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Hilbert symbols, class groups and quaternion algebras

Ted ChinburgEduardo Friedman — 2000

Journal de théorie des nombres de Bordeaux

Let B be a quaternion algebra over a number field k . To a pair of Hilbert symbols { a , b } and { c , d } for B we associate an invariant ρ = ρ R [ 𝒟 ( a , b ) ] , [ 𝒟 ( c , d ) ] in a quotient of the narrow ideal class group of k . This invariant arises from the study of finite subgroups of maximal arithmetic kleinian groups. It measures the distance between orders 𝒟 ( a , b ) and 𝒟 ( c , d ) in B associated to { a , b } and { c , d } . If a = c , we compute ρ R ( [ 𝒟 ( a , b ) ] , [ 𝒟 ( c , d ) ] ) by means of arithmetic in the field k ( a ) . The problem of extending this algorithm to the general case leads to studying a finite graph associated...

Raabe’s formula for p -adic gamma and zeta functions

Henri CohenEduardo Friedman — 2008

Annales de l’institut Fourier

The classical Raabe formula computes a definite integral of the logarithm of Euler’s Γ -function. We compute p -adic integrals of the p -adic log Γ -functions, both Diamond’s and Morita’s, and show that each of these functions is uniquely characterized by its difference equation and p -adic Raabe formula. We also prove a Raabe-type formula for p -adic Hurwitz zeta functions.

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