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Let K = Q(ζp) and let hp be its class number. Kummer showed that p divides hp 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.

Bicyclotomic polynomials and impossible intersections

David Masser, Umberto Zannier (2013)

Journal de Théorie des Nombres de Bordeaux

In a recent paper we proved that there are at most finitely many complex numbers $t \neq 0, 1$ such that the points $(2, \sqrt{2 (2 - t)})$ and $(3, \sqrt{6 (3 - t)})$ are both torsion on the Legendre elliptic curve defined by $y^{2} = x (x - 1) (x - t)$ . In a sequel we gave a generalization to any two points with coordinates algebraic over the field $Q (t)$ and even over $C (t)$ . Here we reconsider the special case $(u, \sqrt{u (u - 1) (u - t)})$ and $(v, \sqrt{v (v - 1) (v - t)})$ with complex numbers $u$ and $v$ .

Binomial squares in pure cubic number fields

Franz Lemmermeyer (2012)

Journal de Théorie des Nombres de Bordeaux

Let $K = ℚ (ω)$ , with $ω^{3} = m$ a positive integer, be a pure cubic number field. We show that the elements $α \in K^{\times}$ whose squares have the form $a - ω$ for rational numbers $a$ form a group isomorphic to the group of rational points on the elliptic curve $E_{m} : y^{2} = x^{3} - m$ . This result will allow us to construct unramified quadratic extensions of pure cubic number fields $K$ .

Bounds for arithmetic multiplicities.

Duke, W. (1998)

Documenta Mathematica

Calculating Root Numbers of Elliptic curves over Q.

Ian Connell (1994)

Manuscripta mathematica

Class fields of abelian extensions of Q.

B. Mazur, A. Wiles (1984)

Inventiones mathematicae

Class invariants of Mordell-Weil groups.

M.J. Taylor, A. Agboola (1994)

Journal für die reine und angewandte Mathematik

Class numbers of quadratic fields and Shimura's correspondence.

Jan Nekovár (1990)

Mathematische Annalen

CM liftings of supersingular elliptic curves

Ben Kane (2009)

Journal de Théorie des Nombres de Bordeaux

Assuming GRH, we present an algorithm which inputs a prime $p$ and outputs the set of fundamental discriminants $D < 0$ such that the reduction map modulo a prime above $p$ from elliptic curves with CM by $𝒪_{D}$ to supersingular elliptic curves in characteristic $p$ is surjective. In the algorithm we first determine an explicit constant $D_{p}$ so that $| D | > D_{p}$ implies that the map is necessarily surjective and then we compute explicitly the cases $| D | < D_{p}$ .

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Displaying 41 – 60 of 395

Average distributions and products of special values of L-series

Average Frobenius distribution of elliptic curves

Average Frobenius distributions for elliptic curves over abelian extensions

Average Frobenius distributions for elliptic curves with nontrivial rational torsion

Average size of 2-Selmer groups of elliptic curves, II

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

Bicyclotomic polynomials and impossible intersections

Binomial squares in pure cubic number fields

Bounds for arithmetic multiplicities.

Calculating Root Numbers of Elliptic curves over Q.

Class fields of abelian extensions of Q.

Class invariants of Mordell-Weil groups.

Class numbers of quadratic fields and Shimura's correspondence.

CM liftings of supersingular elliptic curves

Compatible families of elliptic type

Completely faithful Selmer groups over Kummer extensions.

Computation of the Selmer groups of certain parametrized elliptic curves

Computer-aided serendipity

Computing integral points on elliptic curves

Computing integral points on Mordell's elliptic curves.

Currently displaying 41 – 60 of 395