245 research outputs found

    Algebraic curves with many automorphisms

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    Let XX be a (projective, geometrically irreducible, nonsingular) algebraic curve of genus g2g \ge 2 defined over an algebraically closed field KK of odd characteristic pp. Let Aut(X)Aut(X) be the group of all automorphisms of XX which fix KK element-wise. It is known that if Aut(X)8g3|Aut(X)|\geq 8g^3 then the pp-rank (equivalently, the Hasse-Witt invariant) of XX is zero. This raises the problem of determining the (minimum-value) function f(g)f(g) such that whenever Aut(X)f(g)|Aut(X)|\geq f(g) then XX has zero pp-rank. For {\em{even}} gg we prove that f(g)900g2f(g)\leq 900 g^2. The {\em{odd}} genus case appears to be much more difficult although, for any genus g2g\geq 2, if Aut(X)Aut(X) has a solvable subgroup GG such that G>252g2|G|>252 g^2 then XX has zero pp-rank and GG fixes a point of XX. Our proofs use the Hurwitz genus formula and the Deuring Shafarevich formula together with a few deep results from finite group theory characterizing finite simple groups whose Sylow 22-subgroups have a cyclic subgroup of index 22. We also point out some connections with the Abhyankar conjecture and the Katz-Gabber covers

    Curves with more than one inner Galois point

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    Let C\mathcal{C} be an irreducible plane curve of PG(2,K)\text{PG}(2,\mathbb{K}) where K\mathbb{K} is an algebraically closed field of characteristic p0p\geq 0. A point QCQ\in \mathcal{C} is an inner Galois point for C\mathcal{C} if the projection πQ\pi_Q from QQ is Galois. Assume that C\mathcal{C} has two different inner Galois points Q1Q_1 and Q2Q_2, both simple. Let G1G_1 and G2G_2 be the respective Galois groups. Under the assumption that GiG_i fixes QiQ_i, for i=1,2i=1,2, we provide a complete classification of G=G1,G2G=\langle G_1,G_2 \rangle and we exhibit a curve for each such GG. Our proof relies on deeper results from group theory
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