Weyl groups and abelian varieties

We show that every finite abelian group occurs as the group of rational points of an ordinary abelian variety over F 2 \mathbb {F}_2 , F 3 \mathbb {F}_3 and F 5 \mathbb {F}_5 . We produce partial results for abelian varieties over a general finite field F q \mathbb {F}_q . In particular, we show that certain abelian groups cannot occur as groups of rational points of abelian varieties over F q \mathbb {F}_q when q q is large. Finally, we show that every finite cyclic group arises as the group of rational points of infinitely many simple abelian varieties over F 2 \mathbb {F}_2 .

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

An automorphism of an abelian variety induces a decomposition of the variety up to isogeny. There are two such results, namely the isotypical decomposition and Roan’s decomposition theorem. We show that they are essentially the same. Moreover, we generalize in a sense this result to abelian varieties with action of an arbitrary finite abelian group. An early version of this article was inadvertently published before all the revisions had been completed and then retracted [ https://doi.org/10.1007/s00013-018-1244-3 ]. This article is the final peer reviewed version.

Given a root system Φ of rank r and a global field F containing the n-th roots of unity, it is possible to define a Weyl group multiple Dirichlet series whose coefficients are n-th order Gauss sums. It is a function of r complex variables, and it has meromorphic continuation to all of C, with functional equations forming a group isomorphic to the Weyl group of Φ. Weyl group multiple Dirichlet series and their residues unify many examples that have been studied previously in a case-bycase basis, often with applications to analytic number theory. (Examples may be found in the final section of the paper.) We believe these Weyl group multiple Dirichlet series are fundamental objects. The goal of this paper is to define these series for any such Φ and F , and to indicate how to study them. We will note the following points. • The coefficients of the Weyl group multiple Dirichlet series are multiplicative, but the multiplicativity is twisted , so the Dirichlet series is not an Euler product. • Due to the multiplicativity, description of the coefficients reduces to the case where the parameters are powers of a single prime p. There are only finitely many such coefficients (for given p). • In the “stable case” where n is sufficiently large (depending on Φ), the number of nonzero coefficients in the p-part is equal to the order of the Weyl group. Indeed, these nonzero coefficients are parametrized in a natural way by the Weyl group elements. • The p-part coefficient parametrized by a Weyl group element w is a product of l(w) Gauss sums, where l is the length function on the Weyl group. We note a curious similarity between this description and the coefficients of the generalized theta series on the n-fold cover of GL(n) and GL(n − 1); these coefficients are determined in Kazhdan and Patterson [17] and discussed further in Patterson [21]. See Bump and Hoffstein [11] or Hoffstein [15] for a “classical” description of these coefficients. The noted similarity means that the complete Mellin transform of the theta function would be a multiple Dirichlet series resembling our An+1 multiple Dirichlet series. There is no a priori reason that we are

To each reduced root system Φ of rank r, and each sufficiently large integer n, we define a family of multiple Dirichlet series in r complex variables, whose group of functional equations is isomorphic to the Weyl group of Φ. The coefficients in these Dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime p. For each p, the number of nonzero such coefficients is equal to the order of the Weyl group, and each nonzero coefficient is a product of n-th order Gauss sums. The root system plays a basic role in the combinatorics underlying the proof of the functional equations.

In our previous study of duality for complete discrete valuation fields with perfect residue field, we treated coefficients in finite flat group schemes. In this paper, we treat abelian varieties. This, in particular, implies Grothendieck’s conjecture on the perfectness of his pairing between the Néron component groups of an abelian variety and its dual. The point is that our formulation is well suited to Galois descent. From the known case of semistable abelian varieties, we deduce the perfectness in full generality. We also treat coefficients in tori and, more generally, 1-motives.

We point out that the moduli spaces of all known 3d mathcal{N} = 8 and mathcal{N} = 6 SCFTs, after suitable gaugings of finite symmetry groups, have the form ℂ4r/Γ where Γ is a real or complex reflection group depending on whether the theory is mathcal{N} = 8 or mathcal{N} = 6, respectively. Real reflection groups are either dihedral groups, Weyl groups, or two sporadic cases H3,4 Since the BLG theories and the maximally supersymmetric Yang-Mills theories correspond to dihedral and Weyl groups, it is strongly suggested that there are two yet-to­be-discovered 3d mathcal{N} = 8 theories for H3,4. We also show that all known mathcal{N} = 6 theories correspond to complex reflection groups collectively known as G(k, x, N). Along the way, we demonstrate that two ABJM theories (SU(N)k x SU(N)-k)/ℤN and (U(N)k x U(N)-k) /ℤk are actually equivalent.

