On primes of degree one in function fields
On primes of degree one in function fields
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关于函数域中的一次素数
DOI：
10.1090/s0002-9939-1985-0781050-4
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发表时间：
1985
期刊：
影响因子：
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通讯作者：
R. Indik
中科院分区：
文献类型：
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作者：
G. Anderson;R. Indik
证明了在有限域的代数闭包上，被素数I的幂零化的曲线的雅可比的每一点都是曲线像中一点的I-主分支。设X是定义在Q元域Fq的代数闭包F上的亏格g>；0的一条光滑的、射影的几何连通曲线。固定Fq中X的一个基点x，设p：x-*J表示将x与x0之差的除数类赋给X的每个点x的嵌入。设I是任意素数，X：J(FQ)J(FQ)表示扭群J(FQ)在其I-主分支上的投影。本笔记的目的是证明下面的定理。映射X o Q：X(FQ)J(F)是满射的。为了证明，我们需要一个引理来控制算术级数中一次素数的分布。设K/k是具有正特征的整体场的非分支扩张。设g是k的亏格，fq是k和k的常数域，对k的每个素数v，设Ft，EGal(K/k)表示相应的算术Frobenius。莱玛。如果存在0 E Gal(K/k)使得对k次k的所有素数v都有F，/a0，则Q<；(2g[K：k]+3)2.证明.假设(*)E=-E4(Co)，(Ftj)C14其中v遍历k的所有一次素数(即剩余域与Fq重合)，4遍历Gal(K/k)的所有非平凡复值特征标.现在，根据黎曼假设(见[W]的附录5)，(*)的左侧以q+1 2g/F为界；右侧以([K：K]1)(2g2)v/vj为界。随之而来的是想要的结论。现在转到定理的证明，假设某个点d E J(Fq)不在XoT的像中，假设X是定义在Fq上的光滑射影曲线X0的基变化，X0是Fq有理的，d是X0的Jacobian J0的Fq有理点。设k为x0的函数域，k为k的代数闭包，v0为1984年4月3日编辑收到的k的一次素数。1980年数学学科分类。小学12A80；中学14H99。1985年美国数学学会0002-9939/85$1.00+$0.25每页
We show that over the algebraic closure of a finite field, every point of the jacobian of a curve annihilated by a power of a prime I is the I-primary component of a point in the image of the curve. Let X be a smooth, projective, geometrically connected curve of genus g > 0 defined over the algebraic closure F of the field Fq of q elements. Fixing a basepoint x( of X in Fq, let p: X -* J denote the embedding assigning to each point x of X the divisor class of the difference of x and x0. Let I be any prime number and let X: J(Fq) J(Fq), denote the projection of the torsion group J(Fq) onto its i-primary component. The object of this note is to prove the following THEOREM. The map X o q: X(Fq) J(F) is surjective. For the proof we need a lemma giving control over the distribution of primes of degree one in arithmetic progressions. Let K/k be an abelian unramified extension of global fields of positive characteristic. Let g be the genus of k and suppose that Fq is the field of constants of both K and k. For each prime v of k let Ft, E Gal(K/k) denote the corresponding arithmetic Frobenius. LEMMA. If there exists a0 E Gal(K/k) such that F,, / a0 for all primes v of k of degree one, then q < (2g[ K: k] + 3) 2. PROOF. By hypothesis (* )E = -E 4(co) ,(Ftj) C1 C1 4 where v runs through all the primes of k of degree one (i.e., of residue field coinciding with Fq), and 4 runs through all the nontrivial complex-valued characters of Gal(K/k). Now by the Riemann Hypothesis (see Appendix 5 of [W]) the left side of (*) is bounded below by q + 1 2g~/F; the right side is bounded above by ([K: k] 1)(2g 2)v/Vj. The desired conclusion follows immediately. Turning now to the proof of the theorem, suppose that some point d E J(Fq) fails to be in the image of X o T. Assume, as is permissible, that X is the base-change of a smooth projective curve X0 defined over Fq, x0 is Fq-rational, and d is an Fq-rational point of the jacobian J0 of X0. Fix a rational prime r distinct from 1. Let k denote the function field of X0, k an algebraic closure of k, and v0 the prime of degree one of k Received by the editors April 3, 1984. 1980 Mathemnatics Subject Classificcation. Primary 12A80; Secondary 14H99. 'A1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page