Modular Parametrizations of Neumann-Setzer Elliptic Curves
| dc.creator | Stein, William | |
| dc.creator | Watkins, Mark | |
| dc.date | 2004-04-19 | |
| dc.date.accessioned | 2026-07-07T05:07:32Z | |
| dc.date.available | 2026-07-07T05:07:32Z | |
| dc.description | Suppose $p$ is a prime of the form $u^2+64$ for some integer $u$, which we take to be 3 mod 4. Then there are two Neumann--Setzer elliptic curves $E_0$ and $E_1$ of prime conductor $p$, and both have Mordell--Weil group $\Z/2\Z$. There is a surjective map $X_0(p)\xrightarrowπ E_0$ that does not factor through any other elliptic curve (i.e., $π$ is optimal), where $X_0(p)$ is the modular curve of level $p$. Our main result is that the degree of $π$ is odd if and only if $u \con 3\pmod{8}$. We also prove the prime-conductor case of a conjecture of Glenn Stevens, namely that that if $E$ is an elliptic curve of prime conductor $p$ then the optimal quotient of $X_1(p)$ in the isogeny class of $E$ is the curve with minimal Faltings height. Finally we discuss some conjectures and data about modular degrees and orders of Shafarevich--Tate groups of Neumann--Setzer curves. | |
| dc.identifier | https://arxiv.org/abs/math/0404333 | |
| dc.identifier | http://arxiv.org/abs/math/0404333 | |
| dc.identifier | IMRN 2004, no. 27, 1395-1405 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70895 | |
| dc.subject | Number Theory | |
| dc.title | Modular Parametrizations of Neumann-Setzer Elliptic Curves | |
| dc.type | text |