$h^1\ne h_1$ for Anderson t-motives

Abstract

Let $M$ be an Anderson t-motive of dimension $n$ and rank $r$. Associated are two ${\mathbb{F}}_q[T]$-modules $H^1(M)$, $H_1(M)$ of dimensions $h^1(M)$, $h_1(M)\le r$ - analogs of $H^1(A,\mathbb{Z})$, $H_1(A,\mathbb{Z})$ for an abelian variety $A$. There is a theorem (Anderson): $h^1(M)=r⟺h_1(M)=r$; in this case $M$ is called uniformizable. It is natural to expect that always $h^1(M)=h_1(M)$. Nevertheless, we explicitly construct a counterexample. Further, we answer a question of D.Goss: is it possible that two Anderson t-motives that differ only by a nilpotent operator $N$ are of different uniformizability type, i.e. one of them is uniformizable and other not? We give an explicit example that this is possible.

arXiv comment31 pages. Minor improvements

References in corpus (4)

• Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms
• Periods of $t$-modules as special values
• Duality of Anderson T-motives
• Anderson T-motives and abelian varieties with MIQF: results coming from an analogy

Cited by in corpus (2)

• Lattice of the dual of an Anderson $t$-motive in terms of a map to a flag variety
• Introduction to Anderson t-motives: a survey