Very simple 2-adic representations and hyperelliptic jacobians
By Yuri G. Zarhin. 2001
58 citations recorded by the source corpus.
Let $K$ be a number field, $n>4$ an integer, $f(x)$ an irreducible polynomial over $K$ of degree $n$, whose Galois group is either the full symmetric group $S_n$ or the alternating group $A_n$. Suppose $C:y^2=f(x)$ is the corresponding hyperelliptic curve and $J$ its jacobian defined over $K$. For each prime $\ell$ we write $V_{\ell}(J)$ for the $Q_{\ell}$-Tate module of $J$ and $e_{\ell}$ for the Riemann form on $V_{\ell}(J)$ attached to the theta divisor. (Here $Q_{\ell}$ is the field of $\ell$-adic numbers.) We write $sp(V_{\ell}(J))$ for the $Q_{\ell}$-Lie algebra of the symplectic group of $e_{\ell}$. We write $g_{\ell}$ for the Lie algebra of the image of the Galois group $Gal(K)$ of $K$ in $Aut(V_{\ell}(J))$. We prove that $g_{\ell}$ coincides with the direct sum $Q_{\ell}I \oplus sp(V_{\ell}(J))$ where $I$ is the identity operator.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1f1c6028fdee4307743641121e77ea40a25441a3
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Citations in the corpus
- Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication (2007, background): “l family of hyperelliptic curves If the middle part Vm 2 is of type (1,1), one obtains Mon0(Vm 2 ) = SpQ(2), since SpR(2) ∼= SU(1,1), and Mon0 R(Vm 2 ) = SU(1,1) as one has by Theorem 5.1.1. By using [52] Theorem 10.1 and Remark 10.2, one can conclude that the Hodge group Hg(Vm 2 ) of an arbitrary m”
- ON THE DISTRIBUTION OF GALOIS GROUPS (2010, background): “This is a situation occurring sometimes in applications, for instance Zarhin [20] proved that the Galois action on the Tate modules of a Jacobian variety of a curve C/Q of the type y(2) = f(x) is ‘big’ provided deg(f) ≥ 5 and the Galois group of f contains the alternating group An on n letters.”