Noncommutative geometry and motives: The thermodynamics of endomotives

We combine aspects of the theory of motives in algebraic geometry with noncommutative geometry and the classification of factors to obtain a cohomological interpretation of the spectral realization of zeros of L-functions. The analogue in characteristic zero of the action of the Frobenius on ?-adic cohomology is the action of the scaling group on the cyclic homology of the cokernel (in a suitable category of motives) of a restriction map of noncommutative spaces. The latter is obtained through the thermodynamics of the quantum statistical system associated to an endomotive (a noncommutative generalization of Artin motives). Semigroups of endomorphisms of algebraic varieties give rise canonically to such endomotives, with an action of the absolute Galois group. The semigroup of endomorphisms of the multiplicative group yields the Bost-Connes system, from which one obtains, through the above procedure, the desired cohomological interpretation of the zeros of the Riemann zeta function. In the last section we also give a Lefschetz formula for the archimedean local L-factors of arithmetic varieties.     

The construction of numerically Calabi–Yau orders on projective surfaces

In this paper, we construct a vast collection of maximal numerically Calabi–Yau orders utilising a noncommutative analogue of the well-known commutative cyclic covering trick. Such orders play an integral role in the Mori program for orders on projective surfaces and although we know a substantial amount about them, there are relatively few known examples.     

Wild tame-by-cyclic extensions

Suppose G is a semi-direct product of the form Z/pⁿ?Z/m where p is prime and m is relatively prime to p. Suppose K is a complete discrete valuation field of characteristic p>0 with algebraically closed residue field. The main result states necessary and sufficient conditions on the ramification filtrations that occur for wildly ramified G-Galois extensions of K. In addition, we prove that there exists a parameter space for G-Galois extensions of K with given ramification filtration, and we calculate its dimension in terms of the ramification filtration. We provide explicit equations for wild cyclic extensions of K of degree p³.     

Generic noncommutative surfaces

We study a class of noncommutative surfaces, and their higher dimensional analogs, which come from generic subalgebras of twisted homogeneous coordinate rings of projective space. Such rings provide answers to several open questions in noncommutative projective geometry. Specifically, these rings R are the first known graded algebras over a field k which are noetherian but not strongly noetherian: in other words, R⊗_kB is not noetherian for some choice of commutative noetherian extension ring B. This answers a question of Artin, Small, and Zhang. The rings R are also maximal orders, but they do not satisfy all of the χ conditions of Artin and Zhang. In particular, they satisfy the χ₁ condition but not χ_i for i?2, answering a question of Stafford and Zhang and a question of Stafford and Van den Bergh. Finally, we show that the noncommutative scheme R-proj has finite global dimension.     

On the invertibility of ideals in orders

The problem of invertibility of ideals in orders has been studied by a number of authors. The commutative case has been considered by Dade, Taussky, and Zassenhaus; Frolich; and Singer. Ballew gives a generalization of Frolich's results to a class of noncommutative orders. We examine some of the possible extensions of the results of Dade et al. to noncommutative orders.     

Maximal ergodic theorems for some group actions

We prove maximal ergodic inequalities for a sequence of operators and for their averages in the noncommutative Lp-space. We also obtain the corresponding individual ergodic theorems. Applying these results to actions of a free group on a von Neumann algebra, we get noncommutative analogues of maximal ergodic inequalities and pointwise ergodic theorems of Nevo-Stein.     

The Rank of Abelian Varieties over Infinite Galois Extensions

G. Frey and M. Jarden (1974, Proc. London Math. Soc.28, 112-128) asked if every Abelian variety A defined over a number field k with dim A>0 has infinite rank over the maximal Abelian extension k^ab of k. We verify this for the Jacobians of cyclic covers of P¹, with no hypothesis on the Weierstrass points or on the base field. We also derive an infinite rank criterion by analyzing the ramification of division points of an Abelian variety. As an application, we show that any d -dimensional Abelian variety A over k with a degree n projective embedding over k has infinite rank over the compositum of all extensions of k of degree <n(4d+2).     

