Maximal curves with trivial automorphism group

Source abstract

We construct algebraic curves with trivial geometric automorphism group by taking cyclic scalar quotients of curves of the form $$y^r-y=xR(x),\qquad r=p^e.$$ We give a condition on the genus under which the quotient has no nontrivial automorphism. The proof uses the ramification of a degree-$r$ morphism to the projective line, Castelnuovo--Severi, and explicit automorphisms of the covering curve. The quotient also has gonality $r$, with a unique morphism of degree $r$ to the projective line up to an automorphism of the latter. Applying the construction to maximal additive curves gives explicit maximal curves with trivial geometric automorphism group in every positive characteristic, including families defined over the prime field and of unbounded genus. To the best of our knowledge, these are the first examples of maximal curves with trivial geometric automorphism group. Every maximal curve in these families is covered by the Hermitian curve. For every odd prime $p$ and every $n\ge2$, we prove that the curve $\X_{p,n}$, maximal over $\F_{p^{4n+2}}$, is not a Galois subcover of the corresponding Hermitian curve, even over the algebraic closure. Thus every odd characteristic yields an infinite family of curves covered but not Galois covered by the Hermitian curve. To the best of our knowledge, these are the first such infinite families.