International Congress of Mathematicians 2022 July 6–14 Proceedings: Sections 1-4

Front cover
Front matter
Contents
1. Logic
G. Binyamini and D. Novikov: Tameness in geometry and arithmetic: beyond o-minimality
1. Tame geometry and arithmetic
1.1. O-minimal structures
1.2. Pila–Wilkie counting theorem
1.3. Transcendence methods, auxiliary polynomials
1.4. Beyond Pila–Wilkie theorem: the Wilkie conjecture
2. Sharply o-minimal structures
2.1. Examples and nonexamples
2.1.1. The semialgebraic structure
2.1.2. The analytic structure R_{an}
2.1.3. Pfaffian structures
2.2. Cell decomposition in sharply o-minimal structures
2.3. Yomdin–Gromov algebraic lemma in sharply o-minimal structures
2.4. Pila–Wilkie theorem in sharply o-minimal structures
2.5. Polylog counting in sharply o-minimal structures
3. Complex analytic theory
3.1. Complex cells
3.1.1. Basic fibers and their extensions
3.1.2. The definition of a complex cell
3.1.3. The real setting
3.2. Cellular parametrization
3.3. Analytically generated structures
3.4. Complex cells, hyperbolic geometry, and preparation theorems
3.5. Unrestricted exponentials
4. Sharply o-minimal structures arising from geometry
4.1. Abel–Jacobi maps
4.2. Uniformizing maps of abelian varieties
4.3. Noetherian functions
4.4. Bezout-type theorems and point counting with foliations
4.5. Q-functions
5. Applications in arithmetic geometry
5.1. Geometry governs arithmetic
5.2. The Pila–Zannier strategy
5.3. Point counting and Galois lower bounds
5.4. Effectivity and polynomial time computability
References
N. Dobrinen: Ramsey theory of homogeneous structures: current trends and open problems
1. Introduction
2. The questions
3. Case study: the rationals
4. Historical highlights, recent results, and methods
5. Exact big Ramsey degrees
5.1. Exact big Ramsey degrees with a simple characterization
5.2. Big Ramsey degrees for free amalgamation classes
6. Open problems and related directions
References
A. S. Marks: Measurable graph combinatorics
1. Introduction
2. Measurable colorings
3. Connections with hyperfiniteness
4. Measurable equidecompositions
References
K. Yokoyama: Paris–Harrington principle in second-order arithmetic
1. Introduction
2. First- and second-order arithmetic and the Ramsey theorem
2.1. The Paris–Harrington principle in first-order arithmetic
2.2. Second-order arithmetic and the infinite Ramsey theorem
3. The Paris–Harrington principle in second-order arithmetic
3.1. Second-order formulations of PH
3.2. PH and the notion of α-largeness
4. Generalizations of \mathrm{PH}
4.1. Phase transition
4.2. PH with generalized largeness
4.3. Iterations of generalized PH and correctness statements
5. Indicators and correctness statements
5.1. Models of first- and second-order arithmetic
5.2. Indicators
5.3. Indicators corresponding to largeness notions
References
D. Zhuk: Constraint Satisfaction Problem: what makes the problem easy
1. Introduction
2. Constraint Satisfaction Problem
2.1. Examples
2.2. Reduction from one language to another
2.3. Polymorphisms as invariants
2.4. Local consistency
2.5. CSP over a 2-element domain
2.6. CSP solvable by local consistency checking
2.7. CSP Dichotomy Conjecture
2.8. How to solve CSP if pp-definable relations are simple
2.9. System of linear equations in Z_2
2.10. 2-satisfability
2.11. Strong subuniverses and a proof of the CSP Dichotomy Conjecture
2.12. Conclusions
3. Quantified CSP
3.1. PGP reduction for ∏_2 restrictions
3.2. A general PGP reduction
3.3. Does EGP mean hard?
3.4. Surprising constraint language and the QCSP on a 3-element domain
3.5. Conclusions
4. Other variants of CSP
4.1. CSP over an infinite domain
4.2. Surjective Constraint Satisfaction Problem
4.3. Promise CSP
References
2. Algebra
P.-E. Caprace and G. A. Willis: A totally disconnected invitation to locally compact groups
1. Introduction
2. Decomposition theory
2.1. Normal subgroup structure
2.2. Elementary groups
2.3. More on chief factors
3. Simple groups
3.1. Dense embeddings and local structure
3.2. Applications to lattices
3.3. Applications to commensurated subgroups
4. Scale methods
4.1. Contraction and other groups
4.2. Flat groups of automorphisms
5. Future directions
References
