Mass formula of division algebras over global function fields

派生Hecke作用在除代数平凡上同调上的推广

Derived Hecke action on the trivial cohomology of division algebras

本文将Venkatesh的结构定理推广到所有数域上的除代数，证明Hecke平凡上同调的稳定子模是严格派生Hecke代数作用下的自由模，并建立保持有理结构的理性形式。

循环除代数中极大序上的矩阵作为平方和与立方和

Matrices over maximal orders in cyclic division algebras over Q as sums of squares and cubes

本文研究有理数域上奇素数次循环除代数中极大序上的矩阵，证明在特定条件下，任意阶数大于等于2的矩阵可表示为平方和与立方和，推广了已有结果。

Maass space for lifting to GL(2) over a division quaternion algebra

Muto, Narita and Pitale construct counterexamples to the Generalized Ramanujan Conjecture for GL(2,B) over the division quaternion algebra B with discriminant two via a lift from SL(2). In this paper, we try to exactly characterize the image of this lift. The previous methods of Maass, Kohnen or Kojima do not apply here, hence we approach this problem via a combination of classical and representation theory techniques to identify the image. Crucially, we use the Jacquet Langlands correspondence described by Badulescu and Renard to characterize the representations.

Lifting to GL(2) over a quaternion division algebra and an explicit construction of CAP representations

The aim of this paper is to carry out an explicit construction of CAP representations of GL(2) over a division quaternion algebra with discriminant two. We first construct cusp forms on such group explicitly by lifting from Maass cusp forms for the congruence subgroup of level 2. We show that this lifting is non-zero and Hecke-equivariant. This allows us to determine each local component of such a cuspidal representation. We then know that our cuspidal representations provide examples of CAP representations, and in fact, counterexamples of the Generalized Ramanujan conjecture.

Entropy bounds and quantum unique ergodicity for Hecke eigenfunctions on division algebras

We prove the arithmetic quantum unique ergodicity (AQUE) conjecture for non-degenerate sequences of Hecke eigenfunctions on quotients $Γ\backslash G/K$, where $G\simeq\mathrm{PGL}_{d}(\mathbb{R})$, $K$ is a maximal compact subgroup of $G$ and $Γ<G$ is a lattice associated to a division algebra over $\mathbb{Q}$ of prime degree $d$. More generally, we introduce a new method of proving positive entropy of quantum limits, which applies to higher-rank groups. The result on AQUE is obtained by combining this with a measure-rigidity theorem due to Einsiedler-Katok, following a strategy first pioneer…

On division algebras having the same maximal subfields

We address the problem of when two finite dimensional central division algebras over the same field are necessarily isomorphic given that they have the same maximal subfields.

Some examples of division symbol algebras of degree 3 and 5

In this paper we provide an algorithm to compute the product between two elements in a symbol algebra of degree n and we find an octonion non-division algebra in a symbol algebra of degree three. Starting from this last idea, we try to find an answer to the question if there are division symbol algebras of degree three. The answer is positive and we provide, using MAGMA software, some examples of division symbol algebras of degree 3 and of degree 5. Moreover, we will give some interesting applications of the symbol algebras in number theory.

Division Algebras With Infinite Genus

To what extent does the maximal subfield spectrum of a division algebra determine the isomorphism class of that algebra? It has been shown that over some fields a quaternion division algebra's isomorphism class is largely if not entirely determined by its maximal subfield spectrum. However in this paper, we show that there are fields for which the maximal subfield spectrum says little to nothing about a quaternion division algebra's isomorphism class. We give an explicit construction of a division algebra with infinite genus. Along the way we introduce the notion of a "linking field extension,…

Tamely ramified subfields of division algebras

For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long standing Q-admissibility conjecture. We extend a theorem of Neukirch on embedding problems with local constraints in order to determine for every number field K, what finite solvable groups G appear as Galois groups of tame maximal subfields of K-division algebras, generalizing Liedahl's theorem for metacyclic G and Sonn's solution of the Q-admissibility conj…

Cyclic division algebras and maximal tori

We show the cardinality of a particular nonabelian cohomology associated to cyclic division algebras is equal to a certain partition number. This computation helps us interpret the number of conjugacy classes of maximal tori defined over an abelian extension.

