A Brief Introduction to Modular Forms Computation
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- ゆきさ すえがら
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1 A Brief Introduction to Modular Forms Computation Magma Supported by GCOE Program Math-For-Industry Education & Research Hub
2 What s this? Definitions and Properties Demonstration H := H P 1 (Q) some conditions k Z: f (g(z)) = (cz + d) k f (z) for all g Γ(N) Γ(N): Congruence subgroup N: with Nebentypus ε M k (Γ(N), ε): (N, k, ε) dim C M k (Γ(N)) < + f = n 0 a nq n q = e 2πiz/N : q- S k (Γ(N), ε): (N, k, ε) f M k (Γ(N), ε) s.t. a 0 = 0
3 Background Definitions and Properties Demonstration eigenform Fermat - Serre mod p Galois etc. Algebraic topology String theory Algebraic combinatorics : e.g. Kissing Number Problem
4 on Number Theory Definitions and Properties Demonstration X 0 (N) e.g. N = 39 Hecke T n q- T n f = a n f a n C (Atkin-Lehner-Li, Miyake) S k (Γ 1 (N)) = M N d (N/M) α d (Sk new (Γ 1 (M)))
5 Construct Space and Hecke Action T 2 on M 2 (Γ 0 (41)). Definitions and Properties Demonstration > M41 := ModularForms(Gamma0(41),2); > T2 := HeckeOperator(M41,2); T2; [ ] [ ] [ ] [ ] > Parent(T2); Full Matrix Algebra of degree 4 over Integer Ring > Ch2 := CharacteristicPolynomial(T2); Ch2; x^4-2*x^3-8*x^2 + 14*x +3 > Factorization(Ch2); [ <x - 3, 1>, <x^3 + x^2-5*x - 1, 1> ]
![HeckeOperator(M41,2); T2; [ 3 0 12 6] [ 0 0 3-2] [ 0 1-2 0] [ 0 0-2 1] > Parent(T2); Full Matrix](/docs-images/44/20583738/images/page_5.jpg "Algebra of degree 4 over Integer Ring > Ch2 := CharacteristicPolynomial(T2); Ch2; x^4-2*x^3-8*x^2 +")
6 Compute Newforms Definitions and Properties Demonstration S new 2 (Γ 0 (11)). > S := CuspForms(Gamma0(11),2); > N := Newforms(S); N; [* [* q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6-2*q^7-2*q^9-2*q^10 + q^11 + O(q^12) *], [* 5/12 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 15*q^8 + 13*q^9 + 18*q^10 + q^11 + O(q^12) *] *] > Newforms("G0N11k2A"); // LABELS [* [* q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6-2*q^7-2*q^9-2*q^10 + q^11 + O(q^12) *] *]
![2*q^6-2*q^7-2*q^9-2*q^10 + q^11 + O(q^12) *], [* 5/12 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 +](/docs-images/44/20583738/images/page_6.jpg "15*q^8 + 13*q^9 + 18*q^10 + q^11 + O(q^12) *] *] > Newforms(\"G0N11k2A\"); // LABELS [* [* q - 2*q^2 - q^3 +")
7 SMC: Algebraic vs. Analytic Theorem (Khare-Wintenberger, 2007) Q detρ(c) = 1 2 mod p Galois ρ : Gal(Q/Q) GL 2 (F p ) (N(ρ), k(ρ), ε(ρ)). f = a n q n ( q = e 2πiz ) S k(ρ) (Γ, ε(ρ)) n 1
8 Verification Galois Tr(ρ(Frob l )) a l (mod p) for all l pn(ρ): prime Galois (N, k, ε) (N, k, ε) conductor, Serre weight, character level, weight, character
9 Matching example Q, Galois. E : y 2 + xy + y = x 3 + 1, E E. E Galois ρ E,l N(ρ E,l ) = p l,l ord p( E ) { 2 (l ordl ( p, k(ρ E,l ) = E )) l + 1 (otherwise), ε(ρ E,l ) = 1. Serre > E := EllipticCurve([1,0,1,0,1]); Elliptic Curve defined by y^2 + x*y + y = x^3 + 1 over Rational Field > D := Discriminant(E); D; -639 > Factorization(D); [ <3, 2>, <71, 1> ]
10 Matching example E = Galois ρ 3 = ρ E,3 (N(ρ 3 ), k(ρ 3 ), ε(ρ 3 )) = (71, 4, 1) Tr(Frob p (ρ 3 )) E [ 1, 1, 2, 2, 0, -2, 0, 0, 0, -2, -10, -6,... ], (71, 4, 1). S new 4 (Γ 0 (71)). > S71 := CuspForms(Gamma0(71),4); > f := Newforms(S71,1); > [Coefficient(f,p) : p in [1..50] IsPrime(p)]; [ 1, 1, -16, -1, 24, 7, 72, -153, -213, 232, 149, -204, -432, 71, 273 ]
