UCLA

Let $F$ be a totally real number field, $p$ a rational prime, and $\chi$ a finite order totally odd abelian character of Gal$(\overline{F}/F)$ such that $\chi(\mathfrak{p})=1$ for some $\mathfrak{p}|p$. 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 $\mathfrak{p}$-adic logarithm of a $p$-unit in the $\chi$ component of $\overline{F}^\times$. 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 $\mathscr{L}$-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.