# Pursuing polynomial bounds on torsion

## TL;DR

This paper establishes polynomial bounds on the torsion subgroup size of elliptic curves over number fields, depending on the degree of the field, with various conditions and unconditional results for specific cases.

## Contribution

It provides new polynomial bounds on torsion subgroups of elliptic curves over number fields, including unconditional bounds for certain j-invariants and conditional bounds under GRH.

## Key findings

- Bound #E(F)[tors] by C(epsilon)[F:Q]^{5/2+epsilon} for all epsilon > 0.
- Unconditional polynomial bounds for j(E) in fixed quadratic fields not of imaginary class number one.
- Conditional bounds assuming GRH and strong boundedness of isogenies for non-CM elliptic curves.

## Abstract

We show that for all epsilon > 0, there is a constant C(epsilon) > 0 such that for all elliptic curves E defined over a number field F with j(E) in Q we have #E(F)[tors] \leq C(epsilon)[F:Q]^{5/2+epsilon}. We pursue further bounds on the size of the torsion subgroup of an elliptic curve over a number field E/F that are polynomial in [F:Q] under restrictions on j(E). We give an unconditional result for j(E) lying in a fixed quadratic field that is not imaginary of class number one as well as two further results, one conditional on GRH and one conditional on the strong boundedness of isogenies of prime degree for non-CM elliptic curves.