11 Matching example p ρ 1 * f 1 (1) mod 3 p ρ 1 * f 1 (1) ,.
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n=360 28.6% 34.4% 36.9% n=360 2.5% 17.8% 19.2% n=64 0.8% 0.3% n=69 1.7% 3.6% 0.6% 1.4% 1.9% < > n=218 1.4% 5.6% 3.1% 60.6% 0.6% 6.9% 10.8% 6.4% 10.3% 33.1% 1.4% 3.6% 1.1% 0.0% 3.1% n=360 0% 50%
[Oc, Proposition 2.1, Theorem 2.4] K X (a) l (b) l (a) (b) X [M3] Huber adic 1 Huber ([Hu1], [Hu2], [Hu3]) adic 1.1 adic A I I A {I n } 0 adic 2
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On the action of the Weil group on the l-adic cohomology of rigid spaces over local fields (Yoichi Mieda) Graduate School of Mathematical Sciences, The University of Tokyo 0 l Galois K F F q l q K, F K,
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Fermat s Last Theorem Hajime Mashima November 19, 2018 Abstract About 380 years ago, Pierre de Fermat wrote the following idea to Diophantus s Arithmetica. Cubum autem in duos cubos, aut quadratoquadratum
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Part3 j1701 March 15, 2013 (j1701) Part3 March 15, 2013 1 / 46 table of contents 1 2 3 (j1701) Part3 March 15, 2013 2 / 46 f : R 2 R 2 ( ) x f = y ( 1 1 1 1 ) ( x y ) = ( ) x y y x, y = x ( x y) 0!! (
2 Riemann Im(s) > 0 ζ(s) s R(s) = 2 Riemann [Riemann]) ζ(s) ζ(2) = π2 6 *3 Kummer s = 2n, n N ζ( 2) = 2 2, ζ( 4) =.3 2 3, ζ( 6) = ζ( 8)
![2 Riemann Im(s) > 0 ζ(s) s R(s) = 2 Riemann [Riemann]) ζ(s) ζ(2) = π2 6 *3 Kummer s = 2n, n N ζ( 2) = 2 2, ζ( 4) =.3 2 3, ζ( 6) = ζ( 8)](/thumbs/91/105920054.jpg "2 Riemann Im(s) > 0 ζ(s) s R(s) = 2 Riemann [Riemann]) ζ(s) ζ(2) = π2 6 *3 Kummer s = 2n, n N ζ( 2) = 2 2, ζ( 4) =.3 2 3, ζ( 6) = ζ( 8)")
(Florian Sprung) p 2 p * 9 3 p ζ Mazur Wiles 4 5 6 2 3 5 2006 http://www.icm2006.org/video/ eighth session [ ] Coates [Coates] 2 735 Euler n n 2 = p p 2 p 2 = π2 6 859 Riemann ζ(s) = n n s = p p s s ζ(s)
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Andrew Wiles 1953, 20 Fermat.. Fermat 10,. 1 Wiles. 19 20., Fermat 1. (Fermat). p 3 x p + y p =1 xy 0 x, y 2., n- t n =1 ζ n Q Q(ζ n ). Q F,., F = Q( 5) 6=2 3 = (1 + 5)(1 5) 2. Kummer Q(ζ p ), p Fermat
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Centralizers of Cantor minimal systems
Centralizers of Cantor minimal systems 1 X X X φ (X, φ) (X, φ) φ φ 2 X X X Homeo(X) Homeo(X) φ Homeo(X) x X Orb φ (x) = { φ n (x) ; n Z } x φ x Orb φ (x) X Orb φ (x) x n N 1 φ n (x) = x 1. (X, φ) (i) (X,
( ) 1., ([SU] ): F K k., Z p -, (cf. [Iw2], [Iw3], [Iw6]). K F F/K Z p - k /k., Weil., K., K F F p- ( 4.1).,, Z p -,., Weil..,,. Weil., F, F projectiv
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