Recent Preprint Research Articles

Galois representations attached to Hilbert–Siegel modular forms

This article is a spinoff of the book of Harris and Taylor [HT], in which they prove the local Langlands conjecture for \operatorname{GL}(n) , and its companion paper by Taylor and Yoshida [TY] on local-global compatibility. We record some consequences in the case of genus two Hilbert–Siegel modular forms. In other words, we are concerned with cusp forms \pi on \operatorname{GSp}(4) over a totally real field, such that \pi_{\infty} is regular algebraic (that is, \pi is cohomological). When \pi is globally generic (that is, has a non-vanishing Fourier coefficient), and \pi has a Steinberg component at some finite place, we associate a Galois representation compatible with the local Langlands correspondence for \operatorname{GSp}(4) defined by Gan and Takeda in a recent preprint [GT]. Over \mathbb Q , for \pi as above, this leads to a new realization of the Galois representations studied previously by Laumon, Taylor and Weissauer. We are hopeful that our approach should apply more generally, once the functorial lift to \operatorname{GL}(4) is understood, and once the so-called book project is completed. An application of the above compatibility is the following special case of a conjecture stated in [SU]: If \pi has nonzero vectors fixed by a non-special maximal compact subgroup at v , the corresponding monodromy operator at v has rank at most one.

On [formula omitted]-factorial terminalizations of nilpotent orbits

In a recent preprint, Y. Namikawa proposed a conjecture on Q -factorial terminalizations of nilpotent orbit closures and he proved his conjecture for classical simple Lie algebras. In this paper, we prove his conjecture for exceptional simple Lie algebras. For the birational geometry, contrary to the classical case, two new types of Mukai flops appear. We also give classifications of nilpotent orbit closures in simple Lie algebras whose normalization has only Q -factorial or terminal singularities.

ON WEIL NUMBERS IN CYCLOTOMIC FIELDS

In this paper, we study the p-adic behavior of Weil numbers in the cyclotomic ℤp-extension of the pth cyclotomic field. We determine the characteristic ideal of the quotient of semi-local units by Weil numbers in terms of the characteristic ideals of some classical modules that appear in the Iwasawa theory. In a recent preprint [9] by Nguyen Quang Do and Nicolas, a generalization of this result to a semi-simple case was obtained.

Comment on “Ergodicity and central-limit theorem in systems with long-range interactions” by Figueiredo A. et al.

In the present paper we refute the criticism advanced in a recent preprint by Figueiredo et al [1] about the possible application of the $q$-generalized Central Limit Theorem (CLT) to a paradigmatic long-range-interacting many-body classical Hamiltonian system, the so-called Hamiltonian Mean Field (HMF) model. We exhibit that, contrary to what is claimed by these authors and in accordance with our previous results, $q$-Gaussian-like curves are possible and real attractors for a certain class of initial conditions, namely the one which produces nontrivial longstanding quasi-stationary states before the arrival, only for finite size, to the thermal equilibrium.

The ℓ1 -norm of the Fourier transform on compact vector spaces

Suppose that G is a compact Abelian group. If A ⊂ G, then how small can ||χA||A(G) be? In general, there is no non-trivial lower bound. In a recent preprint, Green and Konyagin show that if G ^ has sparse small subgroup structure and A has density α with α(1 − α) ≫ 1, then ||χA||A(G) does admit a non-trivial lower bound. In this paper we address the complementary case of groups with duals having rich small subgroup structure, specifically the case when G is a compact vector space over 𝔽2. The results themselves are rather technical to state, but the following consequence captures their essence: if A ⊂ F 2 n is a set of density as close to 1/3 as possible, then we show that ‖ χ A ‖ A ( F 2 n ) ≫ log n . We include a number of examples and conjectures which suggest that what we have shown is very far from a complete picture.

