The Chabauty–Coleman bound at a prime of bad reduction and Clifford bounds for geometric rank functions
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Bibliographic record
Abstract
Abstract Let $X$ be a curve over a number field $K$ with genus $g\geq 2$ , $\mathfrak{p}$ a prime of ${ \mathcal{O} }_{K} $ over an unramified rational prime $p\gt 2r$ , $J$ the Jacobian of $X$ , $r= \mathrm{rank} \hspace{0.167em} J(K)$ , and $\mathscr{X}$ a regular proper model of $X$ at $\mathfrak{p}$ . Suppose $r\lt g$ . We prove that $\# X(K)\leq \# \mathscr{X}({ \mathbb{F} }_{\mathfrak{p}} )+ 2r$ , extending the refined version of the Chabauty–Coleman bound to the case of bad reduction. The new technical insight is to isolate variants of the classical rank of a divisor on a curve which are better suited for singular curves and which satisfy Clifford’s theorem.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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