Uniform distribution of special homology classes mod-p - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T14:19:22Z http://mathoverflow.net/feeds/question/55856 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/55856/uniform-distribution-of-special-homology-classes-mod-p Uniform distribution of special homology classes mod-p David Hansen 2011-02-18T14:16:35Z 2011-02-18T14:16:35Z <p>Let $X_0(N)$ be the usual modular curve. For $\chi$ a quadratic Dirichlet character of conductor $D$, define the homology class $c(\chi)=\sum_{i=0}^{|D|-1}\chi(i)\{\frac{i}{|D|}{\infty} \}$; here $\{a,b\}$ is the path joining the points $a,b$. When $D$ and $N$ are coprime, this construction yields a well-defined class in $H_1(X_0(N),\mathbf{Z})$ (see Vatsal's paper "Canonical periods and congruence formulae"). Reducing modulo a fixed prime $p$ gives a class in $H_1(X_0(N),\mathbf{F}_p$). My question is:</p> <blockquote> <p>As $\chi$ varies over quadratic characters of discriminant prime to $N$, is it reasonable to expect that the classes $c(\chi)$ become uniformly distributed in $H_1(X_0(N),\mathbf{F}_p)$ with respect to counting measure?</p> </blockquote> <p>My motivation for asking this is that pairing $c(\chi)$ with certain mod-$p$ cohomology classes essentially gives mod-$p$ special values of L-functions, so knowing this uniform distribution would imply a lot of things about vanishing and nonvanishing of L-functions modulo $p$.</p>