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Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is perfect and that $mathrm{char} k > 2g+1$. Let $S = mathrm{Spec} R$. Let $X$ be the minimal proper regular model of $C$ over $S$. Let $mathrm{Art} (C/K)$ denote the Artin conductor of the $S$-scheme $X$ and let $ u (Delta_C)$ denote the minimal discriminant of $C$. We prove that $-mathrm{Art} (C/K) leq u (Delta_C)$. The key ingredients are a combinatorial refinement of the discriminant introduced in this paper (called the metric tree) and a recent refinement of Abhyankars inversion formula for studying plane curve singularities. We also prove combinatorial restrictions for $-mathrm{Art} (C/K) = u (Delta_C)$.
Let $C$ be a hyperelliptic curve of genus $g$ over the fraction field $K$ of a discrete valuation ring $R$. Assume that the residue field $k$ of $R$ is perfect and that $mathop{textrm{char}} k eq 2$. Assume that the Weierstrass points of $C$ are $K$
We compute the Galois groups for a certain class of polynomials over the the field of rational numbers that was introduced by S. Mori and study the monodromy of corresponding hyperelliptic jacobians.
Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C:y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve
We provide a simple approach for the crystalline comparison of Ainf-cohomology, and reprove the comparison between crystalline and p-adic etale cohomology for formal schemes in the case of good reduction.
We prove that the jacobian of a hyperelliptic curve $y^2=(x-t)h(x)$ has no nontrivial endomorphisms over an algebraic closure of the ground field $K$ of characteristic zero if $t in K$ and the Galois group of the polynomial $h(x)$ over $K$ is very bi