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

We consider anomaly cancellation for SU(N) x SU(2) x 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.