whereX0(N) is the complete curve overQ which classifies pairs of elliptic curves related by a cyclic N -isogeny. The curve E is equipped with the collection of Heegner points defined over ring class fields of suitable imaginary quadratic fields. More precisely, let K be an imaginary quadratic field in which all rational primes dividing N are split and let O be the order of K of conductor c prim...

Let Fq be a finite field and let b and N be integers. We study the probability that the number of points on a randomly chosen elliptic curve E over Fq equals b modulo N . We prove explicit formulas for the cases gcd(N, q) = 1 and N = char(Fq). In the former case, these formulas follow from a random matrix theorem for Frobenius acting on the Ntorsion part of E, obtained by applying density resul...

We compute the PSL(2, N)-module structure of the Riemann-Roch space L(D), where D is an invariant non-special divisor on the modular curve X(N), with N ≥ 7 prime. This depends on a computation of the ramification module, which we give explicitly. These results hold for characteristic p if X(N) has good reduction mod p and p does not divide the order of PSL(2, N). We give as examples the cases N...

We compute the PSL(2, N)-module structure of the RiemannRoch space L(D), where D is an invariant non-special divisor on the modular curve X(N), with N ≥ 7 prime. This depends on a computation of the ramification module, which we give explicitly. These results hold for characteristic p if X(N) has good reduction mod p and p does not divide the order of PSL(2, N). We give as examples the cases N ...

There is an algorithm that takes as input a global field k and produces a curve over k violating the local-global principle. Also, given a global field k and a nonnegative integer n, one can effectively construct a curve X over k such that #X(k) = n.

AIM To identify in-house assessment criteria to predict outcomes of driving tests in patients with stroke who wish to resume driving. MATERIAL AND METHODS A total of 181 patients with stroke who attended Fukui General Hospital (as outpatients or inpatients) between 2003 and 2015 and who had no obvious motor impairment were included in the study. All subjects underwent a neuropsychological exa...

It is shown that N points on a real algebraic curve of degree n in R always determine &n,d N 1+ 1 4 distinct distances, unless the curve is a straight line or the closed geodesic of a flat torus. In the latter case, there are arrangements of N points which determine . N distinct distances. The method may be applied to other quantities of interest to obtain analogous exponent gaps. An important ...