# Counting Rational Points on Kummer surfaces

## TL;DR

This paper investigates the enumeration of rational points on Kummer surfaces linked to two elliptic curves, establishing Manin's conjecture for this family and revealing new identities between counting functions.

## Contribution

It proves Manin's unity for a family of Kummer surfaces and derives new identities between different rational point-counting functions.

## Key findings

- Proves Manin's conjecture for the considered Kummer surfaces.
- Establishes a new identity between two-parameter counting functions.
- Provides explicit rational point counts via elliptic fibrations.

## Abstract

We consider the problem of counting the number of rational points on the family of Kummer surfaces associated with two non-isogenous elliptic curves. For this two-parameter family we prove Manin's unity, using the presentation of the Kummer surfaces as isotrivial elliptic fibration and as double cover of the modular elliptic surface of level two. By carrying out the rational point-count with respect to either of the two elliptic fibrations explicitly, we obtain an interesting new identity between two-parameter counting functions.