A Brief Introduction to Modular Forms Computation

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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> ]

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) *] *]

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) ,.

12 for Generalization Magma Hilbert / totally real case - Quaternion algebra Bianchi / imaginary quadratic case - Sharbly complex, Voronoi polyhedron. Bianchi Ver Bianchi.

13 Final Remark System for Algebra and Geometry Experimentation. W. Stein. Magma interpreter sage: magma.setdefaultrealfieldprecision(50) # magma >= v2.12; optional - magma sage: magma.eval( 1.1 ) # optional - magma (omitted)

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