|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'500'096
Articles rated: 2609
19 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
Common Divisors of Elliptic Divisibility Sequences over Function Fields | | Joseph H. Silverman
; | | Date: |
2 Feb 2004 | | Subject: | Number Theory; Algebraic Geometry MSC-class: 11D61; 11G35 | math.NT math.AG | | Abstract: | Let E/k(T) be an elliptic curve defined over a rational function field of characteristic zero. Fix a Weierstrass equation for E. For points R in E(k(T)), write x_R=A_R/D_R^2 with relatively prime polynomials A_R(T) and D_R(T) in k[T]. The sequence {D_{nR}) for n ge 1 is called the ``elliptic divisibility sequence of R.’’ Let P,Q in E(k(T)) be independent points. We conjecture that deg (gcd(D_{nP},D_{mQ})) is bounded for m,n ge 1, and that gcd(D_{nP},D_{nQ}) = gcd(D_{P},D_{Q}) for infinitely many n ge 1. We prove these conjectures in the case that j(E) is in k. More generally, we prove analogous statements with k(T) replaced by the function field of any curve and with P and Q allowed to lie on different elliptic curves. If instead k is a finite field of characteristic p, and again assuming that j(E) is in k, we show that deg (gcd(D_{nP},D_{nQ})) > n + O(sqrt{n}) for infinitely many n satisfying gcd(n,p) = 1. | | Source: | arXiv, math.NT/0402016 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser claudebot
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER