Bounds for Integral j -Invariants and Cartan Structures on Elliptic Curves

TL;DR

This paper establishes bounds for the j-invariant of integral points on modular curves based on the defining congruence group, and applies these bounds to demonstrate the absence of non-trivial rational points on certain modular curves for large primes.

Contribution

It provides explicit bounds for the j-invariant of integral points on modular curves and proves the non-existence of rational points on Xsplit(p^3) for large primes, assuming GRH.

Findings

Bounded the j-invariant in terms of the congruence group

Proved Xsplit(p^3) has no non-trivial rational points for large p

Under GRH, the bound improves to p^2

Abstract

We bound the j -invariant of integral points on a modular curve in terms of the congruence group defining the curve. We apply this to prove that the modular curve Xsplit (p3) has no non-trivial rational point if p is a sufficiently large prime number. Assuming the GRH, one can replace p3 by p2 .