Let F be a totally real number field, p a rational prime, and chi a finite order totally odd abelian character of Gal(F/F) such that chi(p)=1 for some p|p. Motivated by a conjecture of Stark, Gross conjectured a relation between the derivative of the p-adic L-function associated to chi at its exceptional zero and the p-adic logarithm of a p-unit in the chi component of Fx. In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture assuming two conditions: that Leopoldt's conjecture holds for F and p, and that if there is only one prime of F lying above p, a certain relation holds between the L-invariants of chi and chi-1. The main result of this work removes both of these conditions, thus giving an unconditional proof of the conjecture. We also describe some applications towards simplifying the Iwasawa Main Conjecture at the weight one and Leopoldt zeroes, as well as some partial results on Gross's conjecture in the higher rank setting.
展开▼
