Curves over finite fields attain the Hallouin-Perret quadratic-extension bound exactly when all real parts of their Frobenius eigenvalues coincide, which yields new coefficient bounds and low-genus classifications.
Diophantine stability for curves over finite fields
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abstract
We carry out a survey on curves defined over finite fields that are Diophantine stable; that is, with the property that the set of points of the curve is not altered under a proper field extension. First, we derive some general results of such curves and then we analyze several families of curves that happen to be Diophantine stable.
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Maximal curves with respect to quadratic extensions over finite fields
Curves over finite fields attain the Hallouin-Perret quadratic-extension bound exactly when all real parts of their Frobenius eigenvalues coincide, which yields new coefficient bounds and low-genus classifications.