Get Analytic Theory of Continued Fractions III: Proceedings of a PDF

By Lisa Jacobsen

ISBN-10: 0387518304

ISBN-13: 9780387518305

ISBN-10: 3540518304

ISBN-13: 9783540518303

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Extra resources for Analytic Theory of Continued Fractions III: Proceedings of a Seminar Workshop

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Thus f [σδγ]k = Tn,k (f ) = σδγ∈Xn f [δγ]k = Tn,k (f )[γ]k . δ∈Xn A finite sum of meromorphic function is meromorphic, so Tn,k (f ) is weakly modular. If f is holomorphic on h, then each f [δ]k is holomorphic on h for δ ∈ Xn . A finite sum of holomorphic functions is holomorphic, so Tn,k (f ) is holomorphic. We will frequently drop k from the notation in Tn,k , since the weight k is implicit in the modular function to which we apply the Hecke operator. Henceforth we make the convention that if we write Tn (f ) and if f is modular, then we mean Tn,k (f ), where k is the weight of f .

We will also consider the relative homology group H1 (X0 (N ), Z; {cusps}) of X0 (N ) relative to the cusps; it is the same as usual homology, but in addition we allow paths with endpoints in the cusps instead of restricting to closed loops. Modular symbols provide a “combinatorial” presentation of H1 (X0 (N ), Z) in terms of paths between elements of P1 (Q). 2. Modular Symbols Let M2 be the free abelian group with basis the set of symbols {α, β} with α, β ∈ P1 (Q) modulo the 3-term relations {α, β} + {β, γ} + {γ, α} = 0 above and modulo any torsion.

2 Prove that if n > 1 is odd, then the Bernoulli number Bn is 0. 3) to write down the coefficients of 1, q, q 2 , and q 3 of the Eisenstein series E8 . 4 Suppose k is a positive integer with k ≡ 0 (mod 12). Suppose a, b ≥ 0 are integers with 4a + 6b = k. (a) Prove 3 | a. k (b) Show that Ga4 · Gb6 / G66 = G34 /G26 a 3 . 5 Compute the Miller basis for M28 (SL2 (Z)) with precision O(q 8 ). 23. 6 Consider the cusp form f = q 2 +192q 3 −8280q 4 +· · · in S28 (SL2 (Z)). 25). 1). Let c be the complex number so that the constant coefficient of the q-expansion of g = c · Gk is 1.

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Analytic Theory of Continued Fractions III: Proceedings of a Seminar Workshop by Lisa Jacobsen


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