# A universal Torelli theorem for elliptic surfaces

## TL;DR

This paper proves a universal Torelli theorem for elliptic surfaces, showing that compatible isometries of their Néron-Severi lattices imply isomorphisms of the surfaces, and characterizes their automorphism groups.

## Contribution

It establishes a Torelli-type theorem for semistable elliptic surfaces over a curve, extending isometries to isomorphisms and describing their automorphism groups.

## Key findings

- Compatible isometries induce surface isomorphisms.
- Automorphisms include Picard-Lefschetz transformations.
- Family of Weyl group homomorphisms constructed.

## Abstract

Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{\'e}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $B\to C$ arises from an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic. We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes showing that the Picard-Lefschetz transformations corresponding to an irreducible component of a singular fibre, can be extended as universal isometries. In the process, we get a family of homomorphisms of the affine Weyl group associated to $\tilde{A}_{n-1}$ to that of $\tilde{A}_{dn-1}$, indexed by natural numbers $d$, which are closed under composition.