Genus two curves with everywhere twisted good reduction

Houria Baaziz , John Boxall (1)

1 LMNO - Laboratoire de Mathématiques Nicolas Oresme

We construct examples of genus two curves C over quadratic fields K with everywhere twisted good reduction , i.e., for any finite prime p of K, C has a twist that has good reduction at p. An analogous construction for ellip-tic curves enables us to recover Setzer's family of curves with everywhere good reduction over an imaginary quadratic field. 1. Introduction. Let K be a number field, and let X/K be a smooth projective variety. We say that X has good reduction at a finite prime p of K if X has a smooth model X p over the local ring at p. It is well known that X has good reduction outside a finite set Σ(X) of primes p; we say that X has everywhere good reduction if Σ(X) is empty. A well-known theorem of Fontaine [4] and Abrashkin [1] asserts that there are no abelian varieties with everywhere good reduction over Q. On the other hand many authors have given examples of elliptic curves having everywhere good reduction over quadratic fields. By taking products, one obtains abelian varieties of arbitrary dimension with everywhere good reduction over quadratic fields.

Domains: Mathematics [math]; Mathematics [math]; Number Theory [math.NT]

Houria Baaziz, John Boxall. Genus two curves with everywhere twisted good reduction. Rocky Mountain Journal of Mathematics, 2013, 43 (1), pp.55-73. ⟨10.1216/RMJ-2013-43-1-55⟩. ⟨hal-02153441⟩