Murty and J. Esmonde, Problems in algebraic number theory, Grad. Texts Math. \bf190, Springer-Verlag, 2005.
  • Y. Namikawa and K. Ueno, The complete classification of fibres in pencils of curves of genus two, Manuscr. Math. \bf9 (1973), 143-186.
  • A.N. Par\usin, Minimal models of curves of genus $2$ and homomorphisms of abelian varieties defined over a field of finite characteristic, Math. USSR \bf6 (1972), 65-108.
  • J. Silverman, Advanced topics in the arithmetic of elliptic curves, Grad. Texts Math. \bf151, Springer-Verlag, 1995.
  • H. Xue, Minimal resolution of Atkin-Lehner quotients of $X_0(N)$, J. Number Theor. \bf129 (2009), 2072-2092. \noindentstyle