N. Gupta: The Zariski cancellation problem and related problems in affine algebraic geometry
1. Introduction
2. Cancellation problem
3. Characterization problem
4. Affine fibrations
5. Epimorphism problem
6. A^n-forms
7. An example of Bhatwadekar and Dutta
References
S. Kato: The formal model of semi-infinite flag manifolds
1. Introduction
2. Flag manifolds via representation theory
3. Kac–Moody flag varieties
3.1. Reminder on Kac–Moody algebras and their quantum groups
3.2. Thin and thick flag varieties
4. Global Weyl modules and their projectivity
5. Semi-infinite flag manifolds
6. Frobenius splittings
7. Connection to the space of rational maps
8. K-theoretic Peterson isomorphism
9. Functoriality of quantum K-groups
10. Some perspectives
10.1. Categorifications of the coordinate rings
10.2. Peterson isomorphism in quantum cohomology
10.3. Constructible sheaves on semi-infinite flags
10.4. Tensor product decompositions
10.5. The cotangent bundle of semi-infinite flags
References
M. J. Larsen: Character estimates
1. Introduction
2. Symmetric and alternating groups
3. Groups of Lie type
4. Products of conjugacy classes
5. Waring's problem
6. Fuchsian groups
References
A. Neeman: Finite approximations as a tool for studying triangulated categories
1. Introduction
2. Approximable triangulated categories—the formal definition as a variant on Fourier series
3. Examples of approximable triangulated categories
4. Applications: strong generation
5. The freedom in the choice of compact generator and t-structure
6. Structure theorems in approximable triangulated categories
7. Future directions
References
I. Peeva: Syzygies over a Polynomial Ring
1. Introduction
2. Free resolutions
3. Betti numbers
4. Projective dimension
5. Regularity
6. Regularity of prime ideals
7. Regularity of the radical
8. Shifts
9. The EGH conjecture
References
3. Number Theory – Special lecture
J. H. Silverman: Survey lecture on arithmetic dynamics
1. Introduction
2. Definitions and terminology
3. A dictionary for arithmetic dynamics
4. Topic #1: Dynamical uniform boundedness
5. Topic #2: Dynamical moduli spaces
6. Topic #3: Dynamical unlikely intersections
7. Topic #4: Dynatomic and arboreal representations
7.1. Topic #4(a): Dynatomic representations
7.2. Topic #4(b): Arboreal representations
8. Topic #5: Dynamical and arithmetic complexity
References
3. Number Theory
R. Beuzart-Plessis: Relative trace formulae and the Gan–Gross–Prasad conjectures
1. The local conjectures and multiplicity formulae
1.1. The branching problem
1.2. Approach through local trace formulae
2. The global Gan–Gross–Prasad conjectures and Ichino–Ikeda refinements
2.1. Statements and results
2.2. The approach of Jacquet–Rallis
3. Comparison: Local transfer and fundamental lemma
4. Global analysis of Jacquet–Rallis trace formulae
5. Looking forward
References
A. Caraiani: The cohomology of Shimura varieties with torsion coefficients
1. Introduction
2. A vanishing conjecture for locally symmetric spaces
3. The Hodge–Tate period morphism
4. Cohomology with mod ℓ coefficients
5. Cohomology with mod p and p-adic coefficients
6. Applications beyond Shimura varieties
References
S. Dasgupta and M. Kakde: On the Brumer–Stark conjecture and refinements
1. Background and motivation
2. Stark's conjecture
3. Stark's conjectures at finite places
4. The Brumer–Stark conjecture
4.1. Annihilation of class groups
4.2. Our results
4.3. Explicit formula for Brumer–Stark units
5. Refinements of Stark's conjecture
5.1. The conjecture of Kurihara
5.2. The conjecture of Atsuta–Kataoka
5.3. The conjecture of Burns–Kurihara–Sano
6. Ritter–Weiss modules and Ribet's method
6.1. Ritter–Weiss modules
6.2. Inclusion implies equality
6.3. Ribet's method
6.3.1. L-functions to Eisenstein series
6.4. Eisenstein series to cusp forms
6.5. Cusp forms to Galois representations
6.6. Galois representations to Galois cohomology classes
6.7. Galois cohomology classes to class groups
7. Explicit formula for Brumer–Stark units
7.1. Shintani's method
7.2. The formula
7.3. Horizontal Iwasawa theory
7.4. The Greenberg–Stevens L-invariant
7.5. The method of Darmon–Pozzi–Vonk
References
A. Gamburd: Arithmetic and dynamics on varieties of Markoff type
1. Introduction
1.1.
