Results 11 to 20 of about 161 (55)

Tate module and bad reduction [PDF]

, 2020
Let C/K be a curve over a local field. We study the natural semilinear action of Galois on the minimal regular model of C over a field F where it becomes semistable.
Dokchitser, Tim, Dokchitser, Vladimir, Morgan, Adam   +2 more
core   +3 more sources

On the distribution of Atkin and Elkies primes for reductions of elliptic curves on average [PDF]

, 2015
For an elliptic curve E/Q without complex multiplication we study the distribution of Atkin and Elkies primes l, on average, over all good reductions of E modulo primes p.
Andrew, Igor E. Shparlinski, V. Sutherland   +2 more
core   +2 more sources

Supercongruences and Complex Multiplication [PDF]

, 2012
We study congruences involving truncated hypergeometric series of the form_rF_{r-1}(1/2,...,1/2;1,...,1;\lambda)_{(mp^s-1)/2} = \sum_{k=0}^{(mp^s-1)/2} ((1/2)_k/k!)^r \lambda^k where p is a prime and m, s, r are positive integers.
Kibelbek, Jonas, Long, Ling, Moss, Kevin, Sheller, Benjamin, Yuan, Hao   +4 more
core   +1 more source

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

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

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

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

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

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

Average Analytic Ranks of Elliptic Curves over Number Fields

Forum of Mathematics, Sigma
We give a conditional bound for the average analytic rank of elliptic curves over an arbitrary number field. In particular, under the assumptions that all elliptic curves over a number field K are modular and have L-functions which satisfy the ...
Tristan Phillips
doaj   +1 more source