Let X be a smooth projective curve of genus g>= 2 over an algebraically closed field K, and suppose that Aut(X) contains a cyclic subgroup G of order N>2g - 2. The cases N>= 2g + 1 are known. We treat the two boundary values N = 2g and N = 2g - 1. In the tame case we obtain an explicit finite list of ramification signatures. In odd characteristic we prove that the Sylow p-subgroup of G has order p and determine all wild boundary cases by a direct ramification analysis; the resulting curves are Artin-Schreier-Kummer fiber products. In characteristic two we prove that the 2-part has order at most four and classify all wild boundary cases, including Artin-Schreier-Witt normal forms for the C4 and C12 cases.
Let p be an odd prime. We study the Artin-Schreier analogue of Poonen's problem of constructing curves without extra automorphisms. An Artin-Schreier curve carries an unavoidable deck group Cp, so the smallest possible full geometric automorphism group inside the Artin-Schreier locus is Cp. We determine the exact numer...
We investigate how much arithmetic information about a variety can be detected by sections of the geometrically abelian fundamental groups of sufficiently small open subvarieties. Over a \(p\)-adic field, such a section determines a rational point on the Albanese torsor, and, under a conjectural assumption, this point...
We consider the problem of constructing maximal curves with few automorphisms over finite fields. Our main ingredient is the determination of the \(\operatorname{PGL}_2\!\left(\overline{\mathbb F}_p\right)\)-stabilizer of the roots of the Chebyshev polynomials, apart from two exceptional cases already treated in the li...
Let D_N be the determinant of the generic symmetric N x N matrix, and let differential operators act by ordinary differentiation over a field of odd characteristic p. We determine the apolar ideal of D_N in every size. In addition to the classical quadratic relations, it is generated by one (p-1) x (p-1) determinant of...
Let E:y2+y=x3+x over F2, a supersingular curve whose automorphism group is isomorphic to SL(2,3), of order twenty-four. Using an automorphism of order six, we construct the sextic twist of E along the fiber product of an Artin-Schreier and a Kummer covering, obtaining the family Z2+Z=B(t)X3+A(t)+c, in which the constan...
We prove that for a geometrically connected modular curve $X_H$ of prime level larger than $311$, all automorphisms are modular, i.e., they arise from the moduli interpretation of $X_H$. The proof relies on a general criterion involving the gonality of $X_H$ and the dimension of the CM part of the Jacobian of $X_H$. Fo...
V. Dose, G. Lido, P. Mercuri· 0 citations
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