特徵 p 的特殊值之間的代數關係

Algebraic Relations among Special Values in Characteristic p

張介玉(Chieh-Yu Chang)

指導教授 ： 于靖

國立清華大學/理學院/數學系/博士(2007年)

https://doi.org/10.6843/NTHU.2007.00033

摘要

As analogue to special values at positive integers of the Riemann zeta function, we consider Carlitz zeta values at positive integers. By constructing t-motives and using Papanikolas' theory, we prove that the only algebraic relations among this family of characteristic p zeta values are those coming from the Euler-Carlitz relations and the Frobenius p-th power relations. As the constant filed varies, we prove that among these families of zeta values, the Euler-Carlitz relations and the Frobenius p-th power relations still account for all the algebraic relations.

Given a finite field Fq of q elements with odd characteristic, let Fq[t] be the polynomial ring in the variable t over Fq. For any rank two Drinfeld Fq[t]-module ρ defined over a fixed algebraic closure of Fq(t) without complex multiplication, we consider its period matrix P which is analogous to the period matrix of an elliptic curve defined over an algebraic closure of Q without complex multiplication. We prove that the transcendence degree of the period matrix P over Fq(t) is 4. As a consequence, we prove the algebraic independence of the logarithms associated toρof algebraic functions which are linear independent over Fq(t):

關鍵字

代數獨立 ； Algebraic independence ； t-motives ； Zeta values ； Drinfeld modules ； Periods and quasi-periods ； Logarithms

參考文獻

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7. L. Carlitz, On certain functions connected with polynomials in a Galois field, Duke. Math. J 1 (1935), 137-168.