Projective crystalline representations of étale fundamental groups and twisted periodic Higgs-de Rham flow
Let \(k\) be the algebraic closure of a finite field and let \(X\) be a smooth proper scheme over the Witt ring \(W(k)\). \textit{G. Lan} et al. [J. Eur. Math. Soc. (JEMS) 21, No. 10, 3053--3112 (2019; Zbl 1444.14048)] introduced periodic Higgs-de Rham flows on \(X/W(k)\) and they proved that it is equivalent to the category of crystalline representations of the étale fundamental group of the generic fiber of \(X\). Here the authors introduce the twisted version of periodic Higgs-de Rham flows and they prove that they correspond to projective crystalline representations of the étale fundamental group of the generic fiber of \(X\). They also study the base change of these objects over very ramified valuation rings and show that a stable periodic Higgs bundle gives rise to a geometrically absolutely irreducible crystalline representation. These results are also generalized to the logarithmic setting, when \(X\) is replaced by a pair consisting of \(X\) and a simple normal crossing divisor \(D\) relative to \(W(k)\). Finally, the authors study the dynamics of the map provided by Higgs-de Rham flow on the moduli spaces of rank \(2\) stable Higgs bundles of degree \(1\) on \({\mathbb P}^1\) with logarithmic structure on \(n\) marked points \(D\) for \(n\ge 4\). Using Hrushovski's theorem they produce infinitely many geometrically absolutely irreducible \(\mathrm{PGL}_2({\mathbb Z}^{\mathrm{ur}}_p)\)-crystalline representations of \(\pi _1^{\mathrm{ét}} ({\mathbb P}^1-D)\). They also provide an explicit formula for the self-map of the moduli space in case \(m=4\). Then they pose a few conjectures relating periodic Higgs bundles on \({\mathbb P}^1\) with \(4\) marked points to torsion points on the associated elliptic curve. These results and conjectures are remarkably explicit and can be expressed in an elementary way. They also consider some questions relating periodic Higgs bundles in the \(p\)-adic case to the Hecke eigenforms in the \(\ell\)-adic case posing another conjecture that would allow to use Abe's solution of Deligne's companion conjecture.
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