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Further Counterexamples to Zarhin's conjecture about micro weights

Oliver Bültel

TL;DR

This work constructs two families of abelian varieties in positive characteristic that counter Zarhin's conjecture on micro weights. The first family uses a Rosati-invariant $ ext{O}_D$-action and Morita reductions to obtain a monodromy that is a $ ext{Q}_ ext{ell}$-form of $ ext{G}_m imes ext{SO}(3)^2$, with a ghost that is a supersingular abelian surface; the second employs a $G_2$-type construction from a CM field of degree $16$ to produce $56$-dimensional abelian varieties whose $ ext{ell}$-adic monodromy and exterior-power endomorphisms realize a $G_2$-structure, with a ghost carrying CM by the field $L$. The authors deploy Dieudonné theory (including $ ext{Z}/2 ext{Z}$-graded modules and windows), Serre–Tate deformations, and carefully designed moduli spaces to control Newton polygons and formal isogeny types, thereby producing explicit counterexamples and illuminating the interaction between endomorphisms, monodromy, and Cadoret–Tamagawa ghosts. These results emphasize the nuanced landscape of monodromy in positive characteristic and suggest refined obstructions to Zarhin's conjecture via ghost data and $G$-type structures.

Abstract

We present two new families of Abelian varieties which contradict Zarhin's conjecture about micro weights in positive characteristics. For each of these examples we determine the dimension and the Newton slopes of the ghost Abelian variety in the sense of Cadoret and Tamagawa.

Further Counterexamples to Zarhin's conjecture about micro weights

TL;DR

This work constructs two families of abelian varieties in positive characteristic that counter Zarhin's conjecture on micro weights. The first family uses a Rosati-invariant -action and Morita reductions to obtain a monodromy that is a -form of , with a ghost that is a supersingular abelian surface; the second employs a -type construction from a CM field of degree to produce -dimensional abelian varieties whose -adic monodromy and exterior-power endomorphisms realize a -structure, with a ghost carrying CM by the field . The authors deploy Dieudonné theory (including -graded modules and windows), Serre–Tate deformations, and carefully designed moduli spaces to control Newton polygons and formal isogeny types, thereby producing explicit counterexamples and illuminating the interaction between endomorphisms, monodromy, and Cadoret–Tamagawa ghosts. These results emphasize the nuanced landscape of monodromy in positive characteristic and suggest refined obstructions to Zarhin's conjecture via ghost data and -type structures.

Abstract

We present two new families of Abelian varieties which contradict Zarhin's conjecture about micro weights in positive characteristics. For each of these examples we determine the dimension and the Newton slopes of the ghost Abelian variety in the sense of Cadoret and Tamagawa.
Paper Structure (13 sections, 16 theorems, 38 equations)

This paper contains 13 sections, 16 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. First Example
  3. Some Morita equivalences
  4. On $\mathbb Z/2\mathbb Z$-graded symmetric Dieudonné-modules of height $6$
  5. Deformations
  6. Conclusions
  7. On work of Oort and van der Put
  8. Second example
  9. On $\mathbb Z/r\mathbb Z$-graded Frobenius modules with $\operatorname{SL}(2)_{K(\mathbb F_{p^r})}$-structure
  10. Proof of theorem \ref{['G2']}
  11. On two Moduli Spaces
  12. First moduli space
  13. Second moduli space

Key Result

Theorem 1.1

For every $p>2$ there exists an Abelian $6$-fold $Y$ over a finitely generated extension of $\mathbb F_p$ such that $G_{Y,\ell}^\circ$ is a $\mathbb Q_\ell$-form of $\mathbb G_m\times\operatorname{SO}(3)^2$ for every prime $\ell\neq p$, where the tautological representation of $G_{Y,\ell,\overline{\

Theorems & Definitions (33)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • Lemma 2.6
  • ...and 23 more