Class number divisibility for imaginary quadratic fields
نویسندگان
چکیده
منابع مشابه
Divisibility Criteria for Class Numbers of Imaginary Quadratic Fields
In a recent paper, Guerzhoy obtained formulas for certain class numbers as p-adic limits of traces of singular moduli. Using earlier work by Bruinier and the second author, we derive a more precise form of these results using results of Zagier. Specifically, if −d < −4 is a fundamental discriminant and n is a positive integer, then Tr(pd) ≡ 24 p− 1 · ( 1− (−d p )) ·H(−d) (mod p) provided that p...
متن کاملThe Dirichlet Class Number Formula for Imaginary Quadratic Fields
because 2, 3, and 1± √ −5 are irreducible and nonassociate. These notes present a formula that in some sense measures the extent to which unique factorization fails in environments such as Z[ √ −5]. Algebra lets us define a group that measures the failure, geometry shows that the group is finite, and analysis yields the formula for its order. To move forward through the main storyline without b...
متن کاملThe Dirichlet Class Number Formula for Imaginary Quadratic Fields
Z[ √ −5] = {a+ b √ −5 : a, b ∈ Z}, because 2, 3, and 1± √ −5 are irreducible and nonassociate. These notes present a formula that in some sense measures the extent to which unique factorization fails in environments such as Z[ √ −5]. Algebra lets us define a group that measures the failure, geometry shows that the group is finite, and analysis yields the formula for its order. To move forward t...
متن کاملThe Dirichlet Class Number Formula for Imaginary Quadratic Fields
Z[ √ −5] = {a+ b √ −5 : a, b ∈ Z}, because 2, 3, and 1± √ −5 are irreducible and nonassociate. These notes present a formula that in some sense measures the extent to which unique factorization fails in environments such as Z[ √ −5]. The large-scale methodology deserves immediate note, before the reader is immersed in a long succession of smaller attention-filling specifics: • algebra lets us d...
متن کاملClass number formula for certain imaginary quadratic fields
In this note we shall show how Carlitz in 1954 could have reached an analogue of the Voronoi congruence in the more difficult case of p≡1(mod4): h(-4p) ≡B(p+1)/2(x4)(mod p), where B(p+1)/2(x4) is the generalized Bernoulli number with x4 being the Kronecker symbol associated to the Gaussian field Q(√-4).
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ژورنال
عنوان ژورنال: Research in Number Theory
سال: 2020
ISSN: 2522-0160,2363-9555
DOI: 10.1007/s40993-020-0188-4