By Robert A. Rankin

ISBN-10: 052121212X

ISBN-13: 9780521212120

This booklet offers an advent to the idea of elliptic modular services and kinds, a subject matter of accelerating curiosity as a result of its connexions with the idea of elliptic curves. Modular types are generalisations of capabilities like theta services. they are often expressed as Fourier sequence, and the Fourier coefficients usually own multiplicative houses which bring about a correspondence among modular kinds and Dirichlet sequence having Euler items. The Fourier coefficients additionally come up in convinced representational difficulties within the conception of numbers, for instance within the examine of the variety of ways that a good integer will be expressed as a sum of a given variety of squares. The remedy of the speculation awarded this is fuller than is common in a textbook on automorphic or modular varieties, because it isn't restricted exclusively to modular kinds of indispensable weight (dimension). it is going to be of curiosity to expert mathematicians in addition to senior undergraduate and graduate scholars in natural arithmetic.

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Therefore it suﬃces to show that each φν,δ |2k,2m σ0−1 is holomorphic at ∞ and that it has zero at ∞ if ν > 0. First, suppose that δs0 is not a cusp of Γs . Then δ −1 Γs δ ∩ Γs0 coincides with {1} or {±1}, and hence we have 22 1 Mixed Automorphic Forms φν,δ |2k,2m σ0−1 = C · (φν |2k,2m δσ0−1 σ0 ησ0−1 ) η∈Γs0 with C = 1 or 1/2, respectively. 25) for s = δs0 , σ = σ0 δ −1 , we obtain |(φν |2k,2m δσ0−1 )(z)| ≤ M |z|−2k for all z with Im z > λ for some M, λ > 0. 28) α∈Z where b is a positive real number such that σ0 Γs0 σ0−1 · {±1} = ± ( 10 1b ) α α∈Z .

2 associated to the elliptic ﬁbration π : E → X. 9 There is a canonical pairing , : H1 (X, Σ, (R1 π∗ Q)m ) × H 0 (E m , Ω m+1 ⊕ Ω that is nondegenerate on the right. Proof. 1]. 18) is given by [δ], Φ = Φ δ m+1 for all [δ] ∈ H1 (X, Σ, (R1 π∗ Q)m ) and Φ ∈ H 0 (E m , Ω m+1 ⊕Ω ), where δ is a cycle representing the cohomology class [δ]. 7, we obtain the canonical pairing , : H1 (X, Σ, (R1 π∗ Q)m ) × (S2,m (Γ, ω, χ) ⊕ S2,m (Γ, ω, χ)) −→ C. 10 Let ∂ : H1 (X, Σ, (R1 π∗ Q)m ) → H0 (Σ, (R1 π∗Q)m ) be the boundary map for the homology sequence of the pair (X, Σ).

29) we have ξ(e, k) = {γα, γβ, dχ q − cχ p, −bχ q + aχ p}ω,χ , where α = 0, β = i∞, p = 1k and q = 1m − 1k . 24), we see that m γβ ((dχ pi − cχ qi )ω(z) + (−bχ pi + aχ qi ))dz f1 (z) ξ(e, f ), f = γα i=1 m γβ γα i=1 m β = ((dχ pi − cχ qi )ω(z) + (−bχ pi + aχ qi ))dz f2 (z) + ((dχ pi − cχ qi )χ(γ)ω(z) f1 (γz) α i=1 + (−bχ pi + aχ qi ))(cz + d)−2 dz m β + f2 (γz) α ((dχ pi − cχ qi )χ(γ)ω(z) i=1 −2 + (−bχ pi + aχ qi ))(cz + d) dz. Using the relation χ(γ)ω(z) = (aχ ω(z) + bχ )(cχ ω(z) + dχ )−1 , we obtain i∞ i∞ (f1 |2,m γ)ω(z)k dz + ξ(e, k), f = 0 = r(e, k, f ); hence the proposition follows.

### Modular forms and functions by Robert A. Rankin

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