Noncommutative classical invariant theory

In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspace? _d ^m of the noncommutative invariant algebra? _d consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group S _m . We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(? _d,t) whose coefficients are the number of irreducibleS _m -modules in the decomposition of? _d ^m , and show that it is rational. Finally, we develop some analogues of all this for covariants.     

Noncommutative residue for Heisenberg manifolds. Applications in CR and contact geometry

This paper has four main parts. In the first part, we construct a noncommutative residue for the hypoelliptic calculus on Heisenberg manifolds, that is, for the class of ΨHDO operators introduced by Beals-Greiner and Taylor. This noncommutative residue appears as the residual trace on integer order ΨHDOs induced by the analytic extension of the usual trace to non-integer order ΨHDOs. Moreover, it agrees with the integral of the density defined by the logarithmic singularity of the Schwartz kernel of the corresponding ΨHDO. In addition, we show that this noncommutative residue provides us with the unique trace up to constant multiple on the algebra of integer order ΨHDOs. In the second part, we give some analytic applications of this construction concerning zeta functions of hypoelliptic operators, logarithmic metric estimates for Green kernels of hypoelliptic operators, and the extension of the Dixmier trace to the whole algebra of integer order ΨHDOs. In the third part, we present examples of computations of noncommutative residues of some powers of the horizontal sublaplacian and the contact Laplacian on contact manifolds. In the fourth part, we present two applications in CR geometry. First, we give some examples of geometric computations of noncommutative residues of some powers of the horizontal sublaplacian and of the Kohn Laplacian. Second, we make use of the framework of noncommutative geometry and of our noncommutative residue to define lower-dimensional volumes in pseudohermitian geometry, e.g., we can give sense to the area of any 3-dimensional CR manifold endowed with a pseudohermitian structure. On the way we obtain a spectral interpretation of the Einstein-Hilbert action in pseudohermitian geometry.     

Fiercely ramified cyclic extensions of p-adic fields with imperfect residue field

We study the ramification of fierce cyclic Galois extensions of a local field K of characteristic zero with a one-dimensional residue field of characteristic p > 0. Using Kato’s theory of the refined Swan conductor, we associate to such an extension a ramification datum, consisting of a sequence of pairs (δ _i , ω _i ), where δ _i is a positive rational number and ω _i a differential form on the residue field of K. Our main result gives necessary and sufficient conditions on such sequences to occur as a ramification datum of a fierce cyclic extension of K.     

Noncommutative correction to the Cornell potential in heavy-quarkonium atoms

We investigate the effect of space–time noncommutativity on the Cornell potential in heavy-quarkonium systems. It is known that the space–time noncommutativity can create bound states, and we therefore consider the noncommutative geometry of the space–time as a correction in quarkonium models. Furthermore, we take the experimental hyperfine measurements of the bottomium ground state as an upper limit on the noncommutative energy correction and derive the maximum possible value of the noncommutative parameter θ, obtaining θ ≤ 37.94 · 10^?34 m². Finally, we use our model to calculate the maximum value of the noncommutative energy correction for energy levels of charmonium and bottomium in 1S and 2S levels. The energy correction as a binding effect in quarkonium system is smaller for charmonium than for bottomium, as expected.     

On the semicontinuity of ramification invariants in dimension 2

We consider a cyclic extension L/K of degree 2ⁿ of the field K = k[[T,U]] of characteristic 2. It is shown that for all sufficiently large N, the jets of order N of all curves which are not components of the ramification locus and for which the corresponding valuation of the function field has a unique extension, the valuations of the coefficients of the Inaba equation are positive, and the ramification jumps are maximal, is an open set. In the case of a general (not cyclic) extension, it is shown that the set of jets with a fixed value of the kth jump is the intersection of an open and a closed set. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 13–35.     

