arXiv subjects

Pär Kurlberg

Publications and source records attributed to Pär Kurlberg.

20 records · Page 2Linked to original sources

arXiv2012

On the average exponent of elliptic curves modulo p

Given an elliptic curve E/Q and a prime p at which E has good reduction, let e_p be the exponent of the group E_p(F_p) of F_p-rational points on the reduction of E modulo p. Under the Generalized Riemann Hypothesis (GRH) for the Dedekind zeta functions of the division fields of E, we show that there is a certain constant c_E, depending on E and satisfying 0 < c_E < 1, such that e_p/#E_p(F_p) is equal to c_E on average. In the case where E has complex multiplication (CM) the result holds without GRH. If E is a non-CM curve we show that c_E is equal to a rational number depending on E times a universal constant c = \prod_q {1 - q^3/(q^2-1)(q^5-1)} = 0.899..., the product being over all primes q.

math.NT↗

arXiv2011

Periods of rational maps modulo primes

Let $K$ be a number field, let $ϕ\in K(t)$ be a rational map of degree at least 2, and let $α, β\in K$. We show that if $α$ is not in the forward orbit of $β$, then there is a positive proportion of primes ${\mathfrak p}$ of $K$ such that $α\mod {\mathfrak p}$ is not in the forward orbit of $β\mod {\mathfrak p}$. Moreover, we show that a similar result holds for several maps and several points. We also present heuristic and numerical evidence that a higher dimensional analog of this result is unlikely to be true if we replace $α$ by a hypersurface, such as the ramification locus of a morphism $ϕ: {\mathbb P}^{n} \to {\mathbb P}^{n}$.

math.AG↗