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

2011-04-25T18:57:46Z

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).

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

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