By Min Ho Lee

ISBN-10: 3540219226

ISBN-13: 9783540219224

This quantity offers with quite a few themes round equivariant holomorphic maps of Hermitian symmetric domain names and is meant for experts in quantity concept and algebraic geometry. specifically, it includes a finished exposition of combined automorphic types that hasn't ever but seemed in ebook shape. the most objective is to discover connections between complicated torus bundles, combined automorphic varieties, and Jacobi varieties linked to an equivariant holomorphic map. either number-theoretic and algebro-geometric features of such connections and comparable subject matters are discussed.

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This quantity offers with a variety of issues round equivariant holomorphic maps of Hermitian symmetric domain names and is meant for experts in quantity thought and algebraic geometry. particularly, it incorporates a entire exposition of combined automorphic kinds that hasn't ever but seemed in e-book shape. the most target is to discover connections between advanced torus bundles, combined automorphic kinds, and Jacobi varieties linked to an equivariant holomorphic map.

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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.

### Mixed Automorphic Forms, Torus Bundles, and Jacobi Forms by Min Ho Lee

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