- Chris Holden, UW Madison
- Matt Darnall, UW Madison
- Jared Weinstein, UC Berkeley
- Soroosh Yazdani, UC Berkeley
- Ben Kane, UW Madison

- 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.

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 $1 \leq m \leq \textrm{Sturm bound}$.

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.

- 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?

- 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.

- (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?