## Rocky Mountain Journal of Mathematics

### On quadratic twists of hyperelliptic curves

#### 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

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

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