We introduce the concept of quadratic modular symbol and study how these symbols are related to quadratic-adic -functions. These objects were introduced in [3] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic -adic -functions to more general Shimura curves.
We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves over a prime finite field of elements, such that the discriminant of the quadratic number field containing the endomorphism ring of over is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I. E. Shparlinski (2007).
Let be an elliptic curve defined over , the finite field of elements. We show that for some constant depending only on , there are infinitely many positive integers such that the exponent of , the group of -rational points on , is at most . This is an analogue of a result of R. Schoof on the exponent of the group of -rational points, when a fixed elliptic curve is defined over and the prime tends to infinity.
This note gives a survey of some recent results on the stable reduction of covers of the projective line branched at three points.
Let be a positive integer, a finite field of cardinality with . In this paper, inspired by [6, 3, 4] and using a slightly different method, we study the fluctuations in the number of -points on the curve given by the affine model , where is drawn at random uniformly from the set of all monic -th power-free polynomials of degree as . The method also enables us to study the fluctuations in the number of -points on the same family of curves arising from the set of monic irreducible...
The Mordell–Lang conjecture describes the intersection of a finitely generated subgroup with a closed subvariety of a semiabelian variety. Equivalently, this conjecture describes the intersection of closed subvarieties with the set of images of the origin under a finitely generated semigroup of translations. We study the analogous question in which the translations are replaced by algebraic group endomorphisms (and the origin is replaced by another point). We show that the conclusion of the Mordell–Lang...