1.2.
1.3.
1.4.
1.5.
1.6.
1.7.
1.8.
1.9.
1.10.
2. The unreasonable(?) ubiquity of Markoff equation
2.1. The Markoff chain
2.2. Continued fractions and binary quadratic forms
2.3. The geometry of Markoff numbers
2.4. Cohn tree and Nielsen transformations
2.5. Nielsen systems and product replacement graphs
3. Strong approximation
4. Real dynamics on surfaces of Markoff type
4.1.
4.2.
5. An asymptotic formula for integer points on Markoff–Hurwitz varieties
6. Hasse principle on surfaces of Markoff type
References
P. Habegger: The number of rational points on a curve of genus at least two
1. Introduction
1.1. The Mordell Conjecture
1.2. Some remarks on effectivity
1.3. The number of rational points: conjectures and results
2. Heights
2.1. The absolute logarithmic Weil height
2.2. The canonical height on an abelian variety
3. Vojta's approach to the Mordell Conjecture
4. Comparing Weil and Néron–Tate heights
4.1. Degenerate subvarieties and the Betti map
4.2. Comparing the Weil and Néron–Tate heights on a subvariety
5. Application to moderate points on curves
5.1. The Faltings–Zhang morphism
5.2. Bounding the number of moderate points—a sketch
6. Hyperelliptic curves
References
A. Ichino: Theta lifting and Langlands functoriality
1. Introduction
2. Theta lifting
2.1. Basic definitions and properties
2.2. Explicit realization of the Jacquet–Langlands correspondence
2.3. Seesaw identities
3. The Shimura–Waldspurger correspondence
3.1. Modular forms of half-integral weight
3.2. Global correspondence
3.3. Local correspondence
4. Geometric realization of the Jacquet–Langlands correspondence
References
D. Koukoulopoulos: Rational approximations of irrational numbers
1. Diophantine approximation
1.1. First principles
1.2. Improving Dirichlet's approximation theorem
2. Metric Diophantine approximation
2.1. The theorems of Khinchin and Jarník–Besicovitch
2.2. Generalizing Khinchin's theorem
2.3. Progress towards the Duffin–Schaeffer conjecture
3. The main ingredients of the proof of the Duffin–Schaeffer conjecture
3.1. Borel–Cantelli without independence
3.2. A bound on the pairwise correlations
3.3. Generalizing the Erdős–Vaaler argument
3.4. An iterative compression algorithm
3.5. The quality increment argument
3.6. Proof of Theorem 3.5
References
D. Loeffler and S.L. Zerbes: Euler systems and the Bloch–Kato conjecture
1. What is the Bloch–Kato conjecture?
Critical values
Iwasawa theory
2. What is an Euler system?
3. Euler systems for Shimura varieties
Choosing the data
4. Deformation to critical values
5. Constructing p-adic L-functions
Coherent cohomology
Interpolation
6. P-adic regulators
7. Deformation to critical values
Non-regular weights
References
L. B. Pierce: Counting problems: class groups, primes, and number fields
1. Historical prelude
1.1. The class group
2. The ℓ-torsion conjecture
2.1. The Ellenberg–Venkatesh criterion
3. Families of fields
3.1. Dual problems: counting primes, counting fields
4. Counting primes with L-functions
4.1. Families of L-functions
4.2. Further developments
5. Why do we expect the ℓ-torsion conjecture to be true?
5.1. From the perspective of the Cohen–Lenstra–Martinet heuristics
5.2. From the perspective of counting number fields of fixed discriminant
5.3. From the perspective of counting number fields of bounded discriminant
References
S. W. Shin: Points on Shimura varieties modulo primes
1. Introduction
Conventions
2. Shimura varieties with good reduction
3. The Langlands–Rapoport conjecture
3.1. Galois gerbs
3.2. Versions of the LR conjecture
3.3. From the LR conjecture to a stabilized trace formula
4. Applications
4.1. The Hasse–Weil zeta functions and ℓ-adic cohomology
4.2. The global Langlands correspondence
5. Shimura varieties with bad reduction, Part I
6. Shimura varieties with bad reduction, Part II
6.1. The LR conjecture in the parahoric case
6.2. Semisimple zeta functions and Haines–Kottwitz's test function conjecture
