# Rational Points on Curves and Kim's Conjecture

> **NSF 01002526DB NSF RESEARCH & RELATED ACTIVIT** · Trustees of Boston University (MA) · $225,000

## Abstract

Understanding the integral and rational solutions to polynomial equations has remained a question of fundamental importance for centuries.  For instance, a problem formulated by Diophantus in the third century, when translated into the language of a polynomial equation, amounts to determining the rational solutions (x, y) to the equation y^2 = x^6 + x^2 + 1. This problem remained unsolved until the work of Wetherell in 1997.  Studying this algebraic equation from the point of view of geometry yields a curve, and more precisely, a curve of genus 2.  While curves of genus 2 or more are known to have finitely many rational points by the work of Faltings in 1983, as of yet there is no algorithm to determine these finite sets in general. In particular, when the curve’s Jacobian rank is equal to or larger than the genus, there are many challenges that remain. In this project, the PI will study methods for explicitly determining the finite set of rational points on curves of genus 2 or more.  The PI will also organize educational activities to build the mathematical pipeline and mentor students and postdoctoral researchers.

One promising approach for determining the finite set of rational points on curves of genus 2 or more, regardless of Jacobian rank, is through Kim's nonabelian Chabauty program and the computation of Selmer sets.  Kim has conjectured that Selmer sets in depth n are finite for n sufficiently large, and that moreover, that these Selmer sets eventually precisely

## Key facts

- **NSF award ID:** 2502687
- **Awardee organization:** Trustees of Boston University (MA)
- **SAM.gov UEI:** THL6A6JLE1S7
- **PI:** Jennifer S Balakrishnan
- **Primary program:** 01002526DB NSF RESEARCH & RELATED ACTIVIT
- **All programs:** —
- **Estimated total:** $225,000
- **Funds obligated:** $225,000
- **Transaction type:** Standard Grant
- **Period:** 09/01/2025 → 08/31/2027

## Primary source

NSF Award Search: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2502687

## Citation

> US National Science Foundation, Award 2502687, Rational Points on Curves and Kim's Conjecture. Retrieved via AI Analytics 2026-06-08 from https://api.ai-analytics.org/grant/nsf/2502687. Licensed CC0.

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