By Fred Diamond, Jerry Shurman

This ebook introduces the speculation of modular types, from which all rational elliptic curves come up, with an eye fixed 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 idea; Hecke eigenforms and their mathematics houses; the Jacobians of modular curves and the Abelian kinds linked to Hecke eigenforms. because it offers those principles, the booklet states the Modularity Theorem in numerous varieties, bearing on them to one another and relating their purposes to quantity idea. The authors think no historical past in algebraic quantity concept and algebraic geometry. workouts are integrated.

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**Extra info for A First Course in Modular Forms (Graduate Texts in Mathematics)**

**Sample text**

The set of equivalence classes is denoted S0 (N ) = {enhanced elliptic curves for Γ0 (N )}/ ∼ . An element of S0 (N ) is denoted [E, C], the square brackets connoting equivalence class. An enhanced elliptic curve for Γ1 (N ) is a pair (E, Q) where E is a complex elliptic curve and Q is a point of E of order N . ) Two such pairs (E, Q) and (E , Q ) are equivalent if some ∼ isomorphism E −→ E takes Q to Q . The set of equivalence classes is denoted S1 (N ) = {enhanced elliptic curves for Γ1 (N )}/ ∼ .

Ad − bc − kN = 1 for some a, b, and k, and the matrix γ = ac db ∈ M2 (Z) reduces modulo N into SL2 (Z/N Z). Modifying the entries of γ modulo N doesn’t aﬀect Q, so since SL2 (Z) surjects to SL2 (Z/N Z) we may take γ = a b ∈ SL (Z). Let τ = γ(τ ) and let m = cτ + d. 1 for the third equality), and m 1 + Λτ N = cτ + d + Λτ = Q. N This shows that [E, Q] = [C/Λτ , 1/N + Λτ ] where τ ∈ H. Suppose two points τ, τ ∈ H satisfy Γ1 (N )τ = Γ1 (N )τ . Thus τ = γ(τ ) where γ = ac db ∈ Γ1 (N ). Again let m = cτ + d.

4. Let Γ be a congruence subgroup of SL2 (Z) of level N , and let qN = e2πiτ /N for τ ∈ H. 3 and satisﬁes (3 ) In the Fourier expansion f (τ ) = satisfy the condition |an | ≤ Cnr ∞ n=0 n an qN , the coeﬃcients for n > 0 for some positive constants C and r. 3, and so f ∈ Mk (Γ ). For instance, the proposition easily shows that θ(τ, 4) ∈ M2 (Γ0 (4)). 9. 3 reduces to previous ones when Γ = SL2 (Z). As before, the modular forms and the cusp forms are vector spaces and subspaces, and the sums M(Γ ) = Mk (Γ ) and S(Γ ) = Sk (Γ ) k∈Z k∈Z form a graded ring and a graded ideal.