Division Algebra Cyclicity in Prime Degree over a p-Adic Curve

We reprove two results of Saltman, Theorem 5.1 and Corollary 5.2 of [Sa07]: If F is the function field of a smooth p-adic curve and D is an F-division algebra of prime degree l\neq p, then D is Z/l-cyclic, and that if D is an F-division algebra of prime period l\neq p then D has index l if and only if its ramification locus on a suitable 2-dimensional model forF has no "hot points".

Division Algebras and Wireless Communication

We survey the recent use of division algebras in wireless communication.

Lattice packings through division algebras

In this article, we will show the existence of lattice packings in a sparse family of dimensions. This construction will be a generalisation of Venkatesh's lattice packing result. In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rational field. For this, we develop an analogue of Siegel's mean value theorem over lattices that have a prescribed set of symmetries given by a finite non-commutative group inside the multiplicative subgroup of a division algebra. This approach improves the best known lattice packing bounds in many dimensi…

A Direct Approach to Noncrossed Product Division Algebras

A valuation theoretic approach is presented that directly leads to division algebras that are noncrossed products (instead of, e.g., describing Brauer classes of noncrossed products in an abstract manner). While this feature is shared by Amitsur's original construction, the new approach works over small fields. It is further demonstrated how it can be used to obtain very explicit examples of noncrossed products in the form of iterated twisted function fields over division algebras over global fields. The examples allow even to write down structure constants of noncrossed products.

Codes from unit groups of division algebras over number fields

Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, the first author and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in division algebras; they suggested to use unit groups in quaternion algebras but could not completely analyse the resulting codes. We prove that the noncommutative unit group construction yields asymptotically good families of codes for the sum-rank metric from division algebras of any degree, and we estimate the size of the alphabet in terms of the degree.

Division algebras with the same maximal subfields

We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some of its applications. In the last section, we extend the problem to the context of absolutely almost simple algebraic groups.

Tame division algebras of prime period over function fields of $p$-adic curves

Let F be a field of transcendence degree one over a p-adic field, and let l be a prime not equal to p. Results of Merkurjev and Saltman show that H^2(F,μ_l) is generated by Z/l-cyclic classes. We prove the "Z/l-length" in H^2(F,μ_l) equals the l-Brauer dimension, which Saltman showed to be two. It follows that all F-division algebras of period l are crossed products, either cyclic (by Saltman's cyclicity result) or tensor products of two cyclic division algebras. Our result was originally proved by Suresh assuming F contains μ_l.

Global Jacquet-Langlands correspondence for division algebras in characteristic p

We prove a full global Jacquet-Langlands correspondence between GL(n) and division algebras over global fields of non zero characteristic. If $D$ is a central division algebra of dimension $n^2$ over a global field $F$ of non zero characteristic, we prove that there exists an injective map from the set of automorphic square integrable representations of the multiplicative group of $D$ to the set of automorphic square integrable representations of GL_n(F), compatible at all places with the local Jacquet-Langlands correspondence for unitary representations. We characterize the image of the map. …

The genus of a division algebra and the unramified Brauer group

Let D be a finite-dimensional central division algebra over a field K. We define the genus gen(D) of D to be the collection of classes in the Brauer group of K represented by central division K-algebras D' having the same maximal subfields as D. In this paper, we describe a general approach to proving the finiteness of gen(D) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of K.

On the size of the genus of a division algebra

Let D be a central division algebra of degree n over a field K. One defines the genus gen(D) of D as the set of classes [D'] in the Brauer group Br(K) of K represented by central division algebras D' of degree n over K having the same maximal subfields as D. We prove that if the field K is finitely generated and n is prime to its characteristic, then gen(D) is finite, and give explicit estimations of its size in certain situations.

Selfdual representations of division algebras and Weil groups: A contrast

Selfdual representations of any group fall into two classes when they are irreducible: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the irreducible representations σof the Weil group of a local field k of dimension n with the irreducible representations πof the invertible elements of a division algebra D over k of index n, takes selfdual representations to selfdual representations. In this paper we use global methods to study how the Langlands correspondence behaves relative to this distinction…

On the reduction modulo p of representations of a quaternion division algebra over a p-adic field

Let F be a non-Archimedean local field of residue characteristic p. In this paper, we first compute the reduction modulo p of irreducible smooth representations of a quaternion division algebra over F and of two-dimensional irreducible smooth representations of the Weil group of F. It turns out that a natural correspondence of mod p representations of the two groups and the composite of the local Langlands and Jacquet-Langlands correspondence are not compatible with the reduction, except in the cases considered by Vigneras.