7. Igusa varieties
7.1. The LR conjecture for Igusa varieties
7.2. Applications
References
Y. Tian: The congruent number problem and elliptic curves
1. Congruent number problem
2. Heegner point and explicit Gross–Zagier formula
3. Selmer groups: p-converse and distribution
3.1. Distribution of Selmer groups and Goldfeld conjecture
3.2. Recent progress: p-converse
3.2.1. Rank zero CM p-converse
3.3. Rank one CM p-converse
References
X. Zhu: Arithmetic and geometric Langlands program
1. From classical to geometric Langlands correspondence
1.1. The geometric Satake
1.2. Tamely ramified local geometric Langlands correspondence
1.3. Global geometric Langlands correspondence
2. From geometric to classical Langlands program
2.1. Categorical arithmetic local Langlands
2.2. Global arithmetic Langlands for function fields
2.3. Geometric realization of Jacquet–Langlands transfer
3. Applications to arithmetic geometry
3.1. Local models of Shimura varieties
3.2. The congruence relation
3.3. Generic Tate cycles on mod p fibers of Shimura varieties
3.4. The Beilinson–Bloch–Kato conjecture for Rankin–Selberg motives
References
4. Algebraic and Complex Geometry – Special lecture
M. Levine: Motivic cohomology
1. Introduction
2. Background and history
2.1. The conjectures of Beilinson and Lichtenbaum
2.2. Bloch's higher Chow groups and Suslin homology
2.3. Quillen–Lichtenbaum conjectures
2.4. Voevodsky's category DM and modern motivic cohomology
2.5. Motivic homotopy theory
2.6. Motivic cohomology and the rational motivic stable homotopy category
2.7. Slice tower and motivic Atiyah–Hirzebruch spectral sequences
3. Motivic cohomology over a general base
3.1. Cisinski–Déglise motivic cohomology
3.2. Spitzweck's motivic cohomology
3.3. Hoyois' motivic cohomology
4. Milnor–Witt motivic cohomology
4.1. Milnor–Witt K-theory and the Chow–Witt groups
4.2. The homotopy t-structure and Morel's theorem
4.3. Milnor–Witt motivic cohomology
4.4. Milnor–Witt motives
5. Chow groups and motivic cohomology with modulus
5.1. Higher Chow groups with modulus
5.2. 0-cycles with modulus and class field theory
5.3. Categories of motives with modulus
5.4. Logarithmic motives and reciprocity sheaves
6. p-adic étale motivic cohomology and p-adic Hodge theory
6.1. A quick overview of some p-adic Hodge theory
6.2. Étale motivic cohomology
6.3. The theorems of Geisser–Hesselholt
6.4. Integral p-adic Hodge theory and the motivic filtration
References
4. Algebraic and Complex Geometry
M. Aganagic: Homological knot invariants from mirror symmetry
1. Introduction
1.1. Quantum link invariants
1.2. The knot categorification problem
1.2.1.
1.2.2.
1.2.3.
1.3. Homological invariants from mirror symmetry
1.4. The solution
2. Knot invariants and conformal field theory
2.1. Knizhnik–Zamolodchikov equation and quantum groups
2.1.1.
2.1.2.
2.1.3.
2.2. A categorification wishlist
3. Mirror symmetry
3.1. Homological mirror symmetry
3.2. Quantum differential equation and its monodromy
3.2.1.
3.2.2.
4. Homological link invariants from B-branes
4.1. The geometry
4.1.1.
4.1.2.
4.1.3.
4.1.4.
4.1.5.
4.1.6.
4.2. Branes and braiding
4.2.1.
4.2.2.
4.3. Link invariants from perverse equivalences
4.3.1.
4.3.2.
4.3.3.
4.3.4.
4.3.5.
4.3.6.
4.3.7.
4.3.8.
4.3.9.
4.4. Algebra from B-branes
4.4.1.
5. Mirror symmetry for monopole moduli space
5.1. The algebra for homological mirror symmetry
5.1.1.
5.2. The core of the monopole moduli space
5.2.1.
5.2.2.
5.2.3.
5.2.4.
5.2.5.
5.3. Equivariant Fukaya–Seidel category
5.3.1.
5.3.2.
5.3.3.
5.3.4.
5.4. Link invariants and equivariant mirror symmetry
5.4.1.
5.4.2.
5.4.3.
5.4.4.
5.4.5.
5.4.6.
6. Homological link invariants from A-branes
6.1. The algebra of A-branes
6.1.1.
6.1.2.
6.1.3.
6.1.4.
6.1.5.
6.2. The meaning of link homology
6.2.1.
6.2.2.
6.2.3.
6.2.4.
6.3. Projective resolutions from geometry
6.3.1.
6.3.2.
6.3.3.
6.3.4.
6.3.5.
6.3.6.
6.3.7.
