# Notes on anomalies, elliptic curves and the BS-D conjecture

## TL;DR

This paper investigates anomaly cancellation conditions in specific gauge theories with higher $SU(2)$ representations, using advanced number theory and elliptic curve theorems to classify possible hypercharge assignments.

## Contribution

It applies deep results from elliptic curve theory to classify anomaly-free hypercharge assignments in certain gauge theories with higher $SU(2)$ representations.

## Key findings

- Unique hypercharge assignment for $N=3$
- Two hypercharge options for $N=9$
- Infinite solutions for other cases

## Abstract

We consider anomaly cancellation for $SU(N)\times SU(2)\times U(1)$ gauge theories where the left-handed chiral multiplets are in higher $SU(2)$ representations. In particular, if the left-handed quarks and leptons transform under the triplet representation of $SU(2)$ and if the $U(1)$ gauge group is compact then up to an overall scaling there is only one possible nontrivial assignment for the hypercharges if $N=3$, and two if $N=9$. Otherwise there are infinitely many. We use the Mordell-Weil theorem, Mazur's theorem and the Cremona elliptic curve database which uses Kolyvagin's theorem on the Birch Swinnerton-Dyer conjecture to prove these statements.