We build a database of genus 2 curves defined over Q which contains all curves with minimal absolute height h ≤ 5, all curves with moduli height h ≤ 20, and all curves with extra automorphisms in standard form y 2 = f (x 2 ) defined over Q with height h ≤ 101. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli space M 2 for which the field of moduli is a field of definition is discussed and some open problems are presented.

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Classical optimization algorithms in machine learning often take a long time to compute when applied to a multidimensional problem and require a huge amount of CPU and GPU resource. Quantum parallelism has a potential to speed up machine learning algorithms. We describe a generic mathematical model to leverage quantum parallelism to speed-up machine learning algorithms. We also apply quantum machine learning and quantum parallelism applied to a 3-dimensional image that vary with time.Index Terms-quantum machine learning, higher-dimensional data sets, quantum computation, quantum parallelism.

We use the weighted moduli height as defined in [8] to investigate the distribution of fine moduli points in the moduli space of genus two curves. We show that for any genus two curve with equationwhere H(f ) is the minimal naive height of the curve as defined in [9]. Based on the weighted moduli height h we create a database of genus two curves defined over Q with small h and show that for small such height (h < 5) about 30% of points are fine moduli points.

Let C be a genus 2 curve defined over a field K, char K = p ≥ 0, and Jac(C, ι) its Jacobian, where ι is the principal polarization of Jac(C) attached to C. Assume that Jac(C) is (n, n)-geometrically reducible with E 1 and E 2 its elliptic components. We prove that there are only finitely many curves C (up to isomorphism) defined over K such that E 1 and E 2 are N -isogenous for n = 2 and N = 2, 3, 5, 7 with Aut (Jac C) ∼ = V 4 or n = 2, N = 3, 5, 7 with Aut (Jac C) ∼ = D 4 . The same holds if n = 3 and N = 5. Furthermore, we determine the Kummer and the Shioda-Inose surfaces for the above Jac C and show how such results in positive characteristic p > 2 suggest nice applications in cryptography.