On the rank of Leopoldt's and Gross's regulator maps

Abstract

We generalize Waldschmidt's bound for Leopoldt's defect and prove a similar bound for Gross's defect for an arbitrary extension of number fields. As an application, we prove new cases of Gross's finiteness conjecture (also known as the Gross-Kuz'min conjecture) beyond the classical abelian case, and we show that Gross's $p$-adic regulator has at least half of the conjectured rank. We also describe and compute non-cyclotomic analogues of Gross's defect.

20 pages, comments welcome!

References in corpus (1)

• On the Gross order of vanishing conjecture for large vanishing orders

Cited by in corpus (1)

• Applications of representation theory and of explicit units to Leopoldt's conjecture