The moduli space of dormant opers on elliptic curves
Naoka Karube, Yasuhiro Wakabayashi
TL;DR
The paper analyzes dormant opers and their Miura variants on elliptic curves in positive characteristic, providing explicit descriptions via adjoint invariants and proving finiteness and connectedness of the moduli stacks over the elliptic moduli stack. It introduces the Hitchin–Mochizuki morphism and study of $p$-curvature, Hasse invariants, and Frobenius-invariant differentials to establish a detailed global structure, including a Weyl-group–Galois symmetry on the ordinary locus. A key advance is the canonical diagonal lifting framework that relates dormant opers across finite levels and prime-power characteristics, producing bijections with regular-tuple data and enabling diagonal reductions to level-1 objects. The results unify moduli-theoretic, Lie-theoretic, and arithmetic aspects of dormant opers on elliptic curves, with potential implications for positive-characteristic geometric Langlands and related enumerative questions.
Abstract
A dormant oper is a specific type of principal bundle with a flat connection, defined on an algebraic curve in positive characteristic. The moduli spaces of dormant opers and their variants, known as dormant Miura opers, have been studied in various contexts. This paper focuses on the case where the underlying spaces are (possibly nodal) elliptic curves and provides a detailed examination of the geometric structures of their moduli spaces. In particular, we explicitly describe dormant (generic Miura) opers in terms of regular elements in an associated Lie algebra and establish the connectedness of these moduli spaces. We also explore generalizations to higher level and prime-power characteristic.