### 博碩士論文 952201001 詳細資訊

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(On the Distribution of Primes)

 ★ On the Diophantine Equation of (x^m-1)/(x-1)=(y^n-1)/(y-1) ★ On similarity problem of integral matrices

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Hilbert 類體裡最大的整數環，讓 \$psi\$ 是一個定義在 \$mathcal{O}\$ 上秩為 1 的特定 Drinfeld 模。給一個 \$0 eq alpha in mathcal{O}\$ 和一個 \$mathcal{O}\$ 裡的理想 \$frak{M}\$，令
\$f_{alpha}left(frak{M}
ight) = left{f in A : psi_{f}left(alpha
ight) equiv 0 pmod{frak{M}}
ight}\$ 是一個 \$A\$ 裡的理想。 \$omegaig(f_alphaleft(frak{M}
ight)ig)\$ 表示為 \$f_alphaleft(frak{M}
ight)\$ 相異質理想因子的個數。我們可以証明下面這個量有常態分佈的性質：
\$\$
frac{omegaig(f_alphaleft(frak{M}
ight)ig)-frac{1}{2}left(logdegfrak{M}
ight)^2}{frac{1}{sqrt{3}}left(logdegfrak{M}
ight)^{3/2}}.
\$\$

First of all, we consider a commutative algebraic group \$A\$ which is defined over a global field \$K\$. Then, we fix a positive integer \$n\$. For a prime divisor \$wp\$ of \$K\$, let \$F_{wp}\$ denote the residue field. If \$A\$ has good reduction at \$wp\$, let \$ ilde A\$ be the reduction of \$A\$ modulo \$wp\$ and let \$N_{wp,n}\$ be the number of \$n\$-torsion points in \$ ildeAleft(F_wp
ight)\$, the set of \$F_{wp}\$-rational points in \$ ilde A\$. If \$A\$ has bad reduction at \$wp\$, let \$N_{wp,n} = 0\$. Let \$ ormwp\$ denote the norm of \$wp\$, equal to the cardinality of the residue field \$F_wp\$. We are interested in the average value of \$N_{wp, n}\$, where \$wp\$ runs through the prime divisors in \$K\$, namely the limit
\$\$
limlimits_{x
ightarrow infty } frac{1}{pi_{K}(x)}sumlimits_{ ormwp leq x}N_{wp,n},
\$\$
where \$pi_{K}(x)\$ is the number of primes \$wp\$ with \$ ormwp leq x\$. We denote this limit by \$M(Bbb A_{/K}, n)\$. We shall derive explicit formulas for the average value \$M(Bbb A_{/K}, n)\$ when \$A\$ is a commutative algebraic group of dimension one defined over \$K\$.
Secondly, we consider a global function field \$k\$ of positive characteristic containing a prime divisor \$infty\$ of degree one and whose field of constants is \$Bbb F_q\$. Let \$A\$ be the ring of elements of \$k\$ which are regular outside \$infty\$. Let \$psi\$ be a sgn-normalized rank one Drinfeld \$A\$-module defined over \$mathcal{O}\$, the integral closure of \$A\$ in the Hilbert class field of \$A\$. Given any \$0 eq alpha in mathcal{O}\$ and an ideal \$frak{M}\$ in \$mathcal{O}\$, let \$f_{alpha}left(frak{M
ight) = left{f in A : psi_{f}left(alpha
ight)equiv 0 pmod{frak{M}}
ight}\$ be the ideal in \$A\$. We denote by \$omegaig(f_alphaleft(frak{M}
ight)ig)\$ the number of distinct prime ideal divisors of \$f_alphaleft(frak{M}
ight)\$. If \$q eq 2\$, we prove that the following quantity
\$\$
frac{omegaig(f_alphaleft(frak{M}
ight)ig)-frac{1}{2}left(logdegfrak{M}
ight)^2}{frac{1}{sqrt{3}}left(logdegfrak{M}
ight)^{3/2}}
\$\$
distributes normally.