Division algebra codes achieve MIMO block fading channel capacity within a constant gap

This work addresses the question of achieving capacity with lattice codes in multi-antenna block fading channels when the number of fading blocks tends to infinity. In contrast to the standard approach in the literature which employs random lattice ensembles, the existence results in this paper are derived from number theory. It is shown that a multiblock construction based on division algebras achieves rates within a constant gap from block fading capacity both under maximum likelihood decoding and naive lattice decoding. First the gap to capacity is shown to depend on the discriminant of the…

带正对合的可除代数上的斜埃尔米特型格与非典型交

Lattices with skew-Hermitian forms over division algebras and unlikely intersections

本文研究带正对合可除代数上的斜埃尔米特型格，证明Albert I、II型下这类格子格存在有效有界指数的正交基，并将结果用于非典型交领域，在大Galois轨道猜想下证明单PEL I、II型特殊子簇与曲线交的Zilber-Pink猜想，且证得部分II型的该Galois轨道猜想。

Reduced norms of division algebras over complete discrete valuation fields of local-global type

Let $F$ be a complete discrete valuation field whose residue field $k$ is a global field of positive characteristic $p$. Let $D$ be a central division $F$-algebra of $p$-power degree. We prove that the subgroup of $F^*$ consisting of reduced norms of $D$ is exactly the kernel of the cup product map $λ\in F^*\mapsto (D)\cup(λ)\in H^3(F,\,\mathbb{Q}_{p}/\Z_{p}(2))$, if either $D$ is tamely ramified or of period $p$. This gives a $p$-torsion counterpart of a recent theorem of Parimala, Preeti and Surech, where the same result is proved for division algebras of prime-to-$p$ degree.

Towards a complete DMT classification of division algebra codes

This work aims at providing new bounds for the diversity multiplexing gain trade-off of a general class of division algebra based lattice codes. In the low multiplexing gain regime, some bounds were previously obtained from the high signal-to-noise ratio estimate of the union bound for the pairwise error probabilities. Here these results are extended to cover a larger range of multiplexing gains. The improvement is achieved by using ergodic theory in Lie groups to estimate the behavior of the sum arising from the union bound. In particular, the new bounds for lattice codes derived from Q-centr…

Deligne-Lusztig Constructions for Division Algebras and the Local Langlands Correspondence

Let $K$ be a local non-Archimedean field of positive characteristic and let $L$ be the degree-$n$ unramified extension of $K$. Via the local Langlands and Jacquet-Langlands correspondences, to each sufficiently generic multiplicative character of $L$, one can associate an irreducible representation of the multiplicative group of the central division algebra $D$ of invariant $1/n$ over $K$. In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive $p$-adic groups analogous to Deligne-Lusztig theory for finite reductive groups. In this paper we prove th…

Small generators for S-unit groups of division algebras

Let $k$ be a number field, suppose that $B$ is a central simple division algebra over $k$, and choose any maximal order $\mathcal{D}$ of $B$. The object of this paper is to show that the group $\mathcal{D}_S^*$ of $S$-units of $B$ is generated by elements of small height once $S$ contains an explicit finite set of places of $k$. This generalizes a theorem of H.\ W.\ Lenstra Jr., who proved such a result when $B = k$. Our height bound is an explicit function of the number field and the discriminant of a maximal order in $B$ used to define its $S$-units.

The finiteness of the genus of a finite-dimensional division algebra, and some generalizations

We prove that the genus of a finite-dimensional division algebra is finite whenever the center is a finitely generated field of any characteristic. We also discuss potential applications of our method to other problems, including the finiteness of the genus of simple algebraic groups of type $\textsf{G}_2$. These applications involve the double cosets of adele groups of algebraic groups over arbitrary finitely generated fields: while over number fields these double cosets are associated with the class numbers of algebraic groups and hence have been actively analyzed, similar question over more…

超多元结构

Hyperpolyadic structures

本文引入了对应于二元除法代数的超多元代数，通过矩阵多元过程和“虚塔”迭代构造了不同维数的 n 元除法代数，并提出了“半四元数”等新结构及其相关范数恒等式。