On the refined class number formula of Gross
نویسندگان
چکیده
منابع مشابه
Kuroda’s Class Number Formula
Let k be a number field and K/k a V4-extension, i.e., a normal extension with Gal(K/k) = V4, where V4 is Klein’s four-group. K/k has three intermediate fields, say k1, k2, and k3. We will use the symbol N i (resp. Ni) to denote the norm of K/ki (resp. ki/k), and by a widespread abuse of notation we will apply N i and Ni not only to numbers, but also to ideals and ideal classes. The unit groups ...
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Proof. Using that (Z/p)× is a cyclic group of order p − 1 (i.e. the existence of primitive roots), we see that there is a square root of −1 (that is, a non-trivial fourth root of 1) in (Z/p)× if and only if p ≡ 1 mod 4. Suppose now that p ≡ −1 mod 4, and suppose that α and β are two elements of Z[i] such that p|αβ. Then p = N(p)|N(α)N(β), and so (after relabelling if necessary) we may assume th...
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We use the theory of Kolyvagin systems to prove (most of) a refined class number formula conjectured by Darmon. We show that for every odd prime p, each side of Darmon’s conjectured formula (indexed by positive integers n) is “almost” a p-adic Kolyvagin system as n varies. Using the fact that the space of Kolyvagin systems is free of rank one over Zp, we show that Darmon’s formula for arbitrary...
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with a given q ∈ F×. In particular, in the classical case with F = Q and V = Q, we usually assume that φ is Z-valued on Z and q ∈ Z. The purpose of the present article is to present some new ideas on various arithmetical questions on such an equation. We start with some of our basic symbols and terminology. For a set X we denote by #X or #{X} the number (≤ ∞) of elements of X. For an associativ...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2004
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2004.02.002