arXiv subjects
Publications and source records attributed to Yoshiaki Okumura.
The Taelman class groups associated to Drinfeld modules over function fields serve as an analogue of ideal class groups of number fields. In this paper, we establish an analogue of Iwasawa's asymptotic formula for $\mathbb{Z}_p$-extensions in the context of the Taelman class groups. For this purpose, we naturally introduce and investigate Taelman class groups with moduli.
For rank-two $A$-motives defined over local fields with odd characteristic, we give an analogue of a theorem of Imai stating that abelian varieties with good reduction over $p$-adic fields have only finitely many torsion points values in cyclotomic towers. This implies the finiteness of torsion points of abelian Anderson $A$-modules. For rank-two Drinfeld $A$-modules over global function fields, we also give an analogue of a theorem of Ribet on torsion points of abelian varieties values in maximal cyclotomic extensions of number fields.
In this article, we prove that an asymptotic formula for the prime number race with respect to Fermat curves of prime degree is equivalent to part of the Deep Riemann Hypothesis (DRH), which is a conjecture on the convergence of partial Euler products of $L$-functions on the critical line. We also show that such an equivalence holds for some quotients of Fermat curves. As an application, we compute the order of zero at $s=1$ for the second moment $L$-functions of those curves under DRH.
This article investigates congruences of $\mathfrak{p}$-adic representations arising from effective $A$-motives defined over a global function field $K$. We give a criterion for two congruent $\mathfrak{p}$-adic representations coming from strongly semi-stable effective $A$-motives to be isomorphic up to semi-simplification when restricted to decomposition groups of suitable places of $K$. This is a function field analog of Ozeki-Taguchi's criterion for $\ell$-adic representations of number fields. Motivated by a non-existence conjecture on abelian varieties over number fields stated by Rasmussen and Tamagawa, we also show that there exist no strongly semi-stable effective $A$-motives with some constrained.
For an extension $K/\mathbb{F}_q(T)$ of the rational function field over a finite field, we introduce the notion of virtually $K$-rational Drinfeld modules as a function field analogue of $\mathbb{Q}$-curves. Our goal in this article is to prove that all virtually $K$-rational Drinfeld modules of rank two with no complex multiplication are parametrized up to isogeny by $K$-rational points of a quotient curve of the Drinfeld modular curve $Y_0(\mathfrak{n})$ with some square-free level $\mathfrak{n}$. This is an analogue of Elkies' well-known result on $\mathbb{Q}$-curves.
In the arithmetic of function fields, Drinfeld modules play the role that elliptic curves play in the arithmetic of number fields. The aim of this paper is to study a non-existence problem of Drinfeld modules with constrains on torsion points at places with large degree. This is motivated by a conjecture of Christopher Rasmussen and Akio Tamagawa on the non-existence of abelian varieties over number fields with some arithmetic constraints. We prove the non-existence of Drinfeld modules satisfying Rasmussen-Tamagawa type conditions in the case where the inseparable degree of base fields is not divisible by the rank of Drinfeld modules. Conversely if the rank divides the inseparable degree, then we give an example of Drinfeld modules satisfying Rasmussen-Tamagawa-type conditions.