arXiv subjects

Plawan Das

Publications and source records attributed to Plawan Das.

4 recordsLinked to original sources

arXiv2022

A Base Change Version of Rasmussen-Tamagawa Conjecture

We prove a certain uniform version of the Shafarevich Conjecture. As a corollary, we prove the Rasmussen-Tamagawa Conjecture for a particular class of abelian varieties $A$ defined over a number $K$ of dimension $g$ having everywhere potential good reduction, in particular, for any finite place $v$ of $K$ the localization $A_v:=A\times_{\mathrm{Spec}(K)}\mathrm{Spec}(K_v)$ has either good reduction or {\it totally bad reduction} (connected component $\tilde{\mathcal{A}}_v^0$ of the special fibre $\tilde{\mathcal{A}}_v$ of the N\'eron model $\mathcal{A}_v$ at $v$ is an affine group scheme over the residue field $k_v$ at $v$) and has good reduction over a quadratic extension of $K_v$.

math.NT↗

arXiv2022

A finiteness theorem for abelian varieties with totally bad reduction

We show that up to potential isogeny, there are only finitely many abelian varieties of dimension $d$ defined over a number field $K$, such that for any finite place $v$ outside a fixed finite set $S$ of places of $K$ containing the archimedean places, it has either good reduction at $v$, or totally bad reduction at $v$ and good reduction over a quadratic extension of the completion of $K$ at $v$.

math.NT↗

arXiv2021

Finiteness theorems for potentially equivalent Galois representations: extension of Faltings' finiteness criteria

We study the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $ρ$ is determined by an equality of an $m$-power character $g\mapsto Tr(ρ(g^m))$ for some natural number $m$. Using this, we extend Faltings' finiteness criteria to determine the equivalence of two $\ell$-adic, semisimple representations of the absolute Galois group of a number field, to the context of potential equivalence. We also discuss finiteness results for twist unramified representations.

math.NT↗

arXiv2021

A uniform bound for inertially equivalent, pure $\ell$-adic representations: an extension of Faltings' theorem

We introduce a notion of inertial equivalence for integral $\ell$-adic representation of the Galois group of a global field. We show that the collection of continuous, semisimple, pure $\ell$-adic representations of the absolute Galois group of a global field lifting a fixed absolutely irreducible residual representation and with given inertial type outside a fixed finite set of places is uniformly bounded independent of the inertial type.

math.NT↗