On Pseudopoints of Algebraic Curves

TL;DR

This paper generalizes the concept of pseudopoints from integers to algebraic curves, using exponential sum bounds to analyze the limitations of modular methods in finding points on these curves.

Contribution

It extends the notion of pseudopoints to algebraic curves and employs exponential sum bounds to estimate the smallest pseudopoint, highlighting the constraints of modular approaches.

Findings

Estimates the smallest x-pseudopoint using exponential sum bounds

Demonstrates limitations of modular methods for finding points on algebraic curves

Provides a framework for understanding pseudopoints in algebraic geometry

Abstract

Following Kraitchik and Lehmer, we say that a positive integer $n\equiv1\pmod 8$ is an $x$-pseudosquare if it is a quadratic residue for each odd prime $p\le x$, yet is not a square. We extend this defintion to algebraic curves and say that $n$ is an $x$-pseudopoint of a curve $f(u,v) = 0$ (where $f \in \Z[U,V]$) if for all sufficiently large primes $p \le x$ the congruence $f(n,m)\equiv 0 \pmod p$ is satisfied for some $m$. We use the Bombieri bound of exponential sums along a curve to estimate the smallest $x$-pseudopoint, which shows the limitations of the modular approach to searching for points on curves.