Results 11 to 20 of about 130 (32)

On the class numbers of certain number fields obtained from points on elliptic curves II [PDF]

, 2008
Let k be a number field of finite degree and k an algebraic closure of k, and let E/k be an elliptic curve which is given by the Weierstrass equation of the form y2 = f(x), where f(x) 2 k[x] is a cubic polynomial. For a subset Ξ of P1(k) (regarded as k [
Sato, Atsushi
core   +3 more sources

A remark on Tate's algorithm and Kodaira types

, 2013
We remark that Tate's algorithm to determine the minimal model of an elliptic curve can be stated in a way that characterises Kodaira types from the minimum of v(a_i)/i.
Dokchitser, Tim, Dokchitser, Vladimir
core   +1 more source

Distribution of Farey Fractions in Residue Classes and Lang--Trotter Conjectures on Average

, 2007
We prove that the set of Farey fractions of order $T$, that is, the set $\{\alpha/\beta \in \Q : \gcd(\alpha, \beta) = 1, 1 \le \alpha, \beta \le T\}$, is uniformly distributed in residue classes modulo a prime $p$ provided $T \ge p^{1/2 +\eps}$ for any ...
Cojocaru, A. C., Shparlinski, I. E.
core   +1 more source

Root numbers of elliptic curves in residue characteristic 2

, 2006
To determine the global root number of an elliptic curve defined over a number field, one needs to understand all the local root numbers. These have been classified except at places above 2, and in this paper we attempt to complete the classification. At
Dokchitser, T., Dokchitser, V.
core   +2 more sources

The yoga of the Cassels-Tate pairing

, 2007
Cassels has described a pairing on the 2-Selmer group of an elliptic curve which shares some properties with the Cassels-Tate pairing.
Brown, Cassels, Cassels, Milne, Serre, Serre, Serre, Silverman   +7 more
core   +2 more sources

Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves [PDF]

, 2007
Let $\E/\Q$ be a fixed elliptic curve over $\Q$ which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and W.
Shparlinski, Igor E.
core   +3 more sources

The remaining cases of the Kramer-Tunnell conjecture

, 2016
For an elliptic curve $E$ over a local field $K$ and a separable quadratic extension of $K$, motivated by connections to the Birch and Swinnerton-Dyer conjecture, Kramer and Tunnell have conjectured a formula for computing the local root number of the ...
Cesnavicius, Kestutis, Imai, Naoki
core   +2 more sources

The frequency of elliptic curve groups over prime finite fields [PDF]

, 2015
Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field $\mathbb{F}_p$, we study the frequency to which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$.
Chandee, Vorrapan, David, Chantal, Koukoulopoulos, Dimitris, Smith, Ethan   +3 more
core   +1 more source

Products of Small Integers in Residue Classes and Additive Properties of Fermat Quotients [PDF]

, 2015
We show that for any ϵ > 0 and a sufficiently large cube-free q, any reduced residue class modulo q can be represented as a product of 14 integers from the interval [1, q1/4,e1/2 + ϵ].
Balog, Bourgain, Burgess, Friedlander, Glyn Harman, Harman, Igor E. Shparlinski   +6 more
core   +2 more sources

$p$-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

, 2014
Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq 5$ and has complex multiplication by the full ring of integers $\mathcal{O}_K$ of $K$.
Bannai, Kenichi, Furusho, Hidekazu, Kobayashi, Shinichi   +2 more
core   +1 more source