Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to-$p$ rational torsion subgroup on the Jacobian of $C$. We also determine divisibility of line bundles on $C$, including rationality of theta characteristics and higher spin structures. These computations utilize arithmetic on the special fiber of $C$.

Let $C$ be a smooth real quartic curve in ${\mathbb{P}}^2$. Suppose that $C$ has at least $3$ real branches $B_1,B_2,B_3$. Let $B=B_1\times B_2\times B_3$ and let $O\in B$. Let ${\tau }_O$ be the map from $B$ into the neutral component Jac$\left(C\right){\left(ℝ\right)}^0$ of the set of real points of the jacobian of $C$, defined by letting ${\tau }_O\left(P\right)$ be the divisor class of the divisor $\sum P_i-O_i$. Then, ${\tau }_O$ is a bijection. We show that this allows an explicit geometric description of the group law on Jac$\left(C\right){\left(ℝ\right)}^0$. It generalizes the classical geometric description of the group law on the neutral component of the set of real points of...

Let $C$ be a curve of genus $g\ge 2$ defined over the fraction field $K$ of a complete discrete valuation ring $R$ with algebraically closed residue field. Suppose that $char\left(K\right)=0$ and that the characteristic $p$ of the residue field is not $2$. Suppose that the Jacobian $Jac\left(C\right)$ has semi-stable reduction over $R$. Embed $C$ in $Jac\left(C\right)$ using a $K$-rational point. We show that the coordinates of the torsion points lying on $C$ lie in the unique tamely ramified quadratic extension of the field generated over $K$ by the coordinates of the $p$-torsion...

This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) $C_{q,p,a}:y^q=x^p+a$, and its Jacobians $J_{q,p,a}$, where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of $J_{3,5,a}\left(\mathbb{Q}\right)$ (resp. $J_{q,p,a}\left(\mathbb{Q}\right)$). The main tools are computations of the zeta function of $C_{3,5,a}$ (resp. $C_{q,p,a}$) over ${}_l$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l ≡ -1 (mod qp))...

Consider a family of integral complex locally planar curves whose relative Hilbert scheme of points is smooth. The decomposition theorem of Beilinson, Bernstein, and Deligne asserts that the pushforward of the constant sheaf on the relative Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension. It follows that the perverse filtration on the cohomology of the compactified Jacobian of an integral plane curve...

Nous étudions l’action du groupe de Galois sur les périodes des courbes de Mumford qui sont des revêtements cycliques de ${\mathbb{P}}_K^1$. Lorsque le degré de ce revêtement est premier à la caractéristique résiduelle du corps de base, nous décomposons le réseau des périodes en une somme directe de modules monogènes, le nombre de ces modules monogènes étant déduit de la géométrie de la courbe (théorème 4). Ceci nous permet de donner une condition nécessaire et suffisante pour que le module des périodes soit libre...