Displaying similar documents to “A determinant formula for the relative class number of an imaginary abelian number field”

Bernoulli numbers, Hurwitz numbers, p-adic L-functions and Kummer's criterion.

Alvaro Lozano Robledo (2007)

RACSAM

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Let K = Q(ζ) and let h be its class number. Kummer showed that p divides h if and only if p divides the numerator of some Bernoulli number. In this expository note we discuss the generalizations of this type of criterion to totally real fields and quadratic imaginary fields.

A generalization of Scholz’s reciprocity law

Mark Budden, Jeremiah Eisenmenger, Jonathan Kish (2007)

Journal de Théorie des Nombres de Bordeaux

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We provide a generalization of Scholz’s reciprocity law using the subfields K 2 t - 1 and K 2 t of ℚ ( ζ p ) , of degrees 2 t - 1 and 2 t over ℚ , respectively. The proof requires a particular choice of primitive element for K 2 t over K 2 t - 1 and is based upon the splitting of the cyclotomic polynomial Φ p ( x ) over the subfields.

Non-abelian congruences between L -values of elliptic curves

Daniel Delbourgo, Tom Ward (2008)

Annales de l’institut Fourier

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Let E be a semistable elliptic curve over ℚ . We prove weak forms of Kato’s K 1 -congruences for the special values L 1 , E / ℚ ( μ p n , Δ p n ) . More precisely, we show that they are true modulo p n + 1 , rather than modulo p 2 n . Whilst not quite enough to establish that there is a non-abelian L -function living in K 1 ℤ p [ [ Gal ( ℚ ( μ p ∞ , Δ p ∞ ) / ℚ ) ] ] , they do provide strong evidence towards the existence of such an analytic object. For example, if n = 1 these verify the numerical congruences found by Tim and Vladimir Dokchitser.