★ 橢圓曲線
★ Drinfeld 模
★ 函數體

★ ellipeic curves
★ Drinfeld modules
★ function fields

Chapter 1. The Chebotarëv Density Theorem 7
1. The Dirichlet density version for global fields 7
2. The natural density version for number fields 8
3. The natural density version for function fields 10
Chapter 2. On the distribution of torsion points modulo primes: The case of number fields 13
1. Introduction 13
2. The case of one-dimensional tori T 17
3. The case of elliptic curves with complex multiplication19
Chapter 3. On the distribution of torsion points modulo primes: The case of function fields 23
1. Introduction 23
2. The cases of Ga and Gm 24
3. The case of one-dimensional tori 26
4. The case of elliptic curves 28
Chapter 4. Drinfeld Modules 33
1. Definitions 33
2. Drinfeld A-Modules over L with Generic Characteristic 34
3. The average value \$M(psi_{/L}, frak{n})\$ 36
Chapter 5. On an Erdos-Pomerance conjecture for rank one Drinfeld modules 41
1. Introduction 41
2. Preliminaries 45
3. Proofs of Theorem 5.1 and Theorem 5.2 53
4. Equivalent statements of Theorem 5.3 and 5.4 62
5. Proofs of Theorem 5.3 and 5.4 73
Bibliography 81

