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Published online by Cambridge University Press: 25 June 2025
The work described in this article was motivatedby a desire to understand from a general point ofview the results of Felix Klein on the equationsdefining modular curves of prime order, especiallyhis remarkable discovery that the modular curveX(ll) is thesingular locus of the Hessian of the cubicthreefold This same desire has motivated much ofmy work over the years (see references in thebibliography), including the computation in [Adler1981; 1992b] of the ring of invariants of afive-dimensional complex representation of PSL2(𝔽 11) and the joint work [Adler andRamanan 1996] on moduli of abelian varieties. Atthe same time, these efforts have led to otherproblems of interest in their own right.
In this paper, we make our first attempt at a synthesisof what we have learned from our efforts. We beginin Section 2 with some general considerations aboutrings of invariants introduced in [Adler 1981;1992b], specifically the concept of a bicycle. A bicycle is a ringequipped with an additional structure of left moduleover itself. The ring of invariants of aself-adjoint group of operators or, more generally,of a weakly self-adjoint group, as in Definition2.1, is an example of a bicycle. This fact enablesone to generate rings of invariants from a smallnumber of generators using bicycle operations. Afterintroducing the notion of a bicycle, we then state ageneral conjecture (2.4), called The BicycleConjecture, about the bicycle of invariants of afinite group. As an example in Section 2.7 shows,the conjecture is false in general.
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