The so-called class-invariant homomorphism $\psi$ measures the Galois module structure of torsors--under a finite flat group scheme $G$--which lie in the image of a coboundary map associated to an isogeny between (N\'eron models of) abelian varieties with kernel $G$. When the varieties are elliptic curves with semi-stable reduction and the order of $G$ is coprime to 6, is is known that the homomorphism $\psi$ vanishes on torsion points. In this paper, using Weil restrictions of elliptic curves, we give the construction, for any prime number $p>2$, of an abelian variety $A$ of dimension $p$ endowed with an isogeny (with kernel $\mu_p$) whose coboundary map is surjective. In the case when $A$ has rank zero and the $p$-part of the Picard group of the base is non-trivial, we obtain examples where $\psi$ does not vanishes on torsion points.

On elliptic Calogero–Moser systems for complex crystallographic reflection groups

We begin this paper with a short survey on finite reflection groups. First we establish what a reflection in Euclidean space is. Then we introduce a root system, which is then partitioned into two sets: one of positive roots and one with negative roots. Th is articulates our understanding of groups generated by simple reflections. Furthermore, we develop our insight to Weyl groups and crystallographic groups before exploring crystallographic root systems. The section section of this paper examines the twisted group algebra along with the Demazure element Xi and the Demazure-Lusztig element Ti. Lastly, the third section of this paper computes such Xi and Ti in the case of S3 where S3 is the symmetric group Sn, n=3.

Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford–Tate group, $$\ell $$ -adic monodromy groups, and the Sato–Tate group. Assuming the Mumford–Tate conjecture, we show that from two well chosen Frobenius polynomials of our abelian variety, we can recover the identity component of these groups (or at least an inner form), up to isomorphism, along with their natural representations. We also obtain a practical probabilistic algorithm to compute these groups by considering more and more Frobenius polynomials; the groups are connected and reductive and thus can be expressed in terms of root datum. These groups are conjecturally linked with algebraic cycles and in particular we obtain a probabilistic algorithm to compute the dimension of the Hodge classes of our abelian variety for any fixed degree.

Let k be a field finitely generated over ℚ and p a prime. The torsion conjecture (resp. p-primary torsion conjecture) for abelian varieties over k predicts that the k-rational torsion (resp. the p-primary k-rational torsion) of a d-dimensional abelian variety A over k should be bounded only in terms of k and d. These conjectures are only known for d=1. The p-primary case was proved by Y. Manin, in 1969; the general case was completed by L. Merel, in 1996, after a series of contributions by B. Mazur, S. Kamienny and others. Due to the fact that moduli of elliptic curves are 1-dimensional, the d=1 case of the torsion conjecture (resp. p-primary torsion conjecture) is closely related to the following. For any k-curve S and elliptic scheme E→S, the k-rational torsion (resp. the p-primary k-rational torsion) is uniformly bounded in the fibres E s , s∈S(k). In this paper, we extend this result in the p-primary case to arbitrary abelian schemes over curves. More precisely, we prove the following. Denote by Γ k the absolute Galois group of k. For an abelian variety A over k and a character $\chi:\Gamma_{k}\rightarrow\mathbb{Z}_{p}^{\ast}$ , define A[p ∞](χ) to be the module of p-primary torsion of $A(\overline{k})$ on which Γ k acts as χ-multiplication. Assume that χ does not appear as a subrepresentation of the p-adic representation associated with an abelian variety over k. Then A[p ∞](χ) is always finite, but the exponent of A[p ∞](χ) may depend on A, a priori. Our main result is about the uniform boundedness of A[p ∞](χ) when A varies in a 1-dimensional family. More precisely, if S is a curve over k and A is an abelian scheme over S, then there exists an integer N:=N(A,S,k,p,χ), such that A s [p ∞](χ)⊂A s [p N ] holds for any s∈S(k). This arithmetic result is obtained as a corollary of the following geometric result on the p-primary torsion of abelian varieties over function fields of curves, combined with Mordell’s conjecture (Faltings’ theorem). Let K be the function field of a curve over an algebraically closed field of characteristic 0 and A an abelian variety over K. Assume for simplicity that A contains no nontrivial isotrivial subvariety. Then, for any c≥0, there exists an integer N:=N(c,A,S,k,p)≥0 such that A[p ∞](K′)⊂A[p N ] for all finite extension K′/K with K′ of genus ≤c. A key ingredient of the proof of this geometric result is a certain result on the number of points on reduction modulo p n of p-adic analytic homogeneous spaces. Our uniform boundedness result when χ is the trivial character gives the uniform boundedness for the k-rational p-primary torsion in the fibres A s , s∈S(k) alluded to above. When χ is the p-adic cyclotomic character, together with certain descent methods, it also yields a proof of the 1-dimensional case of (a generalized variant of) the modular tower conjecture, which was, actually, the original motivation for this work. This is a conjecture arising from the regular inverse Galois problem, whose original form was posed by M. Fried in the early 1990s.