[2] A. Bandini, I. Longhi, S. Vigni, Torsion points on elliptic curves over function elds and a theorem of Igusa, Expo. Math., 27 (2009), no.3, 175-209.
[3] Y.-M. Chen, Y.-L. Kuan On the distribution of torsion points modulo primes Bull. Aust. Math. Soc., 86 (2012), no.2, 339-347.
[4] J. Coates and A. Wiles, On the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 39 (1977), no.3, 223-251.
[5] A. C. Cojocaru, On the surjectivity of the Galois representations associated to non-CM elliptic curves, Canad. Math. Bull. vol. 48 (2005), no.1, 16-31.
[6] G. Lejeune Dirichlet, Beweis des Satzes, dass jede unbegrenzte arithmetische Progression, deren erstes Glied und Dierenz ganze Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthält, Abh. Akad. Wiss. Berlin, Math. Abh. (1837), 45-71; Werke I, 313-342.
[7] P. Erdös and M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions, Amer. J. Math. 62 (1940), 738-742.
[8] P. Erdös and C. Pomerance, On the normal number of prime factors of phi(n), Rocky Mountain J. Math. 15 (1985), no.2, 343-352.
[9] M. Fried and M. Jarden, Fields Arithmetics, A Series of Modern Surveys in Mathematics, Vol. 11, Springer (2005).
[10] F. G. Frobenius, Über Beziehungen zwischen den Primidealen eines algebraischen Körpers und den Substitutionen seiner Gruppe, Sitz. Akad. Wiss. Berlin(1896), 689-703; Ges. Abh. II, 719-733.
[11] D. Goss, Basic structures of function eld arithmetic, Spring-Verlag, 1996.
[12] R. Gupta, Ramication in the Coates-Wiles tower, Invent. Math. 81 (1985), no.1, 59-69.
[13] G. H. Hardy and S. Ramanujan, The normal number of prime factors of number n, Quar. J. Pure Appl. Math. 48 (1917), 76-97.
[14] D. Hayes, Explicit class eld theory for global function elds, in "Studies in algebra and number theory,
Adv. in Math. Suppl. Studies" (G.-C. Rota, Eds.), vol. 6, 173-217, Academic Press, San Diego, 1979.
[15] C. Hooley, On Artin’s Conjecture, J. reine angew. Math. 225 (1967), 209-220.
[16] H.-L. Huang, The average number of torsion points on elliptic curves, Thesis, National Central University, (2010).
[17] C.-N. Hsu, The Brun-Titchmarsh Theorem in Function Fields, J. Number Theory 79 (1999), no. l, 67-82.
[18] C.-N. Hsu and J. Yu, On Artin’s conjecture for Drinfeld modules of rank one, J. Number Theory 88 (2001), no. 1, 157-174.
[19] Yen-Liang Kuan, Wentang Kuo and Wei-Chen Yao, On Erdos-Pomerance conjecture for rank one Drinfeld modules, preprint.
[20] W. Kuo and Y.-R. Liu, A Carlitz module analogue of a conjecture of Erdos and Pomerance, Trans. Amer. Math. Soc. 361 (2009), no. 9, 4519-4539.
[21] W. Kuo and Y.-R. Liu, Gaussian laws on Drinfeld modules, Int. J. Number Theory 5 (2009), no. 7, 1179-1203.
[22] A. Khrennikov and M. Nilsson, On the number of cycles of p-adic dynamical systems, J. Number Theory 90 (2001), no. 2, 255-264.
[23] S. Lang, Algebraic Number Theory, Springer, Berlin, 1994.
[24] T.-F. Lee, The average of the number of r-periodic points over a quadratic number eld, Master Thesis, National Central University, (2006).
[25] J. C. Lagarias and A. M. Odlyzko, Eective versions of the Chebotarev density theorem, Algebraic Number Fields : L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) pp. 409-464. Academic Press, London 1977.
[26] H. W. Lenstra, Jr., Complex multiplication structure of elliptic curves, J. Number Theory 56 (1996), no. 2, 227-241.
[27] H. W. Lenstra, Jr., and P. Stevenhagen, Chebotarëv and his density theorem, Math. Intelligencer 18 (1996), no. 2, 26-37.
[28] S. Li and C. Pomerance, On generalizing Artin’s conjecture on primitive roots to composite moduli, J. Reine Angew. Math. 556 (2003), 205-224.
[29] Y.-R. Liu, A generalization of the Erd®s-Kac Theorem and its applications, Canad. Math. Bull. 47(2004), no. 4, 589-606.
[30] M. R. Murty, On the supersingular reduction of elliptic curves, Proc. Indian Acad. Sci. Math. Sci. 97(1987), no. 1-3, 247-250.
[31] M. R. Murty, V. K. Murty, and N. Saradha Modular forms and the Chebotarev density theorem, Amer. j. Math. 110 (1998), no. 2, 253-281.
[32] M. R. Murty and F. Saidak, Non-abelian generalizations of the Erd®s-Kac theorem, Canad. J. Math. 56 (2004), no. 2, 356-372.
[33] R. Pink, The Mumford-Tate conjecture for Drinfeld modules, Publ. Res. Inst. Math. Sci., 33 (1997), no.3, 393-425.
[34] R. Pink, E. Rütsche, Adelic openness for Drinfeld modules in generic characteristic, J. Number Theory, 129 (2009), no. 4, 882-907.
[35] M. Rosen, Number theory in function elds, Graduate Texts in Math. 2010, Springer (2002).
[36] J.-P. Serre, Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15(1972), no. 4, 259-331.
[37] J.-P. Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Études Sci. Publ. Math. No. 54 (1981), 323-401.
[38] J.-P. Serre, On a Theorem of Jordan, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 4, 429-440.
[39] J.-P. Serre and J. Tate, Good reduction of abelian varieties, Ann. of Math. (2) 88 (1968), 492-517.
[40] J. H. Silverman, The Arithmetic of Elliptic Curves, Springer, Berlin, 1986.
[41] J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Springer, Berlin, 1994.
[42] P. Turán, On a theorem of Hardy and Ramanujan, J. London Math. Soc. 9 (1934), 274-276.
[43] D. Ulmer, Elliptic Curves over Function Fields, arXiv:1101.1939v1, 2011.
[44] W.-B. Zhang, Probabilistic number theory in additive arithmetic semigroups, Analytic number theory, Vol. 2 (Allerton Park, IL, 1995), 839-885, Progr. Math., 139, Birkhauser Boston, Boston, MA, 1996.