A propos de la propriété de Bergman

In a recent preprint, George M. Bergman has investigated the following property: for any generating set X of the group G there exists an integer n such that any element of G is a product of n elements of X ∪ X −1 . We will say in this case that G has the Bergman property. We have solved some of the questions asked in the above mentioned preprint and have found it suitable to investigate this property in a more general context, in particular for rings (essentially Boolean rings). To cite this article: A. Khelif, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Complete intersection singularities of splice type as universal abelian covers

It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "splice type singularities", which generalize Brieskorn complete intersections. Second, these arise as universal abelian covers of a class of normal surface singularities with Q-homology sphere links, called "splice-quotient singularities". According to the Main Theorem, splice-quotients realize a large portion of the possible topologies of singularities with Q-homology sphere links. As quotients of complete intersections, they are necessarily Q-Gorenstein, and many Q-Gorenstein singularities with Q-homology sphere links are of this type. We conjecture that rational singularities and minimally elliptic singularities with Q-homology sphere links are splice-quotients. A recent preprint of T Okuma presents confirmation of this conjecture.

P-ADIC PROPERTIES FOR TRACES OF SINGULAR MODULI

We examine the p-adic properties of Zagier's traces Tr (d) of the singular moduli of discriminant -d. In a recent preprint, Edixhoven proved that if p is prime and [Formula: see text], then [Formula: see text] We compute an exact formula for Tr (p2nd) which immediately gives Edixhoven's result as a corollary.

A Positive Proof of the Littlewood-Richardson Rule using the Octahedron Recurrence

We define the hive ring, which has a basis indexed by dominant weights for $GL_n({\Bbb C})$, and structure constants given by counting hives [Knutson-Tao, "The honeycomb model of $GL_n$ tensor products"] (or equivalently honeycombs, or BZ patterns [Berenstein-Zelevinsky, "Involutions on Gel$'$fand-Tsetlin schemes$\dots$ "]). We use the octahedron rule from [Robbins-Rumsey, "Determinants$\dots$"] to prove bijectively that this "ring" is indeed associative. This, and the Pieri rule, give a self-contained proof that the hive ring is isomorphic as a ring-with-basis to the representation ring of $GL_n({\Bbb C})$. In the honeycomb interpretation, the octahedron rule becomes "scattering" of the honeycombs. This recovers some of the "crosses and wrenches" diagrams from Speyer's very recent preprint ["Perfect matchings$\dots$"], whose results we use to give a closed form for the associativity bijection.

Note on the Curvature and Index of an Almost Unitary Contraction Operator

In the recent preprint [1] S. Parrott proves the equality between the Arveson’s curvature and the Fredholm index of a “pure” contraction with finite defect numbers. In the present note one derives a similar formula in the ”non-pure” case.

Probing human response times

In a recent preprint (Dialog in e-mail traffic, preprint cond-mat/0304433), the temporal dynamics of an e-mail network has been investigated by Eckmann, Moses and Sergi. Specifically, the time period between an e-mail message and its reply were recorded. It will be shown here that their data agrees quantitatively with the frame work proposed to explain a recent experiment on the response of “internauts” to a news publication (Physica A 296(3–4) (2001) 539) despite differences in communication channels, topics, time-scale and socio-economic characteristics of the two population. This suggest a generalized response time distribution ∼ t −1 for human populations in the absence of deadlines with important implications for psychological and social studies as well the study of dynamical networks.

