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

## TL;DR

This paper investigates the monodromy groups of families of elliptic curves over complex bases, showing they are typically as large as possible, and contrasts this with hyperelliptic curves where monodromy is smaller.

## Contribution

It establishes conditions under which the monodromy group of elliptic curve families equals SL(2,Z) and compares this to hyperelliptic curves where monodromy is strictly less.

## Key findings

- Monodromy equals SL(2,Z) for non-isotrivial elliptic families with connected fibers.
- Monodromy has index at most 2m in SL(2,Z) when fibers have m components.
- Hyperelliptic curves of genus ≥3 have monodromy strictly less than Sp(2g,Z).

## 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 $J\colon B\to\mathbb A^1$ (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 $m\ge2$ 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 $g\ge3$, 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.