Results 31 to 40 of about 966 (79)
Elliptic Curves over Totally Real Cubic Fields are Modular [PDF]
, 2019 We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to
Derickx, Maarten +2 more
core +2 more sources
An interesting family of curves of genus 1
International Journal of Mathematics and Mathematical Sciences, Volume 23, Issue 6, Page 431-434, 2000., 2000 We study the family of elliptic curves y2 = x3 − t2x + 1, both over ℚ(t) and over ℚ. In the former case, all integral solutions are determined; in the latter case, computation in the range 1 ≤ t ≤ 999 shows large ranks are common, giving a particularly simple example of curves which (admittedly over a small range) apparently contradict the once held ...
Andrew Bremner
wiley +1 more source
Criteria for irreducibility of mod p representations of Frey curves [PDF]
, 2014 Let K be a totally real Galois number field and let A be a set of elliptic curves over K. We give sufficient conditions for the existence of a finite computable set of rational primes P such that for p not in P and E in A, the representation on E[p] is ...
Freitas, Nuno, Siksek, Samir
core +3 more sources
The upper bound estimate of the number of integer points on elliptic curves y2=x3+p2rx
Journal of Inequalities and Applications, 2014 Let p be a fixed prime and r be a fixed positive integer. Further let N(p2r) denote the number of pairs of integer points (x,±y) on the elliptic curve E:y2=x3+p2rx with y>0.
Jin Zhang, Xiaoxue Li
semanticscholar +2 more sources
An exact upper bound estimate for the number of integer points on the elliptic curves y2=x3−pkx
Journal of Inequalities and Applications, 2014 Let p be a fixed prime and k be a fixed odd positive integer. Further let N(pk) denote the number of pairs of integer points (x,±y) on the elliptic curve E:y2=x3−pkx with y>0.
S. Gou, Xiaoxue Li
semanticscholar +2 more sources
On the torsion group of elliptic curves induced by D(4)-triples
Analele Stiintifice ale Universitatii Ovidius Constanta: Seria Matematica, 2014 A D(4)-m-tuple is a set of m integers such that the product of any two of them increased by 4 is a perfect square. A problem of extendibility of D(4)-m-tuples is closely connected with the properties of elliptic curves associated with them. In this paper
Dujella Andrej, Mikić Miljen
doaj +1 more source
Figurate primes and Hilbert's 8th problem
, 2014 In this paper, by using the theory of elliptic curves, we discuss several Diophantine equations related with the so-called figurate primes. Meanwhile, we raise several conjectures related with figurate primes and Hilbert's 8th problem, including Goldbach'
Cai, Tianxin +2 more
core +1 more source
Elliptic curves with torsion group $\Z /6\Z $ [PDF]
, 2015 We exhibit several families of elliptic curves with torsion group isomorphic to $ \Z/6\Z$ and generic rank at least $3$. Families of this kind have been constructed previously by several authors: Lecacheux, Kihara, Eroshkin and Woo.
Dujella, A., Peral, J. C., Tadić, P.
core +3 more sources
Average Analytic Ranks of Elliptic Curves over Number Fields
Forum of Mathematics, SigmaWe 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
On the largest element in D(n)-quadruples
, 2019 Let $n$ be a nonzero integer. A set of nonzero integers $\{a_1,\ldots,a_m\}$ such that $a_ia_j+n$ is a perfect square for all $1\leq ...
Dujella, Andrej, Petričević, Vinko
core +1 more source