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研究生:張介玉
研究生(外文):Chieh-Yu Chang
論文名稱:特徵p的特殊值之間的代數關係
論文名稱(外文):Algebraic Relations among Special Values in Characteristic p
指導教授:于靖于靖引用關係
指導教授(外文):Jing Yu
學位類別:博士
校院名稱:國立清華大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2007
畢業學年度:95
語文別:英文
論文頁數:102
中文關鍵詞:代數獨立Algebraic independencet-motivesZeta valuesDrinfeld modulesPeriods and quasi-periodsLogarithms
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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):
Chapter 1. Introduction 1
1. Special zeta values. 1
2. Period matrices and logarithms of Drinfeld modules. 3
3. Outline of this thesis. 6
Chapter 2. Papanikolas' theory revisited 9
1. Notations. 9
2. Tannakian category. 10
3. Galois theory of Frobenius difference equations. 12
Chapter 3. Special zeta values 17
1. Carlitz's theory. 17
2. Constructing t-motives from polylogarithms. 19
3. Algebraic independence 30
Chapter 4. Families of special zeta values 39
1. Algebraic independence of certain special functions 39
2. Some transcendence degree criterions 44
3. Algebraic independence 50
4. Calculation of the transcendence degrees 59
Chapter 5. Period matrices and Drinfeld logarithms 67
1. t-modules and t-motives. 67
2. Period matrices of Drinfeld modules. 72
3. Preliminary Calculations 73
4. End of proof 87
Bibliography 101
1. G. W. Anderson, t-motives, Duke Math. J. 53 (1986), 457-502.
2. G. W. Anderson and M. A. Papanikolas, Period matrices calculations for Drinfeld modules, private communications.
3. G. W. Anderson and D. S. Thakur, Tensor powers of the Carlitz module and zeta values, Ann. of Math. 132 (1990), 159-191.
4. G. W. Anderson, W. D. Brownawell and M. A. Papanikolas, Determination of the algebraic relations among special Gamma-values in positive characteristic, Ann. of Math. 160 (2004), 237-313.
5. G. W. Anderson, W. D. Brownawell and M. A. Papanikolas, Two boilerplates for Rohrlich conjecture project, 2004, private note.
6. W. D. Brownawell and M. A. Papanikolas, Linear independence of Gamma-values in positive characteristic, J. reine angew. Math. 549 (2002), 91-148.
7. L. Carlitz, On certain functions connected with polynomials in a Galois field, Duke. Math. J 1 (1935), 137-168.
8. C.-Y. Chang and M. A. Papanikolas, On algebraic relations among period matrices and logarithms of Drinfeld modules: the non-CM case, 2007, in preparation.
9. C.-Y. Chang and M. A. Papanikolas and J. Yu , Determination of algebraic relations among special zeta values in positive characteristic (II), 2007, in preparation.
10. C.-Y. Chang and J. Yu, Determination of algebraic relations among special zeta values in positive characteristic, to appear in Adv. Math.
11. L. Denis, Independance algebrique de di®erents ¼, C. R. Acad. Sci. Paris Ser. I Math. 327 8 (1998), 711-714.
12. E.-U. Gekeler, On the deRham isomorphism for Drinfeld modules, J. reine angew. Math. 401 (1989), 188-208.
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15. M. A. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, http://arxiv.org/abs/math/0506078,2005.
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17. A. Thiery, Independance algebrique des periodes et quasi-periodes d'un module de Drinfeld, The arithmetic of function fields (Columbus, OH, 1991), Ohio State Univ. Math. Res. Inst. Publ., 2, de Gruyter, Berlin (1992), 265-284.
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20. J. Yu, On periods and quasi-periods of Drinfeld modules, Compositio Math. 74 (1990), 235-245.
21. J. Yu, Transcendence and special zeta values in characteristic p, Ann. of Math. (2) 134 (1991), 1-23.
22. J. Yu, Analytic homomorphisms into Drinfeld modules, Ann. of Math. (2) 145 (1997), 215-233.
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