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The local-global principle for symmetric determinantal representations of smooth plane curves. (English) Zbl 1390.14086
Let \(C\subset {\mathbb P}_K^2\) be a smooth plane curve of degree \(n\geq 1\) over a field \(K\). We say that \(C\) admits a symmetric determinantal representation over \(K\) if there exists a triple of symmetric matrices \((M_0,M_1,M_1)\) of size \(n\) with entries in \(K\) such that \(C\) is defined by the equation \[ \det (X_0,N_0+X_1M_1+X_2M_2) = 0\, . \]
The paper studies the local-global principle for the existence of symmetric determinantal representations of smooth plane curves defined over a global field \(K\) with \(\text{char}\, K\neq 2\).
The main results of the paper are the following: \smallskip
{Theorem 1.} Let \(K\) be a global field of characteristic different from two, and let \(C\subset {\mathbb P}_K^2\) be a smooth plane curve of degree 2 or 3. If \(C\) admits a symmetric determinantal representation over the completion \(K_v\) for each place \(v\) of \(K\), then \(C\) admits a symmetric determinantal representation over \(A\). \smallskip
{Theorem 2.} Let \(K\) be a global field of characteristic different from two, and let \(C\subset {\mathbb P}_K^2\) be a smooth plane curve of degree 4. Assume that the associated mod 2 Galois representation on the 2-torsion points on the Jacobian variety \(\text{Jac(C)}\) \[ \rho_{C,2}\, :\, \text{Gal}(K^{\text{sep}}/K) \to \text{Sp} (\text{Jac}(C)[2](K^{\text{sep}}) \cong \text{Sp}_6({\mathbb F}_2) \] is surjective. The there is a finite extension \(L/K\) such that \(C\) admits a symmetric determinantal representation over \(L_w\) for each place \(w\) of \(L\) but not over \(L\). \smallskip
The proof of Theorem 1 relies on the existence of a \(K\)-rational point on the conic for \(n=2\), and of a non-trivial \(K\)-rational 2-torsion point on the Jacobian variety of the curve for \(m=3\).
The proof of Theorem 2 makes use of a group-theoetic lemma on the action of the subgroups of \(\text{Sp}({\mathbb F}_2)\) on quadratic forms over \({\mathbb F}_2\).

14H50 Plane and space curves
11D41 Higher degree equations; Fermat’s equation
14K30 Picard schemes, higher Jacobians
14K15 Arithmetic ground fields for abelian varieties
14F22 Brauer groups of schemes
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