As mentioned in an answer to Modularity of $E_2$ on congruence subgroups, there exist modular forms $E_{\ell,2}$ of level $\Gamma_{0}(\ell)$ and weight 2, with $q$-expansion $E_{\ell,2}(q)=E_{2}(q)-\ell E_{2}(q^{\ell})$. (Recall that $E_{2}$ is the quasimodular Eisenstein series with $q$-expansion $E_{2}(q)=1-24 \sum_{n\geq 1}{\sigma_{1}(n)q^{n}}$.)

Is there something in the literature that is related to the following question: Let $\mathcal{E}$ be a supersingular curve defined over $\mathbb{F}_{p^2}$, $\omega$ the canonical invariant differential, and $\alpha$ any level $\ell$ structure. Is $E_{\ell,2}$ evaluated on $(\mathcal{E},\omega,\alpha)$ in $\mathbb{F}_{p^2}$? (We can assume that $p>3$ and $\ell\neq p$ is prime.)

Edit: If this is any motivation, it is possible to show this for the case $\ell=2$ using the explicit structure of the level 2 modular ring over $\mathbb{C}$, which is $\mathbb{C}[E_{2,2},E_{4}]$. Instead of level structures, one can use the formula $E_{6}=E_{2,2}(4E_{2,2}^2-3E_{4})$. The nontrivial input in this case is Landweber's result that implies $\Delta^{\frac{p^2-1}{6}}(\mathcal{E})=1$ for $\mathcal{E}$ supersingular, defined over $\mathbb{F}_{p^2}$ and $j(\mathcal{E})\neq 0,1728$ (see "Supersingular elliptic curves and congruences for legendre polynomials" Theorem 1 (1.5')).

  • $\begingroup$ I started to write a long answer here, but got derailed. In short, you might want to take a look at the "q-expansion principle", from Katz, "p-adic properties of modular schemes and modular forms". $\endgroup$ – Charles Rezk Aug 16 '13 at 14:39
  • $\begingroup$ You are right in that I should figure out what exactly the q-expansion principle says when we have a level structure. But are you using the fact that we are evaluating on supersingular curves, or is this fact actually irrelevant? $\endgroup$ – DCT Aug 16 '13 at 16:16

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