By Rick Miranda

During this e-book, Miranda takes the procedure that algebraic curves are top encountered for the 1st time over the complicated numbers, the place the reader's classical instinct approximately surfaces, integration, and different techniques might be introduced into play. accordingly, many examples of algebraic curves are offered within the first chapters. during this manner, the ebook starts as a primer on Riemann surfaces, with advanced charts and meromorphic capabilities taking middle degree. however the major examples come from projective curves, and slowly yet without doubt the textual content strikes towards the algebraic type. Proofs of the Riemann-Roch and Serre Duality Theorems are provided in an algebraic demeanour, through an version of the adelic evidence, expressed thoroughly by way of fixing a Mittag-Leffler challenge. Sheaves and cohomology are brought as a unifying machine within the latter chapters, in order that their application and naturalness are instantly noticeable. Requiring a historical past of a one semester of advanced variable! conception and a 12 months of summary algebra, this is often a great graduate textbook for a second-semester path in complicated variables or a year-long path in algebraic geometry.

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J2 ) ∈ Z2j , ν = (ν 1 , ν 2 ) = (ν11 , . . , νk1 , ν12 , . . , νk2 ) ∈ Z2k , γ= ab cd ∈ Γ, χ(γ) = aχ b χ cχ dχ ∈ SL(2, Z). Then we have (γ, µ, ν) · (γ , µ , ν ) = (γγ , γ · (µ , ν ) + (µ, ν)), where γ · (µ , ν ) = (µ , ν ) with µ = (aµ11 + bµ12 , . . , aµj1 + bµj2 , cµ11 + dµ12 , . . , cµj1 + dµj2 ) ∈ Z2j , ν = (aχ ν11 + bχ ν12 , . . , aχ νk1 + bχ νk2 , cχ ν11 + dχ ν12 , . . , cχ νk1 + dχ νk2 ) ∈ Z2k . 9) −1 = γz, (cz + d) (ξ + zµ1 + µ2 ), (cχ ω(z) + dχ )−1 (ζ + ω(z)ν 1 + ν 2 ) for γ ∈ Γ , z ∈ H, ξ ∈ Cj , ζ ∈ Ck , µ = (µ1 , µ2 ) ∈ Z2j , and ν = (ν 1 , ν 2 ) ∈ Z2k .

L+1 ∈ SL(2, Z). 26) we see that l−1 {α, β, p, q}ω,χ = {α, η1 , p, q}ω,χ + {ηi , ηi+1 , p, q}ω,χ + {ηl , β, p, q}ω,χ . i=1 Thus in order to prove the proposition it suﬃces to consider the case where (α, β) = (γ(0), γ(i∞)) for some element γ in SL(2, Z). 31) {α, β, p, q}ω,χ = γ · {0, i∞, p , q }ω,χ with (p , q ) = (p, q)χ(γ)−1 or (p, q) = (p , q )χ(γ); hence we have p = a1 p + c1 q, for χ(γ) = aχ bχ cχ d χ q = b1 p + d1 q ∈ SL(2, Z). Let p = (p1 , . . , pm ), q = (q1 , . . , qm ), and deﬁne ck (p , q ) by m m ck (p, q)X k Y m−k .

Let f ψ be a solution of the nonhomogeneous equation Λn f = ψ, and suppose that the solution of Λn f = ψ obtained by the analytic continuation of f ψ around a closed path γ ∈ π1 (U, x0 ) is given by ψ ψ ψ t f ψ + aψ γ,1 ω1 + · · · + aγ,n ωn = f + aγ ω, where t ψ aγ ψ = (aψ γ,1 , . . , aγ,n ), t ω = ω 1 , . . 42) ψ with aψ γ,1 , . . , aγ,n ∈ C. If τ ∈ π1 (U, x0 ) is another closed path, then the ψ analytic continuation of f ψ + t aγ ω around τ becomes ψ ψ f ψ + t aτ ω + t aγ M (τ ). On the other hand, since the analytic continuation of f ψ around γτ is f ψ + t ψ aγτ ω, it follows that 30 1 Mixed Automorphic Forms t ψ aγτ ψ ψ = t aτ + t aγ M (τ ).