浦西的猫2024-8-24 02:25 /
打开你的P眼。——田光善
（目录）-------------------------------------------------
Chapter I
p-adic numbers 1
1. Basic concepts 1
2. Metrics on the rational numbers 2
3. Review of building up the complex numbers 8
4. The field of p-adic numbers 10
5. Arithmetic in Qp 14
1度娘：同一性对称三角不等式
域上的范数：非负齐次三角
2def ord_p (x) ps: ord behave alike log
def || ||_p:=1/{1/{p^ord_p (x)}} (prop: || ||_p is a norm)
def non-Archimedean norm: <=max
柯西列、俩度娘等嫁如果序列视俩度娘同可喜
Theo Ostrowski
i等腰三角形性质ii每个人都是世界的中心
一点想法：皮埃蒂科的世界中范数的定义中先验地引入了到实数的映射，依赖R上的序结构而不是在Q_p上自洽建立'新的序'23.08.24止步于此
TO证明思路：case1:存在范数大于1的正整数找最小元组成基，任意整数展在这组基上有||n||<=n^αC → ||n||<=n^α → ||n||=n^α范数等价绝对值case2:所有正整数的范数小于等于1，找最小的范数小于1的整数（存在性由非平凡得到）该整数必为素数→其他素数的范数为1done
习题1tedious2重要性质3可由TO导出4判别式（暂未证出）5基础题6EZ7取一列素数幂即可8α>1则不满足凸性9由TO加一反例易得10欧几里得算法11数数略12-15数数略16分子中有P的幂次17-20暂时跳过
3生词：cumbersomeQ_p的代数闭包不完备(why?)intervene（ Ге́льфонд:characterizing Qp-linear field automorphisms of Ω, remain unanswered.）24.08止步于此
4Q_p(set of equivalence classes)等价类的范数由极限定义，极限的存在性由等腰三角形性质祷出。（Q_p与R完备化的区别：前者范数取值仍然在p的幂次并零，后者已在全R中取值）def +-*,乘的逆元只需取恒不为零的柯西列的戴表元逐项取逆。（想法：取Q代数闭包的| |_p完备化如何？讨论后的结论：Q代数闭包上没有padicnorm）T2特区代表元的唯一性Lemma25.08跳步于此p进制展开写法26.08止/尾项收敛⇔级数收敛Q_p，展开等价⇔逐项等，"Teichmiiller representatives"（Ex13）
5四则运算,平方根2708Hensel's lemma根判别法牛顿法
Chapter II
p-adic interpolation of the Riemann zeta-function 21
1. A formula for ζ(2k) 22
2. p-adic interpolation of the function f(s) =a^s 26
3. p-adic distributions 30
4. Bernoulli distributions 34
5. Measures and integration 36
6. The p-adic ζ-function as a Mellin-Mazur transform 42
7. A brief survey (no proofs) 47
Chapter III
Building up Ω 52
1. Finite fields
2. Extension of norms 57
3. The algebraic closure of Qp 66
4. Ω 71
Chapter IV
p-adic power series 76
1. Elementary functions 76
2. The logarithm, gamma and Artin-Hasse exponential functions 87
3. Newton polygons for polynomials 97
4. Newton polygons for power series 98
Chapter V
Rationality of the zeta-function of a set of equations
over a finite field 109
1. Hypersurfaces and their zeta-functions 109
2. Characters and their lifting 116
3. A linear map on the vector space of power series 118
4. p-adic analytic expression for the zeta-function 122
5. The end of the proof 125
（目录）-------------------------------------------------
Chapter I
p-adic numbers 1
1. Basic concepts 1
2. Metrics on the rational numbers 2
3. Review of building up the complex numbers 8
4. The field of p-adic numbers 10
5. Arithmetic in Qp 14
1度娘：同一性对称三角不等式
域上的范数：非负齐次三角
2def ord_p (x) ps: ord behave alike log
def || ||_p:=1/{1/{p^ord_p (x)}} (prop: || ||_p is a norm)
def non-Archimedean norm: <=max
柯西列、俩度娘等嫁如果序列视俩度娘同可喜
Theo Ostrowski
i等腰三角形性质ii每个人都是世界的中心
一点想法：皮埃蒂科的世界中范数的定义中先验地引入了到实数的映射，依赖R上的序结构而不是在Q_p上自洽建立'新的序'23.08.24止步于此
TO证明思路：case1:存在范数大于1的正整数找最小元组成基，任意整数展在这组基上有||n||<=n^αC → ||n||<=n^α → ||n||=n^α范数等价绝对值case2:所有正整数的范数小于等于1，找最小的范数小于1的整数（存在性由非平凡得到）该整数必为素数→其他素数的范数为1done
习题1tedious2重要性质3可由TO导出4判别式（暂未证出）5基础题6EZ7取一列素数幂即可8α>1则不满足凸性9由TO加一反例易得10欧几里得算法11数数略12-15数数略16分子中有P的幂次17-20暂时跳过
3生词：cumbersomeQ_p的代数闭包不完备(why?)intervene（ Ге́льфонд:characterizing Qp-linear field automorphisms of Ω, remain unanswered.）24.08止步于此
4Q_p(set of equivalence classes)等价类的范数由极限定义，极限的存在性由等腰三角形性质祷出。（Q_p与R完备化的区别：前者范数取值仍然在p的幂次并零，后者已在全R中取值）def +-*,乘的逆元只需取恒不为零的柯西列的戴表元逐项取逆。（想法：取Q代数闭包的| |_p完备化如何？讨论后的结论：Q代数闭包上没有padicnorm）T2特区代表元的唯一性Lemma25.08跳步于此p进制展开写法26.08止/尾项收敛⇔级数收敛Q_p，展开等价⇔逐项等，"Teichmiiller representatives"（Ex13）
5四则运算,平方根2708Hensel's lemma根判别法牛顿法
Chapter II
p-adic interpolation of the Riemann zeta-function 21
1. A formula for ζ(2k) 22
2. p-adic interpolation of the function f(s) =a^s 26
3. p-adic distributions 30
4. Bernoulli distributions 34
5. Measures and integration 36
6. The p-adic ζ-function as a Mellin-Mazur transform 42
7. A brief survey (no proofs) 47
Chapter III
Building up Ω 52
1. Finite fields
2. Extension of norms 57
3. The algebraic closure of Qp 66
4. Ω 71
Chapter IV
p-adic power series 76
1. Elementary functions 76
2. The logarithm, gamma and Artin-Hasse exponential functions 87
3. Newton polygons for polynomials 97
4. Newton polygons for power series 98
Chapter V
Rationality of the zeta-function of a set of equations
over a finite field 109
1. Hypersurfaces and their zeta-functions 109
2. Characters and their lifting 116
3. A linear map on the vector space of power series 118
4. p-adic analytic expression for the zeta-function 122
5. The end of the proof 125
#1 - 2024-8-24 07:52
菲兹克斯喵 (psi!)
#1-1 - 2024-8-24 14:49
浦西的猫
不是正儿八经的笔记，主要图一乐(*^▽^*)