# What are the zeroes of $E_{p+1}$ on the modular curve $X_1(N)_{\overline{\mathbf{F}}_p}$?

If $p$ is prime $>3$, then the $(p+1)$-st Eisenstein series

$E_{p+1}=-\frac{B_k}{2k}+\Sigma_{n\geq 1}\sigma_{p}(n)q^n$

is the $q$-expansion of a modular form of level one and weight $p+1$ ($B_k$ is the $k$-th Bernoulli number, $\sigma_p(n)$ is the sum of the $p$-th powers $d^p$ of all the positive divisors $d$ of $n$).

It can be viewed as a (meromorphic) section of a certain line bundle on the modular curve $X_1(N)$, where $N\geq 1$ is an integer prime to $p$.

Consider the base change $X_1(N)_{\overline{\mathbf{F}}_p}$ to an algebraic closure of the finite field with $p$ elements ($X_1(N)$ can be constructed over $\mathbf{Z}[1/N]$). I would be interested in computing the divisor of $X_1(N)_{\overline{\mathbf{F}}_p}$

defined by the base change of $E_{p+1}$. (Notice that the analogous question for $E_{p-1}$ has a classical interpretation as the Hasse invariant).

• Can you give me one good reason why you expect the divisor of $E_{p+1}$ mod $p$ to have a formula any more explicit than "what it turns out to be"? Let me stress that I am definitely not saying "I am sure that there is no nice formula"---but I am wondering why you think that there is one. Do you think there is a nice formula for the divisor cut out by $E_{p+27}$, for example? My guess is that there isn't. So the "nice formula" idea has to stop somewhere. Why do you think it doesn't stop at $p-1$? But maybe you're lucky with $p+1$... – Kevin Buzzard Apr 25 '11 at 19:34
• PS I have seen $E_{p+1}$ mod $p$ in the literature on mod $p$ modular forms---for example it appears in Edixhoven's "the weight in Serre's conjectures..." Prop 7.2, and also in Coleman-Voloch Prop 5.8. But in both cases they are only considering $E_{p+1}$ as a modular form defined only on the supersingular points, and hence have lost pretty much all information about its divisor. – Kevin Buzzard Apr 25 '11 at 19:45
• $E_{p+1}$ is more or less canonical on the supersingular locus, where it known as "B". It never vanishes there, though. I think Kevin is right and one shouldn't expect a nice description of the zeros. – Felipe Voloch Apr 25 '11 at 19:46
• Thanks for your comments. I believe that the question is related to the problem of characterising those "functions" f on the supersingular locus that can be extended to mod $p$ modular forms of weight $k$. This problem can always be solved for $k\geq p+1$ (as Serre explains in his '87 letter to Tate). Since we know the effect of multiplying "functions" on the supersingular locus by the (restriction) of $E_{p+1}$, a reformulation of the problem posed above is: what are the modular forms $F$ of weight $k+p+1$ such that $F/E_{p+1}$ is still a modular form (of weight $k$)? – Tommaso Centeleghe Apr 25 '11 at 20:18
• Tommaso: I think one can reformulate my "what are the zeros of $E_{p+27}$?" in just the same way and it's not clear (to me) that this helps at all. I'm with Felipe---why should one expect this question to have a nice answer? – Kevin Buzzard Apr 25 '11 at 21:39