Beschreibung:
<jats:title>Abstract</jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline1" /><jats:tex-math>$X$</jats:tex-math></jats:alternatives></jats:inline-formula> be a curve over a number field <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline2" /><jats:tex-math>$K$</jats:tex-math></jats:alternatives></jats:inline-formula> with genus <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline3" /><jats:tex-math>$g\geq 2$</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline4" /><jats:tex-math>$\mathfrak{p}$</jats:tex-math></jats:alternatives></jats:inline-formula> a prime of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline5" /><jats:tex-math>${ \mathcal{O} }_{K} $</jats:tex-math></jats:alternatives></jats:inline-formula> over an unramified rational prime <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline6" /><jats:tex-math>$p\gt 2r$</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline7" /><jats:tex-math>$J$</jats:tex-math></jats:alternatives></jats:inline-formula> the Jacobian of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline8" /><jats:tex-math>$X$</jats:tex-math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline9" /><jats:tex-math>$r= \mathrm{rank} \hspace{0.167em} J(K)$</jats:tex-math></jats:alternatives></jats:inline-formula>, and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline10" /><jats:tex-math>$\mathscr{X}$</jats:tex-math></jats:alternatives></jats:inline-formula> a regular proper model of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline11" /><jats:tex-math>$X$</jats:tex-math></jats:alternatives></jats:inline-formula> at <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline12" /><jats:tex-math>$\mathfrak{p}$</jats:tex-math></jats:alternatives></jats:inline-formula>. Suppose <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline13" /><jats:tex-math>$r\lt g$</jats:tex-math></jats:alternatives></jats:inline-formula>. We prove that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" xlink:type="simple" xlink:href="S0010437X13007410_inline14" /><jats:tex-math>$\# X(K)\leq \# \mathscr{X}({ \mathbb{F} }_{\mathfrak{p}} )+ 2r$</jats:tex-math></jats:alternatives></jats:inline-formula>, 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.</jats:p>