Algebras associated to elliptic curves

This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one.

In particular, we study the elliptic algebras $A(+)$, $A(-)$, and $A({\mathbf {a}})$, where ${\mathbf {a}}\in \mathbb {P}^{2}$, which were first defined in an earlier paper. We omit a set $S\subset \mathbb {P}^2$ consisting of 11 specified points where the algebras $A({\mathbf {a}})$ become too degenerate to be regular.

Theorem. Let $A$ represent $A(+)$, $A(-)$ or $A({\mathbf {a}})$, where ${\mathbf {a}} \in \mathbb {P}^2\setminus S$. Then $A$ is an Artin-Schelter regular algebra of global dimension three. Moreover, $A$ is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables.

This, combined with our earlier results, completes the classification.