It is well known that the Weyl groups play a key role in the theory of the structure of Lie algebras and their linear representations, as well as in the theory of Chevalley groups. Suppose g is a complex semisimple Lie algebra, h its Cartan subalgebra, Σ and Π denote its root system and the basis of the root system, respectively, and w α i indicates the reflection determined by αi ∈ Π. The so-called Weyl group, denoted by W,refers to the finite group generated by all w α i ( αi ∈ Π). There have been many works published on the structure and properties of W [6, 7]. The following Molien identity related to the global structure of W is well known [3]: $$\underset{w\in W}{\mathop{\Sigma}}\,\frac{1}{\det\left( 1-tw \right)}=\frac{1}{\prod{{{\left( 1-{{t}^{{{d}_{i}}}} \right)}^{,}}}}$$ (8.1) where d i signify the Poincare indices of g. In 1956, by topological methods based on Morse’s theory, Bott proved another identity [2]: $$\underset{w\in W}{\mathop \sum }\,{{t}^{l\left(w \right)}}=\prod{\frac{1-{{t}^{{{d}_{i}}}}}{1-{{t}^{,}}}}$$ (8.2) where l(w) refers to the Π-length of w;i.e.,wmay be expressed by the number of minimal products of w αi ( αi ∈ Π). This important formula may shed new light on our understanding the structure of W. In 1966, Solomon offered a purely algebraic proof of (8.2), thus solving the order problem of the Chevalley simple groups [5, 3].

Possible cause(s) of red leaf disease (RLD) of pineapple (Ananas comosus L.) in Ghana were examined through field observations, isolation of suspected organisms from roots of diseased plants, growing plants in potted steam-sterilised and unsterilised natural field soils. Others included detection of the pineapple closterovirus (PCV) from symptomatic and healthy pineapple leaves using tissue blot immunoassay (TBIA). Effects of soil moisture, soil fertility, sucker condition and light intensity on the disease were studied in pots while some factor combinations were studied in mini plots in the field. Diseased plants from the field had reduced root systems and had Neosartorya fischeri. Nematodes of the genera Aphelenchus, Pratylenchus and Helicotylenchus were recovered from roots and infrequently from the rind of the underground stem. Pineapple plants grown outdoors in steam-sterilised field soil reddened just as those grown in unsterilised soil. The PCV was detected in only 53% of all typically symptomatic leaves examined. All non-symptomatic leaves also tested positive for PCV. Plants grown from symptomatic suckers in a plant house with diffuse light intensity (7,440 lm m-2) and at 29 oC, recovered from RLD within 6 months. These plants, however, reddened when grown outdoors with intense light (39,751 lm m-2) at 30 oC. Significantly (P

A finite group W generated by reflections is called a finite reflection group and have been studied deeply in different important contexts such as geometry and group theory, matrix algebra and Lie Theory. In this paper we study a class of fuzzy subgroups of finite reflection groups which are called parabolic fuzzy subgroups using preferential equality. In the classical case parabolic subgroups are associated with a set of generators, each a simple reflection, indexed by elements of subsets of a fixed root system Δ . We establish a one-to-one correspondence between the class of fuzzy parabolic subgroups of W and the class of fuzzy subsets of S where S is a set of simple reflections associated with Δ .