Contemporary Mathematics
Volume: 566;
2012;
344 pp;
Softcover
MSC: Primary 11; 14; 22; 32; 57;
Print ISBN: 978-0-8218-6900-0
Product Code: CONM/566
List Price: $108.00
AMS Member Price: $86.40
MAA Member Price: $97.20
Electronic ISBN: 978-0-8218-8776-9
Product Code: CONM/566.E
List Price: $108.00
AMS Member Price: $86.40
MAA Member Price: $97.20
Zeta Functions in Algebra and Geometry
Share this pageEdited by Antonio Campillo; Gabriel Cardona; Alejandro Melle-Hernández; Wim Veys; Wilson A. Zúñiga-Galindo
The volume contains the proceedings of the “Second
International Workshop on Zeta Functions in Algebra and Geometry” held
May 3–7, 2010 at the Universitat de les Illes Balears, Palma de Mallorca,
Spain.
Zeta functions can be naturally attached to several mathematical objects,
including fields, groups, and algebras. The conference focused on the following
topics: arithmetic and geometric aspects of local, topological, and motivic
zeta functions, Poincaré series of valuations, zeta functions of groups,
rings, and representations, prehomogeneous vector spaces and their zeta
functions, and height zeta functions.
This book is published in cooperation with Real Sociedad Matemática Española (RSME)
Readership
Graduate students and research mathematicians interested in zeta functions in number theory, geometry, and algebra.
Table of Contents
Zeta Functions in Algebra and Geometry
- Preface ix10 free
- protect oindent Part I:\$L$-functions of varieties over finite fields and Artin $L$-functions 118 free
- Computational aspects of Artin 𝐿-functions 320
- Introduction 320
- 1. Modular forms and Maass forms 421
- 2. Weight one holomorphic modular forms 623
- 3. Some character tables 724
- 4. Artin 𝐿-functions 825
- 5. Modular forms of weight one and Artin 𝐿-functions 1027
- 6. Linear and projective Galois representations 1229
- 7. Computation of Artin 𝐿-functions 1330
- References 1835
- Zeta functions for families of Calabi–Yau 𝑛-folds with singularities 2138
- Estimates for exponential sums with a large automorphism group 4360
- 1. Introduction 4360
- 2. Additive character sums for translation invariant polynomials 4562
- 3. Multiplicative character sums for translation invariant polynomials 5067
- 4. Additive character sums for homothety invariant polynomials 5471
- 5. Multiplicative character sums for homothety invariant polynomials 5673
- References 6077
- protect oindent Part II:\Height zeta functions\and arithmetic 6380
- Height zeta functions on generalized projective toric varieties 6582
- Combinatorial cubic surfaces and reconstruction theorems 99116
- Height zeta functions of equivariant compactifications of semi-direct products of algebraic groups 119136
- protect oindent Part III:\Motivic zeta functions, Poincar'e series, complex monodromy\and knots 159176
- Singularity invariants related to Milnor fibers: survey 161178
- Finite families of plane valuations: value semigroup, graded algebra and Poincaré series 189206
- 1. Introduction 189206
- 2. Valuations 190207
- 3. Graded algebra of a valuation 193210
- 4. Plane valuations 195212
- 5. Poincaré series of the graded algebra 199216
- 6. Graded algebra and Poincaré series of finite families valuations 201218
- 7. An application: Poincaré series of multiplier ideals of a plane divisorial valuation 208225
- References 210227
- 𝑞,𝑡-Catalan numbers and knot homology 213230
- Motivic zeta functions for degenerations of abelian varieties and Calabi-Yau varieties 233250
- The lattice cohomology of 𝑆³_{-𝑑}(𝐾) 261278
- 1. Introduction 261278
- 2. The 3–manifold 𝑆³_{-𝑑}(𝐾) 262279
- 3. Review of lattice cohomologies 266283
- 4. The lattice reduction 269286
- 5. The lattice reduction for 𝑆³_{-𝑑}(𝐾) 272289
- 6. The lattice cohomology of 𝑆³_{-𝑑}(𝐾) via the semigroups Γ_{ℓ} 276293
- 7. The normalized Euler–characteristic and the Seiberg–Witten invariant 281298
- 8. Connection with Heegaard–Floer homology 284301
- 9. Example. The case of 𝜈 cusps of type (2,3) 286303
- 10. Another example 290307
- References 291308
- protect oindent Part IV:\Zeta functions for groups\and representations 293310
- Representation zeta functions of some compact 𝑝-adic analytic groups 295312
- 1. Introduction 295312
- 2. Zeta functions as 𝔭-adic integrals 299316
- 3. Explicit formulae for 𝑆𝐿₂(𝔬) and its principal congruence subgroups 304321
- 4. Explicit formulae for subgroups of quaternion groups 𝑆𝐿₁(𝔇) 309326
- 5. Principal congruence subgroups of 𝑆𝐿₃(𝔬) for unramified 𝔬 of residue field characteristic 3 314331
- Appendix A. Adjoint action of 𝐺𝐿₂(𝔽_{𝕢}) on 𝔰𝔩₂(𝔽_{𝕢}) 326343
- Appendix B. Adjoint action of 𝐺𝐿₃(𝔽_{𝕢}) on 𝔰𝔩₃(𝔽_{𝕢}) 326343
- Appendix C. Auxiliary results regarding 𝔰𝔩₃(𝔽_{𝕢}) and 𝔤𝔩₃(𝔽_{𝕢}) for 𝕢≡₃0 327344
- References 329346
- Applications of some zeta functions in group theory 331348