Finite groups scheme actions and incompressibility of Galois covers: beyond the ordinary case

  • Najmuddin Fakhruddin

    School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
  • Rijul Saini

    School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
Finite groups scheme actions and incompressibility of Galois covers: beyond the ordinary case cover
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Abstract

Inspired by recent work of B. Farb et al. [Compos. Math. 157, No. 11, 2407--2432 (2021; Zbl 1490.14042)], we develop a method for using actions of finite group schemes over a mixed characteristic dvrR\mathrm{dvr}\,\,\mathbb{R} to get lower bounds for the essential dimension of a cover of a variety over K=Frac(R)K = \text{Frac}(\mathbb{R}). We then apply this to prove pp-incompressibility for congruence covers of a class of unitary Shimura varieties for primes pp at which the reduction of the Shimura variety (at any prime of the reflex field over p)p) does not have any ordinary points. We also make some progress towards a conjecture of Brosnan on the pp-incompressibility of the multiplication by pp map of an abelian variety.

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Najmuddin Fakhruddin, Rijul Saini, Finite groups scheme actions and incompressibility of Galois covers: beyond the ordinary case. Doc. Math. 27 (2022), pp. 151–182

DOI 10.4171/DM/868