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

By Fred Diamond, Jerry Shurman

This ebook introduces the idea of modular varieties, 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 conception; 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 e-book states the Modularity Theorem in quite a few varieties, concerning them to one another and referring to their functions to quantity idea. The authors imagine no heritage in algebraic quantity concept and algebraic geometry. routines are integrated.

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3. zero out of five stars An exploration of the habit of enormous numbers. July thirteen, 2004
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3(b–d) for more properties of the Weil pairing, in particular that the Weil pairing is preserved under isomorphisms of complex tori. 5. 1. 1. 2. 3. 3. (a) Show that the Weil pairing is independent of which basis {ω1 , ω2 } is used, provided ω1 /ω2 ∈ H. (b) Show that the Weil pairing is bilinear, alternating, and nondegenerate. ) (c) Show that the Weil pairing is compatible with N . This means that for positive integers N and d, the diagram E[dN ] × E[dN ] edN (·,·) / µdN d(·,·)  E[N ] × E[N ]  eN (·,·) ·d / µN commutes, where the vertical maps are suitable multiplications by d.

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 .

N∈Z To show that the Laurent series truncates from the left to a power series it suffices to show that lim ((f [α]k )(τ ) · qN ) = 0. 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).

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