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Register a new userCounting imaginary quadratic points via universal torsors, II
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We prove Manins conjecture for four singular quartic del Pezzo surfaces over imaginary quadratic number fields, using the universal torsor method.
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Let $K/k$ be an extension of number fields, and let $P(t)$ be a quadratic polynomial over $k$. Let $X$ be the affine variety defined by $P(t) = N_{K/k}(mathbf{z})$. We study the Hasse principle and weak approximation for $X$ in three cases. For $[K:k]=4$ and $P(t)$ irreducible over $k$ and split in $K$, we prove the Hasse principle and weak approximation. For $k=mathbb{Q}$ with arbitrary $K$, we show that the Brauer-Manin obstruction to the Hasse principle and weak approximation is the only one. For $[K:k]=4$ and $P(t)$ irreducible over $k$, we determine the Brauer group of smooth proper models of $X$. In a case where it is non-trivial, we exhibit a counterexample to weak approximation.
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 prove that a smooth complete intersection of two quadrics of dimension at least $2$ over a number field has index dividing $2$, i.e., that it possesses a rational $0$-cycle of degree $2$.
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.
We describe the practical implementation of an average polynomial-time algorithm for counting points on superelliptic curves defined over $mathbb Q$ that is substantially faster than previous approaches. Our algorithm takes as input a superelliptic curves $y^m=f(x)$ with $mge 2$ and $fin mathbb Z[x]$ any squarefree polynomial of degree $dge 3$, along with a positive integer $N$. It can compute $#X(mathbb F_p)$ for all $ple N$ not dividing $mmathrm{lc}(f)mathrm{disc}(f)$ in time $O(md^3 Nlog^3 Nloglog N)$. It achieves this by computing the trace of the Cartier--Manin matrix of reductions of $X$. We can also compute the Cartier--Manin matrix itself, which determines the $p$-rank of the Jacobian of $X$ and the numerator of its zeta function modulo~$p$.
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