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admissible change aﬃne algebraic curve algebraic number ap(E ap(f automorphism bijection change of variable Chapter character modulo coeﬃcients compact Riemann surface complex elliptic curve complex tori compute congruence subgroup converges coordinate corresponding curve over Q cusp forms deﬁned Deﬁnition denote diagram Dirichlet character divisors double coset double coset operator Eisenstein series element elliptic curve elliptic points equivalence classes Exercise extends ﬁnite ﬁrst formula Fourier expansion Frobp function ﬁeld G SL2(Z Galois group Galois representation gives half plane Hecke operators hint holomorphic isogeny isomorphism Jacobian Lemma linear matrix maximal ideal meromorphic functions modular curve modular forms Modularity Theorem moduli space morphism multiplicative newform nonzero number ﬁeld number theory polynomial positive integer prime proof Proposition quotient rational Recall reduction relation result ring satisfies Section Show Similarly surjects topology transform vector space Version Weierstrass equation Z/NZ zero
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