On monodromy in families of elliptic curves over $\mathbb{C}$

Abstract

We show that if we are given a smooth non-isotrivial family of elliptic curves over~$\mathbb{C}$ with a smooth base~$B$ for which the general fiber of the mapping $J:B\to {\mathbb{A}}^1$ (assigning $j$-invariant of the fiber to a point) is connected, then the monodromy group of the family (acting on $H^1(\cdot ,\mathbb{Z})$ of the fibers) coincides with $\mathrm{SL}(2,\mathbb{Z})$; if the general fiber has $m\ge 2$ connected components, then the monodromy group has index at most~$2m$ in $\mathrm{SL}(2,\mathbb{Z})$. By contrast, in \emph{any} family of hyperelliptic curves of genus $g\ge 3$, the monodromy group is strictly less than $\mathrm{Sp}(2g,\mathbb{Z})$. Some applications are given, including that to monodromy of hyperplane sections of Del Pezzo surfaces.

21 pages, 3 figures. Version 5: result strengthened

References in corpus (1)

• An open image theorem for a general class of abelian varieties