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Ring of integers
About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.
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Abstract: We compute the indecomposable objects of \dot{U}^+_3 - the category that categorifies the positive half of the quantum sl_3, and we decompose an arbitrary object into indecomposable ones. On decategorified level we obtain the Lusztig's canonical basis of the positive half U^+_q(sl_3) of the quantum sl_3. We also categorify the higher quantum Serre relations in U_q^+(sl_3), by defining a certain complex in the homotopy category of $\dot{U}^+_3$ that is homotopic to zero. We work with the category $\dot{U}^+_3$ that is defined over the ring of integers. This paper is based on the (extended) diagrammatic calculus introduced to categorify quantum groups.
11 citations
TL;DR: In this article, it was shown that the ramification of a torsion element in a regular proper model over the ring of integers in a number field can be split in an extension of degree p^3.
Abstract: Let k be a number field and F a function field in one variable over k. We prove that the ramification of a \(p\)-torsion element in \(Br\)(\(F\)) on a regular proper model over the ring of integers in \(k\) can be split in an extension of degree \(p^3\). Using this result, we show that Colliot-Thelene’s conjecture on 0-cycles of degree 1 implies finiteness for the \(u\)-invariant of the function field of a curve over a totally imaginary number field and period-index bounds for the Brauer groups of arbitrary fields of transcendence degree 1 over the rational numbers.
11 citations
TL;DR: In this article, the authors study the canonical morphism $q\colon \mathrm{Pic}^{0}_{X/S}\to J$ which extends the biduality isomorphism on generic fibres and show that it is pro-algebraic in nature.
Abstract: Let $K$ be a local field with algebraically closed residue field and $X_K$ a torsor under an elliptic curve $J_K$ over $K$. Let $X$ be a proper minimal regular model of $X_K$ over the ring of integers of $K$ and $J$ the identity component of the Neron model of $J_K$. We study the canonical morphism $q\colon \mathrm{Pic}^{0}_{X/S}\to J$ which extends the biduality isomorphism on generic fibres. We show that $q$ is pro-algebraic in nature with a construction that recalls Serre's work on local class field theory. Furthermore we interpret our results in relation to Shafarevich's duality theory for torsors under abelian varieties.
10 citations
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TL;DR: In this paper, the authors obtained an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, with discrete subgroup of Hecke type for a non-zero ideal $I$ in the ring of integers of a real number field.
Abstract: We obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region, for $\SL_2$ over a totally real number field $F$, with discrete subgroup of Hecke type $\Gamma_0(I)$ for a non-zero ideal $I$ in the ring of integers of $F$. The weights are products of Fourier coefficients. This implies in particular the existence of infinitely many cuspidal automorphic representations with multi-eigenvalues in various regions growing to infinity. For instance, in the quadratic case, the regions include floating boxes, floating balls, sectors, slanted strips and products of prescribed small intervals for all but one of the infinite places of $F$. The main tool in the derivation is a sum formula of Kuznetsov type.
10 citations
TL;DR: In this article, the volume of the quotient of three-dimensional hyperbolic space by SL2 over the ring of integers of an imaginary quadratic number field was shown to be the same as that of Humbert.
Abstract: We close a gap in Humbert's classical calculation of the volume of the quotient of three-dimensional hyperbolic space by SL2 over the ring of integers of an imaginary quadratic number field.
10 citations