函数域的K2群的挠元

徐克舰, 刘敏

数学学报 ›› 2010, Vol. 53 ›› Issue (3) : 611-616.

PDF(399 KB)
PDF(399 KB)
数学学报 ›› 2010, Vol. 53 ›› Issue (3) : 611-616. DOI: 10.12386/A2010sxxb0068
论文

函数域的K2群的挠元

    徐克舰1, 刘敏2
作者信息 +

On the Torsion in K2 of a Function Field

    Ke Jian XU1, Min LIU2
Author information +
文章历史 +

摘要

F是域, 令 Gn(F)={{a, Φn(a)} ∈K2(F)|an(a) ∈ F*}, 这里Φn(x) 是n次分圆多项式.使用函数域的ABC定理证明了 若F是常数域为k函数域, l≠ch(k)是素数, 则对 n≥3且l>2或n>3且l=2, Gln(F)不是K2(F)的子群. 由此部分地证实了Browkin的猜想.

 

Abstract

Let F be a field and letGn(F)={{a, Φn(a)} ∈K2(F)|an(a) ∈ F*}, where Φn(x) denotes the n-th cyclotomic polynomial. If F is a function field with perfect constant field k and l≠ch(k) a prime number, it is proved by using the ABC theorem for function fields that Gln(F) is not a subgroup of K2(F) if n≥3 and l>2 or n>3 and l=2, which confirms a conjecture of Browkin partially.

 

关键词

函数域 / 分圆元素 / Milnor K2

Key words

function field / cyclotomic element / Milnor K2-group

引用本文

导出引用
徐克舰, 刘敏. 函数域的K2群的挠元. 数学学报, 2010, 53(3): 611-616 https://doi.org/10.12386/A2010sxxb0068
Ke Jian XU, Min LIU. On the Torsion in K2 of a Function Field. Acta Mathematica Sinica, Chinese Series, 2010, 53(3): 611-616 https://doi.org/10.12386/A2010sxxb0068

参考文献


[1] Milnor J., Introduction to Algebraic K-Theory, In: Ann. of Math. Studies 72. New Jersey: Princeton Uni. Press, 1971.

[2] Tate J., Relations between K2 and galois cohomology, Invent. Math., 1976, 36: 257--274.

[3] Suslin A. A., Torsions in K2 of fields, K-Theory, 1987, 1(1): 5--29.

[4] Browkin J., Elements of Small Order in K2(F), In: Algebraic K-theory, Lecture Notes in Math., Vol. 966. Berlin, Heidelberg, New York: Springer-Verlag, 1982, 1--6.

[5] Qin H. R., The Subgroups of Finite Order in K2(Q),$ In: Bass H., Kuku A. O, Pedrini C., eds, Algebraic K-theory and its application, Singapore: World Scientific, 1999, 600--607.

[6] Xu K. J., Qin H. R., A conjecture on a class of elements of finite order in K2(Fwp), Science in China Ser. A, 2001, 44(4): 484--490.

[7] Xu K. J., Qin H. R., Some diophantine equations over Z
[i] and Z
[√-2] with applications to K2 of a field, Communication in Algebra, 2002, 30(1): 353--367.

[8] Cheng X. Y., Xia J. G., Qin H. R., Some elements of finite order in K2(Q), Acta Mathematica Sinica, English Series, 2007, 23(5): 819--826.

[9] Xu K. J., On the elements of prime power order in K2 of a number field, Acta Arithmetica, 2007, 127(2): 199--203.

[10] Xu K. J., Liu M., On the torsion in K2 of a field, Science in China Series A, Mathematics, 2008, 51(7): 1187--1195.

[11] Faltings G., Endlinchkeitsatze für abelsche varietaten über zahlkorpern, Invent. Math., 1983, 73: 349--366.

[12] Grauert H., Mordell vermtung über rationale punkte auf algebraischen kurven und funktionenkörper, Publ Math., I. H. E. S., 1965.

[13] Manin J. I., Rational points of algebraic curves over function fields, Izv Akad. Nauk. SSSR, Ser. Mat., 1963, 27: 1395--1440.

[14] Li K. Z., A Geometric Proof of Mordell Conjecture for Function Field, http://arxiv.org/abs/math.AG/0701407 (2007).

[15] Rosen M., Number Theory in Function Fields, Berlin, Heidelberg, New York: Springer-Verlag, 2002.

基金

国家自然科学基金资助项目(10871106)

PDF(399 KB)

396

Accesses

0

Citation

Detail

段落导航
相关文章

/