# Frobenian multiplicative functions and rational points in fibrations

## TL;DR

This paper investigates the distribution of rational points on algebraic varieties over $\,\mathbb{Q}$, providing lower bounds and sharp results for specific families and equations, using methods from arithmetic geometry and additive combinatorics.

## Contribution

It introduces new lower bounds and sharp results for counting rational points in families over $\,\mathbb{Q}$, addressing cases where the Hasse principle fails and solving some of Serre's questions.

## Key findings

- Established lower bounds for rational points in certain families.
- Obtained sharp results for multinorm equations.
- Answered specific cases of Serre's question on Brauer group elements.

## Abstract

We consider the problem of counting the number of varieties in a family over $\mathbb{Q}$ with a rational point. We obtain lower bounds for this counting problem for some families over $\mathbb{P}^1$, even if the Hasse principle fails. We also obtain sharp results for some multinorm equations and for specialisations of certain Brauer group elements on higher dimensional projective spaces, where we answer some cases of a question of Serre. Our techniques come from arithmetic geometry and additive combinatorics.