Overview
This research investigates the nature of Jacobians associated with Heisenberg curves. Specifically, it proves that for any odd prime $\ell$, Heisenberg curves of level $\ell^n \neq 3$ do not have complex multiplication (CM) Jacobians.
Research Context
Heisenberg curves are defined as cyclic covers of Fermat curves. They also arise as non-abelian covers of the projective line, characterized by branching over three points by the discrete Heisenberg group modulo an integer. As normal Belyi covers, these curves are known to exhibit numerous automorphisms, a property that led Oort to question whether such curves possess CM Jacobians. In 1986, Ihara proposed a method for studying the pro-$\ell$ Galois representation linked to the thrice-punctured projective line, utilizing towers of curves. To investigate the kernel of this representation, Ihara suggested the use of Heisenberg curves. However, at the time of Ihara's proposal, it remained unknown whether the Jacobians of these curves lacked complex multiplication.
Approach
The study focused on Heisenberg curves defined by their specific characteristics: cyclic covers of Fermat curves and non-abelian covers of the projective line branched over three points by the discrete Heisenberg group modulo an integer. The primary objective was to determine whether the Jacobians of these curves exhibited complex multiplication. The analysis was conducted for any odd prime $\ell$ and for Heisenberg curves of level $\ell^n$, with specific exclusion of the case where $\ell^n = 3$.
Findings
The research establishes that for any odd prime $\ell$, Heisenberg curves of level $\ell^n \neq 3$ do not have complex multiplication Jacobians. This finding directly addresses a previously unresolved aspect of Ihara's original argument regarding the use of Heisenberg curves to study the kernel of the pro-$\ell$ Galois representation associated with the thrice-punctured projective line. Furthermore, this result provides an infinite family of new counterexamples to Oort's question concerning whether curves with many automorphisms possess CM Jacobians.
Why This Matters
This finding completes a crucial part of Ihara's proposed framework from 1986, confirming suitability of Heisenberg curves for studying the kernel of the pro-$\ell$ Galois representation. It also contributes to the understanding of algebraic curves by offering an infinite family of examples that contradict Oort's question concerning complex multiplication in Jacobians of curves with many automorphisms.