Dipòsit Digital de Documents de la UAB
https://ddd.uab.cat
Dipòsit Digital de Documents de la UAB latest documentscaMon, 18 Feb 2019 20:05:09 GMTInvenio 1.1.6ddd.bib@uab.cat3605125https://ddd.uab.cat/img/uab/ddd_logo.gifDipòsit Digital de Documents de la UAB
https://ddd.uab.cat
Search Search this site:p
https://ddd.uab.cat/search
On periodic solutions of 2-periodic Lyness difference equations
https://ddd.uab.cat/record/150643
We study the existence of periodic solutions of the non–autonomous periodic Lyness’recurrence un+2 = (an + un+1)/un, where {an}n is a cycle with positive values a,b and with positive initial conditions. It is known that for a = b = 1 all the sequences generated by this recurrence are 5–periodic. We prove that for each pair (a, b) 6= (1, 1) there are infinitely many initial conditions giving rise to periodic sequences, and that the family of recurrences have almost all the even periods. If a 6= b, then any odd period, except 1, appears. Bastien, GuyFri, 06 May 2016 08:54:16 GMThttps://ddd.uab.cat/record/1506432013Rational Periodic Sequences for the Lyness Recurrence
https://ddd.uab.cat/record/150557
Consider the celebrated Lyness recurrence xn+2 = (a + xn+1)/xn with a ∈ Q. First we prove that there exist initial conditions and values of a for which it generates periodic sequences of rational numbers with prime periods 1, 2, 3, 5, 6, 7, 8, 9, 10 or 12 and that these are the only periods that rational sequences {xn}n can have. It is known that if we restrict our attention to positive rational values of a and positive rational initial conditions the only possible periods are 1, 5 and 9. Moreover 1-periodic and 5-periodic sequences are easily obtained. We prove that for infinitely many positive values of a, positive 9-period rational sequences occur. This last result is our main contribution and answers an open question left in previous works of Bastien & Rogalski and Zeeman. We also prove that the level sets of the invariant associated to the Lyness map is a two-parameter family of elliptic curves that is a universal family of the elliptic curves with a point of order n, n ≥ 5, including n infinity. This fact implies that the Lyness map is a universal normal form for most birational maps on elliptic curves. Gasull, ArmengolFri, 06 May 2016 08:49:23 GMThttps://ddd.uab.cat/record/1505572012Volcanoes of ℓ-isogenies of elliptic curves over finite fields :: the case ℓ = 3
https://ddd.uab.cat/record/138508
This paper is devoted to the study of the volcanoes of ℓ-isogenies of elliptic curves over a finite field, focusing on their height as well as on the location of curves across its different levels. The core of the paper lies on the relationship between the ℓ-Sylow subgroup of an elliptic curve and the level of the volcano where it is placed. The particular case ℓ = 3 is studied in detail, giving an algorithm to determine the volcano of 3-isogenies of a given elliptic curve. Experimental results are also provided. Miret, Josep M.Wed, 30 Sep 2015 09:58:23 GMThttps://ddd.uab.cat/record/1385082007Thetanullwerte :: from periods to good equations
https://ddd.uab.cat/record/138505
We will show the utility of the classical Jacobi Thetanullwerte for the description of certain period lattices of elliptic curves, providing equations with good arithmetical properties. These equations will be the starting point for the construction of families of elliptic curves with everywhere good reduction. Guardia, JordiWed, 30 Sep 2015 09:07:18 GMThttps://ddd.uab.cat/record/1385052007A p-adic construction of ATR points on Q-curves
https://ddd.uab.cat/record/133145
In this note we consider certain elliptic curves defined over real quadratic fields isogenous to their Galois conjugate. We give a construction of algebraic points on these curves defined over almost totally real number fields. The main ingredient is the system of Heegner points arising from Shimura curve uniformizations. In addition, we provide an explicit p-adic analytic formula which allows for the effective, algorithmic calculation of such points. Guitart, XavierFri, 03 Jul 2015 06:06:03 GMThttps://ddd.uab.cat/record/1331452015