On the rank one abelian Gross–Stark conjecture
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Abstract
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 F_\chi^\times . In a recent work, Dasgupta, Darmon, and Pollack have proven this conjecture in the rank one setting 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 paper removes both of these conditions, thus giving an unconditional proof of the rank one conjecture.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.004 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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