By Fred Diamond, Jerry Shurman

This publication introduces the speculation of modular types, from which all rational elliptic curves come up, with a watch towards the Modularity Theorem. dialogue covers elliptic curves as complicated tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner thought; Hecke eigenforms and their mathematics houses; the Jacobians of modular curves and the Abelian types linked to Hecke eigenforms. because it provides those rules, the e-book states the Modularity Theorem in numerous kinds, bearing on them to one another and referring to their purposes to quantity concept. The authors think no heritage in algebraic quantity thought and algebraic geometry. workouts are incorporated.

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Let Γ be a congruence subgroup of SL2 (Z) and let k be an integer. A function f : H −→ C is a modular form of weight k with respect to Γ if (1) f is holomorphic, (2) f is weight-k invariant under Γ , (3) f [α]k is holomorphic at ∞ for all α ∈ SL2 (Z). If in addition, (4) a0 = 0 in the Fourier expansion of f [α]k for all α ∈ SL2 (Z), then f is a cusp form of weight k with respect to Γ . The modular forms of weight k with respect to Γ are denoted Mk (Γ ), the cusp forms Sk (Γ ). The cusp conditions (3) and (4) are phrased independently of the congruence subgroup Γ .

For τ ∈ H, extend the formula + j(γ, τ ) = cτ +d to γ ∈ GL+ 2 (Q), and extend the weight-k operator to GL2 (Q) by the rule (f [γ]k )(τ ) = (det γ)k−1 j(γ, τ )−k f (γ(τ )) for f : H −→ C. 2(b). (b) Show that every γ ∈ GL+ 2 (Q) satisﬁes γ = αγ where α ∈ SL2 (Z) and γ = r a0 db with r ∈ Q+ and a, b, d ∈ Z relatively prime. Use this to show that given f ∈ Mk (Γ ) for some congruence subgroup Γ and given such γ = αγ , since f [α]k has a Fourier expansion, so does f [γ]k . Show that if the Fourier expansion for f [α]k has constant term 0 then so does the Fourier expansion for f [γ]k .

We now develop the deﬁnition of a modular form with respect to a congruence subgroup. Let k be an integer and let Γ be a congruence subgroup of SL2 (Z). A function f : H −→ C is a modular form of weight k with respect to Γ if it is weakly modular of weight k with respect to Γ and satisﬁes a holomorphy condition to be described below. 1. Each congruence subgroup Γ of SL2 (Z) contains a translation matrix of the form 1h :τ →τ +h 01 for some minimal h ∈ Z+ . This is because Γ contains Γ (N ) for some N , but h may properly divide N —for example, the group Γ1 (N ) contains the translation matrix [ 10 11 ].