The disaster of central control

I t is commonplace to say that central control of large complex entities is doomed to fail. We have numerous examples from economies and societies of the past and the present demonstrating inefficiency when the systems exceed a certain size. In the old days, wise emperors were well aware of the problem and answered appropriately by the principle of divide et impera. Nature seems to have an elegant solution the problem too: Modular structure and partial autonomy of modules. The best example is the multicellular organism where the individual cell retains as much autonomy in metabolism as can be tolerated without endangering the whole system: A little more independence of somatic cells, for example, leads to tumor formation. Efficient division of labor is observed in bacterial cells too. No wonder I have thought that we can always learn from biology how to manage successfully the most sophisticated situations and how to handle and control complexity. It was a shocking experience therefore when I read a recent preprint and previous articles by John Mattick and his colleagues [1–3]. They present a plausible interpretation of the limitation in bacterial genome sizes based on DNA sequences: The number of genes in prokaryotes is bounded by an unaffordable regulatory overhead in genomes that are too large. This view is supported by an empirical fit of a power law to the data derived from some 90 fully sequenced prokaryotic genomes. The number of regulatory genes grows approximately with the square of the total number of genes, nr 1.63 10 5 n 1.2 10 5 n, where nr and n are the numbers of regulatory genes and all genes, respectively. Thus, complete regulation of ribosomal protein synthesis—when centrally organized on the DNA level in the spirit of the elegant operon mechanism as discovered by Jaques Monod, Francois Jacob, and Andre Lwow—falls into an inefficiency trap when genomes become too large. The guess is that the critical genome size for prokaryotes is in the range of 10,000 genes and indeed, this appears to be the size limit of bacterial genomes. Eukaryotes—these are all higher organisms from yeast to man—make use of other control mechanism in addition to genetic control at the DNA level. Examples are alternative splicing of precursormessenger RNAs (for reviews see [4, 5]), small interfering RNAs [6], genomic imprinting [7, 8], and other forms of epigenetic regulation of gene expression and silencing. For the purpose of illustration we shall consider here an idealized—and perhaps not yet fully accepted—model view of alternative splicing: The translation product of the same DNA stretch yields slightly or substantially different proteins depending on the individual organism and the particular tissue in which it is expressed. Diverse proteins are obtained by cutting out different sections called introns from the precursor RNA sequence. A well-known example is the protein that controls sex PETER SCHUSTER

ℐ–invariant random strict total orderings on ℤ

AbstractIn a recent preprint ([4]) Saul Jacka and Jon Warren develop further the ideas of Paul Ressel and myself [3], though they work without positive definite functions on semigroups. I approach the results of [4] on ℐ–invariant random strictly total orderings by semigroup methods as in [3]. These orderings are most directly related to card shuffling, cf. the rich work of Persi Diaconis. Although more concrete applications are not yet known to me, the related papers [3], [2] and [1] definitively do have a background in population genetics, so the present work on ℐ–invariant random orderings is certainly not far from real–life applications.

A deterministic (2−2/( k+1)) n algorithm for k-SAT based on local search

Local search is widely used for solving the propositional satisfiability problem. Papadimitriou (1991) showed that randomized local search solves 2-SAT in polynomial time. Recently, Schöning (1999) proved that a close algorithm for k-SAT takes time (2−2/ k) n up to a polynomial factor. This is the best known worst-case upper bound for randomized 3-SAT algorithms (cf. also recent preprint by Schuler et al.). We describe a deterministic local search algorithm for k-SAT running in time (2−2/( k+1)) n up to a polynomial factor. The key point of our algorithm is the use of covering codes instead of random choice of initial assignments. Compared to other “weakly exponential” algorithms, our algorithm is technically quite simple. We also describe an improved version of local search. For 3-SAT the improved algorithm runs in time 1.481 n up to a polynomial factor. Our bounds are better than all previous bounds for deterministic k-SAT algorithms.

On the combinatorics of the graph-complex

In their recent preprint [3] Kontsevich and Shoikhet have introduced two graph-complexes: the complex on the even (resp. odd) space in order to study the cohomology of the Lie algebra Ham 0 (resp. Ham 0 odd) of Hamiltonian vector fields vanishing at the origin on the infinite-dimensional even (resp. odd) space. We construct an isomorphism between those two graph-complexes, proving in particular that their cohomologies coincide. This solves a problem posed by Shoikhet. To cite this article: B. Lass, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1–6

A parabolic Pommerenke–Levin–Yoccoz inequality

In a recent preprint [B], Bergweiler relates the number of critical points contained in the immediate basin of a multiple fixed point $\beta $ of a rational map $f: {\mathbb P}^1\to {\mathbb P}^1$, the number ${N}$ of attracting petals and the residue $\i

Virtual operad algebras and realization of homotopy types

We prove that the category of algebras over a cofibrant operad admits a closed model category structure. This leads to the notion of “virtual operad algebra” – the algebra over a cofibrant resolution of the given operad. In particular, virtual commutative algebras can serve to an algebraic description of homotopy p-types as in the recent preprint of Mandell (M. Mandell, E ∞-algebras and p-adic homotopy theory, Hopf preprint server, October, 1998). Our main result allows one to simplify the proof of Mandell's theorem.

