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Register a new userThe Automorphism Groups of a Family of Maximal Curves
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The Hasse Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal. Giulietti and Korchmaros introduced a curve C_3 which is maximal over F_{q^6} and determined its automorphism group. Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves C_n, indexed by an odd integer n greater than or equal to 3, such that C_n is maximal over F_{q^{2n}}. In this paper, we determine the automorphism group Aut(C_n) when n > 3; in contrast with the case n=3, it fixes the point at infinity on C_n. The proof requires a new structural result about automorphism groups of curves in characteristic p such that each Sylow p-subgroup has exactly one fixed point. MSC:11G20, 14H37.
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Let $p$ be a prime, let $r$ and $q$ be powers of $p$, and let $a$ and $b$ be relatively prime integers not divisible by $p$. Let $C/mathbb F_{r}(t)$ be the superelliptic curve with affine equation $y^b+x^a=t^q-t$. Let $J$ be the Jacobian of $C$. By work of Pries--Ulmer, $J$ satisfies the Birch and Swinnerton-Dyer conjecture (BSD). Generalizing work of Griffon--Ulmer, we compute the $L$-function of $J$ in terms of certain Gauss sums. In addition, we estimate several arithmetic invariants of $J$ appearing in BSD, including the rank of the Mordell--Weil group $J(mathbb F_{r}(t))$, the Faltings height of $J$, and the Tamagawa numbers of $J$ in terms of the parameters $a,b,q$. For any $p$ and $r$, we show that for certain $a$ and $b$ depending only on $p$ and $r$, these Jacobians provide new examples of families of simple abelian varieties of fixed dimension and with unbounded analytic and algebraic rank as $q$ varies through powers of $p$. Under a different set of criteria on $a$ and $b$, we prove that the order of the Tate--Shafarevich group of $J$ grows quasilinearly in $q$ as $q to infty.$
Let $K$ be a field and $f:mathbb{P}^N to mathbb{P}^N$ a morphism. There is a natural conjugation action on the space of such morphisms by elements of the projective linear group $text{PGL}_{N+1}$. The group of automorphisms, or stabilizer group, of a given $f$ for this action is known to be a finite group. In this article, we address two mainly computational problems concerning automorphism groups. Given a finite subgroup of $text{PGL}_{N+1}$ determine endomorphisms of $mathbb{P}^N$ with that group as subgroup of its automorphism group. In particular, we show that every finite subgroup occurs infinitely often and discuss some associated rationality problems. Inversely, given an endomorphism determine its automorphism group. In particular, we extended the Faber-Manes-Viray fixed-point algorithm for $mathbb{P}^1$ to endomorphisms of $mathbb{P}^2$. A key component is an explicit bound on the size of the automorphism group depending on the degree of the endomorphism.
Let $F$ be any field. We give a short and elementary proof that any finite subgroup $G$ of $PGL(2,F)$ occurs as a Galois group over the function field $F(x)$. We also develop a theory of descent to subfields of $F$. This enables us to realize the automorphism groups of finite subgroups of $PGL(2,F)$ as Galois groups.
Fix integers $r,s_1,...,s_l$ such that $1leq lleq r-1$ and $s_lgeq r-l+1$, and let $Cal C(r;s_1,...,s_l)$ be the set of all integral, projective and nondegenerate curves $C$ of degree $s_1$ in the projective space $bold P^r$, such that, for all $i=2,...,l$, $C$ does not lie on any integral, projective and nondegenerate variety of dimension $i$ and degree $<s_i$. We say that a curve $C$ satisfies the {it{flag condition}} $(r;s_1,...,s_l)$ if $C$ belongs to $Cal C(r;s_1,...,s_l)$. Define $ G(r;s_1,...,s_l)=maxleft{p_a(C): Cin Cal C(r;s_1,...,s_l)right }, $ where $p_a(C)$ denotes the arithmetic genus of $C$. In the present paper, under the hypothesis $s_1>>...>>s_l$, we prove that a curve $C$ satisfying the flag condition $(r;s_1,...,s_l)$ and of maximal arithmetic genus $p_a(C)=G(r;s_1,...,s_l)$ must lie on a unique flag such as $C=V_{s_1}^{1}subset V_{s_2}^{2}subset ... subset V_{s_l}^{l}subset {bold P^r}$, where, for any $i=1,...,l$, $V_{s_i}^i$ denotes an integral projective subvariety of ${bold P^r}$ of degree $s_i$ and dimension $i$, such that its general linear curve section satisfies the flag condition $(r-i+1;s_i,...,s_l)$ and has maximal arithmetic genus $G(r-i+1;s_i,...,s_l)$. This proves the existence of a sort of a hierarchical structure of the family of curves with maximal genus verifying flag conditions.
In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $mathbb Q$ whose Jacobians have such maximal adelic Galois representations.
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