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Register a new userThe Distribution of Integers in a Totally Real Cubic Field
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Tianyi Mao
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Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive integers of given trace in a general totally real number field of any degree. When the field is cubic, we show that the asymptotic behavior of a weighted Diophantine sum is related to the structure of the unit group. The main term can be expressed in terms of Grossencharacter $L$-functions.
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We consider very general random integers and (attempt to) prove that many multiplicative and additive functions of such integers have limiting distributions. These integers include, for instance, the curvatures of Apollonian circle packings, trace of Frobenius elements for elliptic curves, the Ramanujan tau-function, Mersenne numbers, and many others.
We compute the etale cohomology ring $H^*(text{Spec } mathcal{O}_K,mathbb{Z}/nmathbb{Z})$ where $mathcal{O}_K$ is the ring of integers of a number field $K.$ As an application, we give a non-vanishing formula for an invariant defined by Minhyong Kim.
Let $K$ be a totally real number field of degree $n geq 2$. The inverse different of $K$ gives rise to a lattice in $mathbb{R}^n$. We prove that the space of Schwartz Fourier eigenfunctions on $mathbb{R}^n$ which vanish on the component-wise square root of this lattice, is infinite dimensional. The Fourier non-uniqueness set thus obtained is a discrete subset of the union of all spheres $sqrt{m}S^{n-1}$ for integers $m geq 0$ and, as $m rightarrow infty$, there are $sim c_{K} m^{n-1}$ many points on the $m$-th sphere for some explicit constant $c_{K}$, proportional to the square root of the discriminant of $K$. This contrasts a recent Fourier uniqueness result by Stoller. Using a different construction involving the codifferent of $K$, we prove an analogue of our results for discrete subsets of ellipsoids. In special cases, these sets also lie on spheres with more densely spaced radii, but with fewer points on each. We also study a related question about existence of Fourier interpolation formulas with nodes $sqrt{Lambda}$ for general lattices $Lambda subset mathbb{R}^n$. Using results about lattices in Lie groups of higher rank, we prove that, if $n geq 2$ and if a certain group $Gamma_{Lambda} leq operatorname{PSL}_2(mathbb{R})^n$ is discrete, then such interpolation formulas cannot exist. Motivated by these more general considerations, we revisit the case of one radial variable and prove, for all $n geq 5$ and all real $lambda > 2$, Fourier interpolation results for sequences of spheres $sqrt{2 m/ lambda}S^{n-1}$, where $m$ ranges over any fixed cofinite set of non-negative integers. The proof relies on a series of Poincare type for Hecke groups of infinite covolume, similarly to the construction previously used by Stoller.
We present a variation of the modular algorithm for computing the Hermite normal form of an $mathcal O_K$-module presented by Cohen, where $mathcal O_K$ is the ring of integers of a number field $K$. An approach presented in (Cohen 1996) based on reductions modulo ideals was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature. In this paper, we present a modification of the approach of Cohen to prevent the coefficient swell and we rigorously assess its complexity with respect to the size of the input and the invariants of the field $K$.
We study a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes (resp. unit groups of maximal orders). For each finite noncyclic group $G$, we give an explicit formula for the number of conjugacy classes of maximal orders whose unit groups modulo center are isomorphic to $G$, and write down a representative for each conjugacy class. This leads to a complete recipe (even explicit formulas in special cases) for the refined class number formula for all finite groups. As an application, we prove the existence of superspecial abelian surfaces whose endomorphism algebras coincide with $mathbb{Q}(sqrt{p})$ in all positive characteristic $p otequiv 1pmod{24}$.
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