# 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:
• Jared Weinstein, UC Berkeley
• Soroosh Yazdani, UC Berkeley

# 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?
• Problem 2: Formulate a level raising conjecture modulo $\lambda^n$. Provide computational and theoretical evidence.
• Problem 3: Formulate a level lowering conjecture modulo $\lambda^n$. Provide computational and theoretical evidence.

# 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)).

• At $p=13$, $n$ is divisible by 9 and in fact there is $g \in S_2^{new}(\Gamma_0(11*13))$so that $f_E \equiv g \ (\textrm{mod }9)$ with coefficients in $\mathbb Z[x]/(x^4-3x^3-x^2+5x+1)$ so that $f_E \equiv g \ (\textrm{mod }9)$.

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:

• For the elliptic curve 19a, we use our method to find a $g \in S_2^{new}(\Gamma_0(19*5))$ with coefficients in $\mathbb Z[x]/(x^4-2x^3-6x^2-8x+9)$ so that $f_E \equiv g \ (\textrm{mod }9)$. We began by noticing that 9 divides $n^+$. In this case, $w_p$ completely decomposes the space of newforms so that $$S_2^{new,+}(\Gamma_0(19*5))= (g).$$ Finally we established that $a_m(f_E) \equiv a_m(g) \ (\textrm{mod } 9)$ for $1 \leq m \leq \textrm{Sturm bound}$.

# 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?
• Idea: Do we really need to generate $I_E$? We are only interested in it for creating $\# \mathbb T/I_E$, which is $(T_1)$. As long as we can determine the order of $T_1$, we're done. Does Matt's intersection of lattices idea work to prove we're not getting a multiple of the order we're after?

# Notes on our code

• Added sample SAGE code (levels.py). The main function is findindex(elliptic curve, space of modular symbols). For an elliptic curve of level $N$ and the hecke algebra of the space of modular symbols of level $Np$ for the space M, it calculates the index $n= \#(\mathbb T/I_E)$.
• levels3.py is more comprehensive; the main function is testrun(elliptic curve, range of primes). For an elliptic curve of conductor $N$ and each prime $p$ in the range, it computes $n^\pm = \#(\mathbb{T}^{p\text{-new},\pm}/I_E)$ and their correspoinding quotients (in 7.1.4) $t_p^\pm$, returning: $p \ [n^+,\ n^-]\ t_p^+ \ t_p^-$.
• levels4.py is basically the same as levels3.py, save that it clears the memory cache in between primes. We hope this helps the program run through primes more quicky and without crashing.
• levels5.py extends the function findindex to allow the elliptic curve to be level Np and the space to be level N. Additionally, some small optimizations have been done to this function, and additional driver functions (testlower, lowerone, testrunnocusp) allow additional functionality for testing.

# 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.
• (8/26/06) This can't literally be true because the elliptic curve $E=X_0(11)$ has multiplicative reduction at $11$ with Tamagawa number $c_{11}(E)=5$, but you certainly can't lower 11 from the level mod 5! Here, the problem is that the Galois representation on 5-division points is reducible. How can we fix our conjecture to reflect this issue?