Monodromy of cyclic coverings of the projective line

We show that the image of the pure braid group under the monodromy action on the homology of a cyclic covering of degree d of the projective line is an arithmetic group provided the number of ramification points is sufficiently large compared to the degree d and the ramification degrees are co-prime to d.     

A new family of noncommutative 2-spheres

We give new examples of noncommutative manifolds. In particular we construct a “strong” deformation of C(S²), consisting of a family of noncommutative 2-spheres, and study their analytic and topological properties.     

Maximal Quadratic Modules on ∗-rings

We generalize the notion of and results on maximal proper quadratic modules from commutative unital rings to ?-rings and discuss the relation of this generalization to recent developments in noncommutative real algebraic geometry. The simplest example of a maximal proper quadratic module is the cone of all positive semidefinite complex matrices of a fixed dimension. We show that the support of a maximal proper quadratic module is the symmetric part of a prime ?-ideal, that every maximal proper quadratic module in a Noetherian ?-ring comes from a maximal proper quadratic module in a simple artinian ring with involution and that maximal proper quadratic modules satisfy an intersection theorem. As an application we obtain the following extension of Schmüdgen’s Strict Positivstellensatz for the Weyl algebra: Let c be an element of the Weyl algebra \(\mathcal{W}(d)\) which is not negative semidefinite in the Schrödinger representation. It is shown that under some conditions there exists an integer k and elements \(r_1,\ldots,r_k \in \mathcal{W}(d)\) such that ∑ _j=1 ^k r _j c r _j ^? is a finite sum of hermitian squares. This result is not a proper generalization however because we don’t have the bound k ≤d.     

Faddeev invariants for central simple algebras over rational function fields

Let p be a prime, k a field, containing a primitive pth root of unity, char k ≠ p. We give an upper bound for the Faddeev index of a central simple algebra of exponent p over the rational function field k(t) in the case where the ramification set of the algebra consists of rational points. This bound depends only on the number of ramification points and in certain cases turns out to be strict. In the case where p =  2 and the ramification set in \({\mathbb{A}_k^1}\) consists of three rational points we compute the Faddeev index, using the language of quadratic forms. Let X be a smooth geometrically irreducible complete curve over k. We show that there exist algebras of exponent p over k(X) with the prescribed Faddeev index, provided there are algebras of exponent p and arbitrarily large index over k. In the last section of the paper we consider another invariant of a central simple algebra of prime exponent p over k(t), the so called Faddeev cyclic length. In certain cases we compute this invariant, using triviality of the divided power operations on central simple cyclic algebras of exponent p.     

Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Let M be aσ-finite von Neumann algebra and let AM be a maximal subdiagonal algebra with respect to a faithful normal conditional expectationΦ.Based on the Haagerup’s noncommutative Lpspace Lp(M)associated with M,we consider Toeplitz operators and the Hilbert transform associated with A.We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space H2(M)is just the right analytic Toeplitz algebra.Furthermore,the Hilbert transform on noncommutative Lp(M)is shown to be bounded for 1p∞.As an application,we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative H1(M)as a concrete space of operators.     

Analogue of the Hyodo Inequality for the Ramification Depth in Degree <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript> Extensions

Ramification in complete discrete valuation fields is studied. For the case of a perfect residue field, there is a well-developed theory of ramification groups. Hyodo introduced the concept of ramification depth associated with the different of an extension and obtained an inequality that combines the concept of ramification depth in a degree p² cyclotomic extension with the concept of ramification depth in a degree p subextension. The paper gives a detailed consideration of the structure of degree p² extensions that can be obtained by a composite of two degree p extensions. In this case, it is not required that the residue field be perfect. Using the concepts of wild and ferocious extensions and the defect of the main unit, degree p² extensions are classified and more accurate estimates for the ramification depth are obtained. In a number of cases, exact formulas for ramification depth are presented.