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Register a new userOn Manins conjecture for a certain singular cubic surface over imaginary quadratic fields
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We prove Manins conjecture over imaginary quadratic number fields for a cubic surface with a singularity of type E_6.
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We prove Manins conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for a spherical threefold with canonical singularities and two infinite families of spherical threefolds with log terminal singularities. Moreover, we show that one of these families does not satisfy a conjecture of Batyrev and Tschinkel on the leading constant in the asymptotic formula. Our proofs are based on the universal torsor method, using Brions description of Cox rings of spherical varieties.
Manins conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson formula. An alternative approach to Manins conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manins conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xyz.
We introduce the split torsor method to count rational points of bounded height on Fano varieties. As an application, we prove Manins conjecture for all nonsplit quartic del Pezzo surfaces of type $mathbf A_3+mathbf A_1$ over arbitrary number fields. The counting problem on the split torsor is solved in the framework of o-minimal structures.
A positive definite even Hermitian lattice is called emph{even universal} if it represents all even positive integers. We introduce a method to get all even universal binary Hermitian lattices over imaginary quadratic fields $Q{-m}$ for all positive square-free integers $m$ and we list optimal criterions on even universality of Hermitian lattices over $Q{-m}$ which admits even universal binary Hermitian lattices.
We will introduce a method to get all universal Hermitian lattices over imaginary quadratic fields over $mathbb{Q}(sqrt{-m})$ for all m. For each imaginary quadratic field $mathbb{Q}(sqrt{-m})$, we obtain a criterion on universality of Hermitian lattices: if a Hermitian lattice L represents 1, 2, 3, 5, 6, 7, 10, 13,14 and 15, then L is universal. We call this the fifteen theorem for universal Hermitian lattices. Note that the difference between Conway-Schneebergers fifteen theorem and ours is the number 13.
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