An introduction to decoupling and harmonic analysis over $\mathbb{Q}_p$

TL;DR

This paper introduces decoupling and harmonic analysis over the p-adic numbers, demonstrating how simpler p-adic settings facilitate rigorous proofs and yield number theoretic insights.

Contribution

It provides an accessible introduction to decoupling in the p-adic context, highlighting the advantages over real analysis for establishing decoupling theorems.

Findings

Decoupling theorems over $\\mathbb{Q}_p$ are rigorously established.

Simpler heuristics in $\\mathbb{Q}_p$ enable stronger analytical results.

Applications to number theory are derived from p-adic decoupling results.

Abstract

The goal of this expository paper is to provide an introduction to decoupling by working in the simpler setting of decoupling for the parabola over $\mathbb{Q}_p$. Over $\mathbb{Q}_p$, commonly used heuristics in decoupling are significantly easier to make rigorous over $\mathbb{Q}_p$ than over $\mathbb{R}$ and such decoupling theorems over $\mathbb{Q}_p$ are still strong enough to derive interesting number theoretic conclusions.