Maarten Derickx
Number theory seminar on March 18, 2011 at 3:30pm in Padelford C401.
Title
Torsion points on elliptic curves over number fields of small degree.
Abstract
Merel has shown that if we fix nt(2f)20110318/latex_a351f1a78c577a198050a8394e24d844b107a745_p1.png "$d \geq 1$")
, then there are only finitely many primes nt(2f)20110318/latex_07e287db17d9adad69e31ec444cbc74de07b09ac_p1.png "$p$")
occurring as the order of a torsion point on an elliptic curve nt(2f)20110318/latex_31d7e8355d2f5ef9534573ec397c222a2f528d79_p1.png "$E/K$")
where nt(2f)20110318/latex_28754e5c590fc69cca5bbc447032d803c39c1e00_p1.png "$K$")
is a number field of degree nt(2f)20110318/latex_94ddfad7ca354f398f9c625847815015550699d5_p1.png "$\leq d$")
. Oesterle has proven that the largest such is smaller then nt(2f)20110318/latex_220024eed175215dfb07e6412a7570dc25f831d5_p1.png "$(3^{d/2}+1)^2$")
. Both upper bounds use a criterion by Kammienny. In my talk I will show several variations on this criterion. I will also show how to bring down the bound for nt(2f)20110318/latex_71a7e050d1f3893ee86c12235b5bacdd68861138_p1.png "$d=5$")
and nt(2f)20110318/latex_d9836b2c35e2b8f39fa078eb6ba752d8a5f09cd0_p1.png "$d=6$")
using this criterion using a computer program based on William Steins work on nt(2f)20110318/latex_7bdb0e785af2278c7cef067311623ca32d6e9f85_p1.png "$d=4$")
.