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Automorphic Forms (Universitext) by Anton Deitmar PDF

By Anton Deitmar

ISBN-10: 144714435X

ISBN-13: 9781447144359

Automorphic varieties are a huge complicated analytic software in quantity conception and smooth mathematics geometry. They performed for instance a necessary position in Andrew Wiles's facts of Fermat's final Theorem. this article presents a concise advent to the area of automorphic kinds utilizing methods: the vintage hassle-free conception and the fashionable viewpoint of adeles and illustration conception. The reader will study the real goals and result of the idea via focussing on its crucial features and proscribing it to the 'base field' of rational numbers. scholars for instance in mathematics geometry or quantity idea will locate that this publication presents an optimum and simply obtainable creation into this subject.

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Note that for N ≥ 3 the group Γ (N) does not contain the element −1. Therefore, for such a group Γ there can exist non-zero modular forms of odd weight. 2 (a) The intersection of two congruence groups is a congruence group. (b) Let Γ be a congruence subgroup and let α ∈ GL2 (Q). Then Γ ∩ αΓ α −1 is also a congruence subgroup. Proof (a) Let Γ, Γ ⊂ Γ0 be congruence subgroups. By definition, there are M, N ∈ N with Γ (M) ⊂ Γ , Γ (N ) ⊂ Γ . Then Γ (MN) ⊂ (Γ (M) ∩ Γ (N )) ⊂ (Γ ∩ Γ ). (b) Fix N ≥ 2 such that Γ (N) ⊂ Γ .

0 Since h ∈ C ∞ (R/Z), the series k ck (h) converges absolutely as well and c (g) = (c (h) − c (h)) = 0. k k−1 k k k Now for the uniqueness of the Fourier coefficients. Let (ak )k∈Z be as in the proposition. By locally uniform convergence the following interchange of integration and summation is justified. For l ∈ Z we have 1 cl (g) = g(t)e−2πilt dt = 0 1 ak k=−∞ ∞ ak e2πkt e−2πilt dt 0 k=−∞ ∞ = 1 e2πkt e−2πilt dt. 0 One has 1 1 e2πkt e−2πilt dt = 0 0 e2π(k−l)t dt = 1 if k = l, 0 otherwise. This implies cl (g) = al .

Set M = M1 M2 N . We claim that Γ (M) ⊂ αΓ0 α −1 or equivalently α −1 Γ (M)α ⊂ Γ0 . For γ ∈ Γ (M) we write γ = I + Mg with g ∈ M2 (Z). It follows that α −1 γ α = I + N (M2 α −1 )g(M1 α) ∈ Γ (N ) ⊂ Γ . 19. The cusps of the fundamental domain DΓ lie in the set Γ (1)∞ = Q ∪ {∞}. The stabilizer group Γ (1)∞ of the point ∞ in Γ (1) is ± 1Z 1 . 3 Let Γ be a subgroup of finite index in Γ0 = SL2 (Z). For every c ∈ Q ∪ {∞} there exists a σc ∈ GL2 (Q)+ such that • σc ∞ = c and • σc−1 Γc σc = 1Z 1 ± 1 Z1 if −1 ∈ / Γ, if −1 ∈ Γ.

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Automorphic Forms (Universitext) by Anton Deitmar

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