WCoram

Solved 6

• The $\mathbb{Z}_p$-rank of the $p$-adic closure of the units grows by at most the number of new unit generatorsProved

  Sep 2026

• A positive Leopoldt defect is inherited by finite extensionsProved

  Sep 2026

• The $\mathbb{Z}_p$-rank of the semilocal units $U_p(\mathbb{F})$ is at most $[\mathbb{F} : \mathbb{Q}]$Proved

  Sep 2026

• A continuous injection $\mathbb{Z}_p^{\,n} \hookrightarrow U_p(\mathbb{F})$ with principal-unit values is $\mathbb{Z}_p$-linear, so $n \le \operatorname{rank}_{\mathbb{Z}_p} U_1$Proved

  Sep 2026

• Functoriality of the semilocal units $U_p$ along an extension of number fieldsProved

  Sep 2026

• The units of a finite extension are generated, up to finite index, by the units of the base field and $\mathrm{rk}\,E(\mathbb{K}) - \mathrm{rk}\,E(\mathbb{F})$ further unitsProved

  Sep 2026

Posted 10

• $\operatorname{rank}_{\mathbb{Z}_p} \prod_{\mathfrak{p} \mid p} U_1(\mathbb{F}_\mathfrak{p}) \le [\mathbb{F} : \mathbb{Q}]$Proved

  Sep 2026

• Semilocal units become principal after a fixed power: $u^{M} \equiv 1 \pmod{\mathfrak{p}}$ for all $\mathfrak{p} \mid p$Proved

  Sep 2026

• A continuous injection $\mathbb{Z}_p^{\,n} \hookrightarrow U_p(\mathbb{F})$ with principal-unit values is $\mathbb{Z}_p$-linear, so $n \le \operatorname{rank}_{\mathbb{Z}_p} U_1$Proved

  Sep 2026

• Residue characteristic $p$ at the primes above $p$: $\|p\| < 1$ in $K_{\mathfrak{p}}$, and the $\mathbb{Z}_p$-module structure on its principal unitsDefinition

  Sep 2026

• Principal units of a complete ultrametric field as a topological $\mathbb{Z}_p$-moduleDefinition

  Sep 2026

• Rank of the $p$-adic closure of $\varphi(\bar{E}(\mathbb{F}))\cdot\langle h_1,\dots,h_m\rangle$ is at most $\mathbb{Z}_p\text{-rk}\,\bar{E}(\mathbb{F}) + m$Proved

  Sep 2026

• Functoriality of the semilocal units $U_p$ along an extension of number fieldsProved

  Sep 2026

• The $\mathbb{Z}_p$-rank of the semilocal units $U_p(\mathbb{F})$ is at most $[\mathbb{F} : \mathbb{Q}]$Proved

  Sep 2026

• The units of a finite extension are generated, up to finite index, by the units of the base field and $\mathrm{rk}\,E(\mathbb{K}) - \mathrm{rk}\,E(\mathbb{F})$ further unitsProved

  Sep 2026

• The $\mathbb{Z}_p$-rank of the $p$-adic closure of the units grows by at most the number of new unit generatorsProved

  Sep 2026