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Read e-book online A First Course in Modular Forms (Graduate Texts in PDF

By Fred Diamond, Jerry Shurman

This booklet 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 advanced 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 homes; the Jacobians of modular curves and the Abelian types linked to Hecke eigenforms. because it offers those rules, the publication states the Modularity Theorem in a number of varieties, referring to them to one another and relating their functions to quantity conception. The authors think no historical past in algebraic quantity conception and algebraic geometry. routines are incorporated.

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11). 7. Show that if two enhanced elliptic curves for Γ of the special type (C/Λτ , (torsion data)) are equal then τ = γ(τ ) for some γ = [ 10 1b ] ∈ Γ . 13) gives a well defined function F on enhanced elliptic curves of the special type. 8. (a) Show that the function F2v from the section is degree-2 homogeneous with respect to Γ (N ). (b) Similarly, show that the functions F0v , F0d , and F0 are degree-0 homogeneous with respect to Γ (N ), Γ1 (N ), and Γ0 (N ) respectively. 2 Modular Curves as Riemann Surfaces For any congruence subgroup Γ of SL2 (Z) the corresponding modular curve has been defined as the quotient space Γ \H, the set of orbits Y (Γ ) = {Γ τ : τ ∈ H}.

The function ∞ (1 − q n )24 , η 24 (τ ) = q q = e2πiτ n=1 is invariant under τ → τ + 1 and satisfies η 24 (−1/τ ) = τ 12 η 24 (τ ). Also, limIm(τ )→∞ η 24 (τ ) = 0, so η 24 ∈ S12 (SL2 (Z)). 1, is 1-dimensional. Comparing the leading terms of the Fourier expansions η 24 (τ ) = q+· · · and ∆(τ ) = (2π)12 q + · · · thus gives the remarkable identity ∆ = (2π)12 η 24 , or 60 (c,d) 1 (cτ + d)4 3 − 27 140 (c,d) 1 (cτ + d)6 2 ∞ (1 − q n )24 . 2 Congruence subgroups 21 More on the eta function, especially its connection with theta functions per the beginning of this section, is in Chapter 1 of [Bum97].

QN →0 If α fixes ∞ then this is immediate from the Fourier series of f itself. 2 Congruence subgroups 23 lim |(f [α]k )(τ ) · qN | ≤ C lim (y r−k |qN |). qN →0 qN →0 Recalling that qN = e , show that y = C log(1/|qN |), and use the fact that polynomials dominate logarithms to complete the proof. 7. 4. 8. (a) Verify the Fourier expansion of G2 . 4) for two particular matrices γ1 , γ2 ∈ SL2 (Z). Show that G2 then satisfies the identity for the product γ1 γ2 and the inverse γ1−1 as well. 4) it suffices to prove the identity for a set of generators for SL2 (Z).

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A First Course in Modular Forms (Graduate Texts in Mathematics) by Fred Diamond, Jerry Shurman

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