Maximal curves and Tate-Shafarevich results for quartic and sextic twists

We study elliptic surfaces corresponding to an equation of the specific type y2=x3+f(t)x, defined over the finite field Fq for a prime power q≡3mod4. It is shown that if s4=f(t) defines a curve that is maximal over Fq2 then the rank of the group of sections defined over Fq on the elliptic surface is determined in terms of elementary properties of the rational function f(t). Similar results are shown for elliptic surfaces given by y2=x3+g(t) using prime powers q≡5mod6 and curves s6=g(t). Finally, for each of the forms used here, existence of curves with the property that they are maximal over Fq2 is discussed, as well as various examples.</p

On the spectrum of genera of quotients of the Hermitian curve

We investigate the genera of quotient curves $\mathcal H_q/G$ of the $\mathbb F_{q^2}$-maximal Hermitian curve $\mathcal H_q$, where $G$ is contained in the maximal subgroup $\mathcal M_q\leq{\rm Aut}(\mathcal H_q)$ fixing a pole-polar pair $(P,\ell)$ with respect to the unitary polarity associated with $\mathcal H_q$. To this aim, a geometric and group-theoretical description of $\mathcal M_q$ is given. The genera of some other quotients $\mathcal H_q/G$ with $G\not\leq\mathcal M_q$ are also computed. Thus we obtain new values in the spectrum of genera of $\mathbb F_{q^2}$-maximal curves. A plane model for $\mathcal H_q/G$ is obtained when $G$ is cyclic of order $p\cdot d$, with $d$ a divisor of $q+1$

New examples of maximal curves with low genus

We investigate the Jacobian decomposition of some algebraic curves over finite fields with genus $4$, $5$ and $10$. As a corollary, explicit equations for curves that are either maximal or minimal over the finite field with $p^2$ elements are obtained for infinitely many $p$'s. Lists of small $p$'s for which maximality holds are provided. In some cases we describe the automorphism group of the curve