Comment on “Spin Transport Properties of the Quantum One-Dimensional Non-Linear Sigma Model”

In a recent preprint (cond-mat/9905415), Fujimoto has used the Bethe ansatz to compute the finite temperature, zero frequency Drude weight of spin transport in the quantum O(3) non-linear sigma model in a magnetic field $H \neq 0$. We show here that, contrary to his claims, the results are in accord with earlier semiclassical results (Sachdev and Damle, cond-mat/9610115). We also comment on his 1/N expansion, and show that it does not properly describe the long-time correlations.

The Stellar Content of Local Group Galaxies: IAU Symposium 19211Conference was held in Cape Town, South Africa, in 1998 September. Proceedings will be edited by P. A. Whitelock and R. D. Cannon and published in the ASP Conference Series.

IAU Symposium 192 was both broader and narrower than its title would suggest. As will be discussed below, this meeting covered many aspects of the Local Group other than stellar content. On the other hand, discussion of the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC) was cut back, since a previous IAU Symposium (190) on the Magellanic Clouds had been held only 2 months earlier in Victoria, Canada. Local Group membership.—A substantial portion of the conference was devoted to the fundamental question of Local Group (LG) membership, with review papers by both E. K. Grebel and T. E. Armandroff touching on this subject. In the last year the number of known companions to M31 has doubled, thanks to work by Armandroff, I. D. Karachentsev, and collaborators. The Armandroff search is based on a matched filter applied to the Digital Sky Survey and has produced many more candidates that are still being investigated. However, all the new LG objects to date are brighter than , M 10 v suggesting that many other fainter galaxies are lurking, waiting to be discovered. (Contrast this work on M31 companions with a search by M. Irwin and collaborators, who have turned up only one new object after searching two-thirds of the sky.) On several occasions it was remarked that NGC 55 (which is usually considered to be a member of the nearby Sculptor Group) lies on the zero velocity surface of the Local Group and may actually be considered a member of our Group. (A recent preprint by H. Jerjen et al. [astro-ph/9809046] suggests that it may be more appropriate to consider the Local and Sculptor Groups as a single supergalactic structure.) A related question is that of the luminosity function of the Local Group. If the inventory of LG galaxies is not complete below 10, then one cannot rule out the possibility of a steep upturn in the LG luminosity function (as observed in some other samples of galaxy populations). Morphology.—Sidney Van den Bergh highlighted the absence of large dwarf spheroidals (such as those in the Virgo cluster or the one known in the M81 Group). K. Freeman pointed out the difference in the structure of bulges in earlytype systems (e.g., M31, the Sombrero galaxy, etc.) that possess

Kinematical Hilbert spaces for fermionic and Higgs quantum field theories

We extend the recently developed kinematical framework for diffeomorphism-invariant theories of connections for compact gauge groups to the case of a diffeomorphism-invariant quantum field theory which includes, besides connections, also fermions and Higgs fields. This framework is appropriate for coupling matter to quantum gravity. The presence of diffeomorphism invariance forces us to choose a representation which is a rather non-Fock-like one: the elementary excitations of the connection are along open or closed strings, while those of the fermions or Higgs fields are at the end points of the string. Nevertheless we are able to promote the classical reality conditions to quantum adjointness relations which, in turn, uniquely fixes the gauge- and diffeomorphism-invariant probability measure that underlies the Hilbert space. Most of the fermionic part of this work is independent of the recent preprint by Baez and Krasnov and earlier work by Rovelli and Morales-Tecótl because we use new canonical fermionic variables, so-called Grassman-valued half-densities, which enable us to solve the difficult fermionic adjointness relations.