On hyperelliptic Jacobians with complex multiplication by Q(−2+2)

Abstract

Consider a one-parameter family of hyperelliptic curves $C_a:y^2=x^5+3a{x}^4-2a^2{x}^3-6a^3{x}^2+3a^{4}x+a^5$ defined over $Q$, and their Jacobians $J_a$ where without loss of generality a is a non-zero squarefree integer. Clearly, the curve $C_a$ is a quadratic twist by a of $C_1$. Note that $J_a$ has complex multiplication by the quartic field $Q\left(\sqrt{-2+\sqrt{2}}\right)$. For a prime $p\nmid 2a$ we obtain types of reduction (supersingular, superspecial, ordinary) of $J_a$ at p in terms of congruences modulo 16 and the exact formulas for the zeta function of $C_a$ over $F_p$ for $p\not\equiv 1,7\left(\mathrm{mod}16\right)$. We deduce as conclusions the complete characterization of torsion subgroups of $J_a\left(Q\right)$, namely $J_a{\left(Q\right)}_{\mathrm{tors}}=J_a\left(Q\right)\left[2\right]\simeq Z/2Z$, and some information about $\mathrm{rank}{J}_a\left(Q\right)$.