This is an Archived copy of a WIKI page from the MSRI 2006 workshop on Modular forms.

This project was proposed by William Stein. Here is the webpage were the initial question was asked, along with an insightful comment from Richard Taylor.

Team Level Raising/Lowering

from left to right:

Background: Level Raising and Lowering

Put the original level lowering and raising theorems and references to the papers here.

The Problems

Problem 1: Let $E$ be the elliptic curve 11a given by the equation $$y^2 + y = x^3 - x^2 - 10x - 20,$$ and let $$f=f_E=q - 2q^{2} - q^{3} + \cdots \in S_2(\Gamma_0(11))$$ be the corresponding newform. For $l^d = 9, 27, 49$, compute all newforms $g \in S_2(\Gamma_0(11p))$ with $p < 500$ prime and $g\equiv f$ mod $l^d$. Is there a pattern?

Level Raising Conjecture

Let $N$ be a level, let $p\not \div N$ be prime, and let $\mathbb{T}^{p\text{-new},\pm}_{pN}$ be the $\pm 1$-eigenspace of the $w_p$ operator on the $p$-new quotient of the cuspidal Hecke algebra of level $pN$. Let $E$ be an elliptic curve of conductor $N$, and let $I_E\subset\mathbb{T}$ the ideal generated by $T_m-a_m(E)$ for $m$ ranging over integers prime to $p$. This idea measures congruences between the cusp form $f_E$ and cusp forms of level $Np$. Then $$ \text{$\mathbb{T}^{p\text{-new},\pm}/I_E\cong\mathbb{Z}/n\mathbb{Z}$},$$ where $n$ divides $p+1\mp a_p$ with quotient $t_p\vert \gcd_{l\not | N}(l+1\mp a_l(E))$. This would generalize Ribet's level raising theorem.

Implications

One implication of this conjecture is that the $n$'s index all possible prime powers $l^d$ for which there can be a newform $g$ of level $p$*N congruent to $f_E$ mod $l^d$; there is such a $g$ only if $n^+$ or $n^-$ are divisible by $l^d$.

We can give a partial answer to Problem 1. We begin with the results from testrun(EllipticCurve('11a'), prime_range(37)).

In Ribet's level raising Theorem, one of the hypotheses is that, if $l$ is the modulus for which congruence is hoped, $E[l]$ must be irreducible. However, in our data we have an example of an elliptic curve with nontrivial rational 3-torsion which is congruent mod 9 to a newform of higher level:

Evidence

Using the program levels3.py (described below), we obtained data indicating support for our conjecture, namely $t_p$ is an integer which divides the order of rational torsion for the curve chosen. level_raising_data.txt, contains about 80 data points for various elliptic curves and small primes.

When $l^d$ divides $n^+$ or $n^-$ for an elliptic curve of conductor N, we have looked for a corresponding newform of level N*p which is congruent to $f_E$ mod $l^p$. We have had some success, illustrated by the examples above.

Questions

When generating $I_E$, we only use $m$ up to some bound (such as the Sturm bound) and assume that's all we need. Can we justify this? How?

Notes on our code

Level lowering conjecture

Say $E$ is an optimal elliptic curve of conductor $pN$, with $p\not | N$ and having split multiplicative reduction at $p$. Let $\mathbb{T}$ be the Hecke algebra on cusp forms of level $N$. Let $I_E$ be the ideal generated by $T_m-a_m(E)$ for $m$ prime to $p$. Then $$ \text{$\mathbb{T}/I_E \overset{?}= \mathbb{Z}/c_p(E)\mathbb{Z}$},$$ where $c_p(E)$ is the Tamagawa number of $E$ at $p$ up to a power of 2.

Attachments