What are the zeroes of $E_{p+1}$ on the modular curve $X_1(N)_{\overline{\mathbf{F}}_p}$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:33:42Z http://mathoverflow.net/feeds/question/62953 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62953/what-are-the-zeroes-of-e-p1-on-the-modular-curve-x-1n-overline-mathbf What are the zeroes of $E_{p+1}$ on the modular curve $X_1(N)_{\overline{\mathbf{F}}_p}$? Tommaso Centeleghe 2011-04-25T18:57:46Z 2011-04-25T18:57:46Z <p>If $p$ is prime $>3$, then the $(p+1)$-st Eisenstein series</p> <p>$E_{p+1}=-\frac{B_k}{2k}+\Sigma_{n\geq 1}\sigma_{p}(n)q^n$</p> <p>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 $n$).</p> <p>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 $p$.</p> <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 $X_1(N)_{\overline{\mathbf{F}}_p}$</p> <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).</p>