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

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 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 homes; the Jacobians of modular curves and the Abelian forms linked to Hecke eigenforms. because it offers those rules, the booklet states the Modularity Theorem in a number of varieties, touching on them to one another and relating their functions to quantity thought. The authors imagine no history in algebraic quantity idea and algebraic geometry. routines are integrated.

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4. Let wN = N0 −10 ∈ GL+ 2 (Q). Show that wN normalizes the group Γ0 (N ) and so gives an automorphism Γ0 (N )τ → Γ0 (N )wN (τ ) of the modular curve Y0 (N ). Show that this automorphism is an involution (meaning it has order 2) and describe the corresponding automorphism of the moduli space S0 (N ). 5. Let N be a positive integer and let p be prime. Define maps π1 , π2 : Y0 (N p) −→ Y0 (N ) to be π1 (Γ0 (N p)τ ) = Γ0 (N )τ and π2 (Γ0 (N p)τ ) = Γ0 (N )(pτ ). How do the corresponding maps π ˆ1 , π ˆ2 : S0 (N p) −→ S0 (N ) act on equivalence classes [E, C]?

In sum, |f (τ )| ≤ C0 + C/y r as y → ∞. (b) For every α ∈ SL2 (Z), the transformed function (f [α]k )(τ ) is holomorphic and weight-k invariant under α−1 Γ α and therefore has a Laurent expansion n (f [α]k )(τ ) = an qN , qN = e2πiτ /N . 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 |).

Recall that a holomorphic isomorphism of complex tori takes the form z + Λ → mz + Λ where Λ = mΛ. Since ℘mΛ (mz) = m−2 ℘Λ (z) and ℘mΛ (mz) = m−3 ℘Λ (z), the corresponding isomorphism of elliptic curves is (x, y) → (m−2 x, m−3 y) or equivalently the substitution (x, y) = (m2 x , m3 y ), changing the cubic equation y 2 = 4x3 − g2 x − g3 associated to Λ to the equation y 2 = 4x3 − m−4 g2 x − m−6 g3 associated to Λ . 3 again) normalize the elliptic curves associated to C/mΛi and C/mΛµ3 to have equations y 2 = 4x(x − 1)(x + 1), y 2 = 4(x − 1)(x − µ3 )(x − µ23 ).

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