2,806 research outputs found
On a theory of the -function in positive characteristic
We present a theory of the -function (or Bernstein-Sato polynomial) in
positive characteristic. Let be a non-constant polynomial with coefficients
in a perfect field of characteristic Its -function is
defined to be an ideal of the algebra of continuous -valued functions on
The zero-locus of the -function is thus naturally
interpreted as a subset of which we call the set of roots of
We prove that has finitely many roots and that they are negative
rational numbers. Our construction builds on an earlier work of Musta\c{t}\u{a}
and is in terms of -modules, where is the ring of Grothendieck
differential operators. We use the Frobenius to obtain finiteness properties of
and relate it to the test ideals of Comment: Final versio
Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers
The space of unitary local systems of rank one on the complement of an
arbitrary divisor in a complex projective algebraic variety can be described in
terms of parabolic line bundles. We show that multiplier ideals provide natural
stratifications of this space. We prove a structure theorem for these
stratifications in terms of complex tori and convex rational polytopes,
generalizing to the quasi-projective case results of Green-Lazarsfeld and
Simpson. As an application we show the polynomial periodicity of Hodge numbers
of congruence covers in any dimension, generalizing results of E. Hironaka and
Sakuma. We extend the structure theorem and polynomial periodicity to the
setting of cohomology of unitary local systems. In particular, we obtain a
generalization of the polynomial periodicity of Betti numbers of unbranched
congruence covers due to Sarnak-Adams. We derive a geometric characterization
of finite abelian covers, which recovers the classic one and the one of
Pardini. We use this, for example, to prove a conjecture of Libgober about
Hodge numbers of abelian covers.Comment: final version, to appear in Adv. Mat
Bernstein-Sato polynomials in positive characteristic
In characteristic zero, the Bernstein-Sato polynomial of a hypersurface can
be described as the minimal polynomial of the action of an Euler operator on a
suitable D-module. We consider the analogous D-module in positive
characteristic, and use it to define a sequence of Bernstein-Sato polynomials
(corresponding to the fact that we need to consider also divided powers Euler
operators). We show that the information contained in these polynomials is
equivalent to that given by the F-jumping exponents of the hypersurface, in the
sense of Hara and Yoshida.Comment: 26 pages; v.2: new section added, treating the decomposition of an
arbitrary D-module under the Euler operators; v.3: final version, to appear
in Journal of Algebr
Statistical Mechanics of 2+1 Gravity From Riemann Zeta Function and Alexander Polynomial:Exact Results
In the recent publication (Journal of Geometry and Physics,33(2000)23-102) we
demonstrated that dynamics of 2+1 gravity can be described in terms of train
tracks. Train tracks were introduced by Thurston in connection with description
of dynamics of surface automorphisms. In this work we provide an example of
utilization of general formalism developed earlier. The complete exact solution
of the model problem describing equilibrium dynamics of train tracks on the
punctured torus is obtained. Being guided by similarities between the dynamics
of 2d liquid crystals and 2+1 gravity the partition function for gravity is
mapped into that for the Farey spin chain. The Farey spin chain partition
function, fortunately, is known exactly and has been thoroughly investigated
recently. Accordingly, the transition between the pseudo-Anosov and the
periodic dynamic regime (in Thurston's terminology) in the case of gravity is
being reinterpreted in terms of phase transitions in the Farey spin chain whose
partition function is just a ratio of two Riemann zeta functions. The mapping
into the spin chain is facilitated by recognition of a special role of the
Alexander polynomial for knots/links in study of dynamics of self
homeomorphisms of surfaces. At the end of paper, using some facts from the
theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we
develop systematic extension of the obtained results to noncompact Riemannian
surfaces of higher genus. Some of the obtained results are also useful for 3+1
gravity. In particular, using the theorem of Margulis, we provide new reasons
for the black hole existence in the Universe: black holes make our Universe
arithmetic. That is the discrete Lie groups of motion are arithmetic.Comment: 69 pages,11 figures. Journal of Geometry and Physics (in press
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