A. Deines: Dembele's Algorithm for Modular Elliptic Curves Over Real Quadratic Fields
May 13, 2011 in Padelford C401 at 3:30pm
Abstract
From the Eichler-Shimura construction we know that for each newform nt(2f)20110513/latex_7b643e3d048a93cd23d29c57357edfda2d68ab16_p1.png "$f$")
of weight 2 and level nt(2f)20110513/latex_22724cd8fbb1f3c0d4aee0b89b59d5ede2c02cdd_p1.png "$n$")
with rational Fourier coefficients, there exists an elliptic curve nt(2f)20110513/latex_74dd541e68458ec8bfc902b16d0c9ddd436dd6da_p1.png "$E_f$")
over nt(2f)20110513/latex_75001aaf3dc902eacb022eec8614ff5454324f35_p1.png "${\mathbf Q}$")
attached to . We can instead work over nt(2f)20110513/latex_6519ff76c3418599430298ce95b9af61e235533d_p1.png "$F$")
, a real quadratic number field with narrow class number one, instead of over nt(2f)20110513/latex_b6fff43ea3ab1273379d045423ec9089fd87d937_p1.png "$\mathbf{Q}$")
. Let be a Hilbert newform of weight 2 and level nt(2f)20110513/latex_5f7f618a6b197b8365106df6b87ca1697a0d7f0f_p1.png "$I$")
with rational Fourier coefficients, where is an integral ideal of . It is a conjecture that for every there is an elliptic curve over attached to . Recently Dembele has developed an algorithm which computes the (candidate) elliptic curve under the assumption that the Eichler-Shimura conjecture is true. I will discuss Dembele's algorithm, give an example, and discuss the status of implementing this algorithm in Sage.