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Register a new userGalois groups of Mori trinomials and hyperelliptic curves with big monodromy
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Authors
Yuri G. Zarhin
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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.
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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 $mathop{textrm{char}} k eq 2$. Assume that the Weierstrass points of $C$ are $K$-rational. Let $S = mathop{textrm{Spec}} R$. Let $mathcal{X}$ be the minimal proper regular model of $C$ over $S$. Let $mathop{textrm{Art}} (mathcal{X}/S)$ denote the Artin conductor of the $S$-scheme $mathcal{X}$ and let $ u (Delta)$ denote the minimal discriminant of $C$. We prove that $-mathop{textrm{Art}} (mathcal{X}/S) leq u (Delta)$. As a corollary, we obtain that the number of components of the special fiber of $mathcal{X}$ is bounded above by $ u(Delta)+1$.
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 over K, and $J$ the jacobian of $C$. We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the identity element of $J$). It is well known that for each $mathfrak{b} in J(K)$ there are exactly $2^{2g}$ elements $mathfrak{a} in J(K)$ such that $2mathfrak{a}=mathfrak{b}$. M. Stoll constructed an algorithm that provides Mumford representations of all such $mathfrak{a}$, in terms of the Mumford representation of $mathfrak{b}$. The aim of this paper is to give explicit formulas for Mumford representations of all such $mathfrak{a}$, when $mathfrak{b}in J(K)$ is given by $P=(a,b) in C(K)subset J(K)$ in terms of coordinates $a,b$. We also prove that if $g>1$ then $C(K)$ does not contain torsion points with order between $3$ and $2g$.
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 $-mathrm{Art} (C/K) = u (Delta_C)$.
We prove that $p$-determinants of a certain class of differential operators can be lifted to power series over $mathbb{Q}$. We compute these power series in terms of monodromy of the corresponding differential operators.
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 big and $deg(h)$ is an even number >8. (The case of odd $deg(h)>3$ follows easily from previous results of the author.)
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