# Algebraische Zahlentheorie by Jürgen Neukirch By Jürgen Neukirch

Die algebraische Zahlentheorie ist eine der traditionsreichsten und gleichzeitig heute besonders aktuellen Grunddisziplinen der Mathematik. In dem vorliegenden Buch wird sie in einem ausf?hrlichen und weitgefa?ten Rahmen abgehandelt, der sowohl die Grundlagen als auch ihre H?hepunkte enth?lt. Die Darstellung f?hrt den Leser in konkreter Weise in das Gebiet ein, l??t sich dabei von modernen Erkenntnissen ?bergeordneter Natur leiten und ist in vielen Teilen neu. Der grundlegende erste Teil ist mit einigen neuen Aspekten versehen, wie etwa einer ausf?hrlichen Theorie der Ordnungen. ?ber die Grundlagen hinaus enth?lt das Buch eine geometrische Neubegr?ndung der Theorie der algebraischen Zahlk?rper durch die Entwicklung einer "Riemann-Roch-Theorie" vom "Arakelovschen Standpunkt", die bis zu einem "Grothendieck-Riemann-Roch-Theorem" f?hrt, ferner lokale und globale Klassenk?rpertheorie und schlie?lich eine Darstellung der Theorie der Theta- und L-Reihen, die die klassischen Arbeiten von Hecke in eine fa?liche shape setzt.

Das Buch wendet sich an Studenten nach dem Vordiplom bzw. Bachelor. Dar?ber hinaus ist es dem Forscher als weiterweisendes Handbuch unentbehrlich.

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We often call s the spectral parameter. Note that Wl (P, n, s) = Wl (P, n, −s). In the literature one often meets the parametrization λ = s(1−s). In comparing results one then needs to make the translation s → s + 12 . 3. The choice λ = s(1 − s) has the advantage of placing the critical line at Re s = 12 , which reminds us of the Riemann zeta function. 7. 2 Diﬀerential equation. 1 is determined by a function f of one variable: F (gP p(z)k(θ)) if P ∈ X ∞ = e2πinx f (y)eilθ if P ∈ PY . 3. This second order linear diﬀerential operator lP,n has real analytic coeﬃcients, so any function f corresponding to an element of Wl (P, n, s) is a real analytic function on (0, ∞).

Product of the subgroups A = { a(y) : y > 0 } = R>0 and N ∼ vzmn(x)x ∈ R = R. This leaves only the deﬁnition of the multiplication of k(θ) ˜ I invite the and p(z). The deﬁnition above lifts the product of p(z) and k(θ) to G. ˜ and that g → gˆ gives a surjective reader to check that we now have a Lie group G, ˜ → G, with kernel central in G. ˜ homomorphism G ˜ It is generated by ζ = k(π). We denote The group Z˜ is the center of G. ˜ Z˜ ∼ ¯ If we use H as a model of hyperbolic plane the quotient G/ = G/{±Id} by G.

4 Iwasawa coordinates. The isomorphism of analytic varieties G ˜ = N ˜ A˜K. ˜ It leads to the Iwasawa corresponds to the Iwasawa decomposition G ˜ coordinates on G: p(z)k(θ) → (x, y, θ). In these coordinates W E = ∂θ ± = e±2iθ (±2iy∂x + 2y∂y ∓ i∂θ ) ω = −y 2 ∂x2 − y 2 ∂y2 + y∂x ∂θ . 5 Polar coordinates. The polar decomposition G ˜ ˜ coordinates on G K: k(η)a(tu )k(ψ) → (u, η, ψ) with η, ψ ∈ R, η + ψ mod πZ, u ∈ (0, ∞) and tu = 1 + 2u + 2 u2 + u > 1, u= (tu − 1)2 > 0. 4tu 30 Chapter 2 Universal covering group We have W = ∂ψ .

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