Invariants of \(\text{SL}_2(\mathbb{F}_q)\cdot\text{Aut}(\mathbb{F}_q)\) acting on \(\mathbb{C}^n\) for \(q=2n\pm 1\) (Q2702554)

According to the author, the work described in this article was motivated by a desire to understand, from a general point of view, the results of Felix Klein on the equations defining modular curves of prime order. From this perspective, the article's primary conclusion is theorem 7.8 which states that for a prime \(p\geq 11\) and not equal to \(13\), the modular curve \(X(p)\) may be constructed geometrically (in the set theoretic sense) from a certain \(3\)-tensor, i.e., the curve is the intersection of the covariants of the \(3\)-tensor. Inspired by Klein's work on \(X(7)\) and \(X(11)\) and their relationship to the invariants and covariants of \(\text{PSL}_2({\mathbb{F}}_7)\) and \(\text{PSL}_2({\mathbb{F}}_{11})\) respectively, the author investigates the ring of invariants of the semi-direct product \(\text{SL}_2({\mathbb{F}}_q)\cdot \text{Aut}({\mathbb{F}}_q)\) (where \(q\) is a power of \(p\)) and the symplectic group \(\text{Sp}({\mathbb{F}}_p^r)\), acting on certain Weil representations. (The Weil representations are described in an extensive appendix.) For any weakly self-adjoint representation, the author describes a pairing on the ring of invariants. The ring of invariants along with this pairing is called the bicycle of invariants. NEWLINENEWLINENEWLINEThe number of invariants required to generate the bicycle of invariants is often fewer than the number required to generate the corresponding ring of invariants. In conjecture 4.2, the author describes a conjectured generating set for the bicycle of invariants of certain Weil representations of \(\text{SL}_2({\mathbb{F}}_q)\). Conjecture 5.2 leads to a conjectured generating set for the bicycle of invariants of even degree for certain Weil representations of \(\text{Sp}({\mathbb{F}}_p^r)\). Both conjectures constitute applications of the bicycle conjecture (Conjecture 2.4). Although the author acknowledges that the bicycle conjecture is not true as stated, he believes that a suitable refinement is true and that the conjecture does constitute a principle for identifying generators for a bicycle of invariants.NEWLINENEWLINEFor the entire collection see [Zbl 0941.00006].