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Computational Perspectives on Number Theory
This volume contains papers presented at the conference "Computational Prespectives on Number Theory" held at the University of Illinois at Chicago in honor of the retirement of A. O. L. Atkin. In keeping with Atkin's interests and work, the papers cover a range of topics, including algebraic number theory, p-adic modular forms and modular curves. Many of the paers reflect Atkin's particular interest in computational and algorithmic questions.
On the Existence of Feller Semigroups with Boundary Conditions
This paper is devoted to the functional analytic approach to the problem of construction of Feller semigroups with Ventcel' (Wentzell) boundary conditions. This paper considers the non-transversal case and solves from the viewpoint of functional analysis the problem of construction of Feller semigroups for elliptic Waldenfels operators. Intuitively, our result may be stated as follows: One can construct a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it "dies" at which time it reaches the set where the absorption phenomenon occurs.
Projective Modules over Lie Algebras of Cartan Type
This paper investigates the question of linkage and block theory for Lie algebras of Cartan type. The second part of the paper deals mainly with block structure and projective modules of Lies algebras of types W and K.
Vertex Algebras and Integral Bases for the Enveloping Algebras of Affine Lie Algebras
We present a new proof of the identities needed to exhibit an explicit [bold]Z-basis for the universal enveloping algebra associated to an affine Lie algebra. We then use the explicit [bold]Z-bases to extend Borcherds' description, via vertex operator representations, of a [bold]Z-form of the enveloping algebras for the simply-laced affine Lie algebras to the enveloping algebras associated to the unequal root length affine Lie algebras.
Arithmetic Geometry, Number Theory, and Computation
This volume contains articles related to the work of the Simons Collaboration “Arithmetic Geometry, Number Theory, and Computation.” The papers present mathematical results and algorithms necessary for the development of large-scale databases like the L-functions and Modular Forms Database (LMFDB). The authors aim to develop systematic tools for analyzing Diophantine properties of curves, surfaces, and abelian varieties over number fields and finite fields. The articles also explore examples important for future research. Specific topics include● algebraic varieties over finite fields● the Chabauty-Coleman method● modular forms● rational points on curves of small genus● S-unit equations and integral points.
Sum of Even Powers of Real Linear Forms
This work initiates a systematic analysis of the representation of real forms of even degree as sums of powers of linear forms and the resulting implications in real algebraic geometry, number theory, combinatorics, functional analysis, and numerical analysis. The proofs utilize elementary techniques from linear algebra, convexity, number theory, and real algebraic geometry and many explicit examples and relevant historical remarks are presented.
Advances in Cryptology – ASIACRYPT 2005
This book constitutes the refereed proceedings of the 11th International Conference on the Theory and Application of Cryptology and Information Security, ASIACRYPT 2005, held in Chennai, India in December 2005.The 37 revised full papers presented were carefully reviewed and selected from 237 submissions. The papers are organized in topical sections on algebra and number theory, multiparty computation, zero knowledge and secret sharing, information and quantum theory, privacy and anonymity, cryptanalytic techniques, stream cipher cryptanalysis, block ciphers and hash functions, bilinear maps, key agreement, provable security, and digital signatures.
Kernel Functions, Analytic Torsion, and Moduli Spaces
This memoir is a study of Ray-Singer analytic torsion for hermitian vector bundles on a compact Riemann surface [italic]C. The torsion is expressed through the trace of a modified resolvent. Thus, one can develop perturbation-curvature formulae for the Green-Szegö kernel and also for the torsion in terms of the Ahlfors-Bers complex structure of the Teichmuller space and Mumford complex structure of the moduli space of stable bundles of degree zero on [italic]C.
Selberg Trace Formulae and Equidistribution Theorems for Closed Geodesics and Laplace
This work is concerned with a pair of dual asymptotics problems on a finite-area hyperbolic surface. The first problem is to determine the distribution of closed geodesics in the unit tangent bundle. The second problem is to determine the distribution of eigenfunctions (in microlocal sense) in the unit tangent bundle.
The Subregular Germ of Orbital Integrals
An integral formula for the subregular germ of a [italic small capital]K-orbital integral is developed. The formula holds for any reductive group over a [italic]p-adic field of characteristic zero. This expression of the subregular germ is obtained by applying Igusa's theory of asymptotic expansions. The integral formula is applied to the question of the transfer of a [italic small capital]K-orbital integral to an endoscopic group. It is shown that the quadratic characters arising in the subregular germs are compatible with the transfer. Details of the transfer are given for the subregular germ of unitary groups.
