3,292 research outputs found
ESTIMATES RELATED TO THE ARITHMETIC OF ELLIPTIC CURVES
This dissertation presents results related to two problems in the arithmetic of elliptic curves. Feng and Xiong equate the nontriviality of the Selmer groups associated with congruent number curves to the presence of certain types of partitions of graphs associated with the prime factorization of n. The triviality of the Selmer groups associated to the congruent number curve implies that the curve has rank zero which in turn implies n is noncongruent. We extend the ideas of Feng and Xiong in order to compute the Selmer groups of congruent number curves. We prove an average version of a generalization of the Lang-Trotter conjecture for elliptic curves over number fields. For the of degree one and degree two primes we calculate an explicit constant
The geometry of efficient arithmetic on elliptic curves
The arithmetic of elliptic curves, namely polynomial addition and scalar
multiplication, can be described in terms of global sections of line bundles on
and , respectively, with respect to a given projective embedding
of in . By means of a study of the finite dimensional vector
spaces of global sections, we reduce the problem of constructing and finding
efficiently computable polynomial maps defining the addition morphism or
isogenies to linear algebra. We demonstrate the effectiveness of the method by
improving the best known complexity for doubling and tripling, by considering
families of elliptic curves admiting a -torsion or -torsion point
Triangular Numbers and Elliptic Curves
Some arithmetic of elliptic curves and theory of elliptic surfaces is used to find all rational solutions (r, s, t) in the function field Q(m, n) of the pair of equations r(r + 1)/2 = ms(s + 1)/2 r(r + 1)/2 = nt(t + 1)/2. It turns out that infinitely many solutions exist. Several examples will be given
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