References
A. Asok and J. Fasel: Vector bundles on algebraic varieties
1. Introduction
2. A few topological stories
2.1. Moore–Postnikov factorizations
2.2. The topological splitting problem
3. A quick review of motivic homotopy theory
3.1. Homotopical sheaf theory
3.2. The motivic homotopy category
3.3. A1-weak equivalences
3.4. Representability results
3.5. Postnikov towers, connectedness and strictly A1-invariant sheaves
3.6. Complex realization
4. Obstruction theory and vector bundles
4.1. The homotopy sheaves of the classifying space of BGL_n
4.2. Splitting bundles, Euler classes, and cohomotopy
4.3. The next nontrivial A1-homotopy sheaf of spheres
4.3.1. The KO-degree map
4.3.2. The motivic J-homomorphism
4.4. Splitting in corank 1
4.5. The enumeration problem
5. Vector bundles: nonaffine varieties and algebraizability
5.1. Descent along a Jouanolou device
5.2. Algebraizability I: obstructions
5.3. Algebraizability II: building motivic vector bundles
References
A. Bayer and E. Macrì: The unreasonable effectiveness of wall-crossing in algebraic geometry
1. Introduction
2. Stability conditions on derived categories
2.1. Bridgeland stability conditions
2.2. Stability conditions as polarizations
2.3. K3 categories
3. Constructions based on K3 categories
3.1. Curves in irreducible holomorphic symplectic manifolds
3.2. Curves
3.3. Surfaces in cubic fourfolds
4. Threefolds
4.1. The generalized Bogomolov–Gieseker inequality
4.2. Tilt-stability methods
4.3. Tilt-stability applications
5. Further research directions
References
V. Delecroix, É. Goujard, P. Zograf, and A. Zorich: Counting lattice points in moduli spaces of quadratic differentials
1. Introduction
2. Moduli spaces of quadratic differentials and square-tiled surfaces
3. Cylinder decomposition, multicurves, and stable graphs
4. Formula for the Masur–Veech volume
5. Random square-tiled surfaces and random multicurves
6. Square-tiled surfaces and enumeration of meanders
References
A. I. Efimov: K-theory of large categories
1. Introduction
2. Preliminaries on DG categories
3. Presentable and dualizable DG categories
4. K-theory of rings, abelian categories, and triangulated categories
5. Continuous K-theory
6. Continuous K-theory of categories of sheaves
6.1. The case of the real line
6.2. Reduction to the hypercomplete case
6.3. Continuous presheaves and continuous partially ordered sets
References
T. Hausel: Enhanced mirror symmetry for Langlands dual Hitchin systems
1. Introduction
2. Background
2.1. Mirror symmetry
2.2. Geometric Langlands correspondence
2.3. SYZ mirror symmetry for Langlands dual Hitchin systems
2.4. Topological mirror symmetry for Langlands dual Hitchin systems
2.5. Geometric Langlands as enhanced homological mirror symmetry
3. Enhanced mirror symmetry at the tip of the nilpotent cone
3.1. Very stable Higgs bundles and mirror symmetry
3.1.1. Białynicki-Birula decomposition of semiprojective varieties
3.1.2. Białynicki-Birula partition for Higgs bundles
3.2. Multiplicity algebra and explicit Hitchin system on Lagrangians
3.3. Explicit Hitchin system for wobbly Lagrangians
3.3.1. Lagrangian closure of W^+_\delta
3.4. Towards a classical limit of the geometric Satake correspondence
References
B. Klingler: Hodge theory, between algebraicity and transcendence
1. Introduction
2. Variations of Hodge structures and period maps
2.1. Polarizable Hodge structures
2.2. Hodge classes and Mumford–Tate group
2.3. Period domains and Hodge data
2.4. Hodge varieties
2.5. Polarized Z-variations of Hodge structures
2.6. Generic Hodge datum and period map
3. Hodge theory and tame geometry
3.1. Variational Hodge theory between algebraicity and transcendence
3.2. O-minimal geometry
3.3. O-minimal geometry and algebraization
3.3.1. Diophantine criterion
3.3.2. Definable Chow and definable GAGA
3.4. Definability of Hodge varieties
3.5. Definability of period maps
3.6. Applications
3.6.1. About the Cattani–Deligne–Kaplan theorem
3.6.2. A conjecture of Griffiths
4. Functional transcendence
4.1. Bialgebraic geometry
4.2. The Ax–Schanuel theorem for period maps
4.3. On the distribution of the Hodge locus
5. Typical and atypical intersections: the Zilber–Pink conjecture for period maps
5.1. The Zilber–Pink conjecture for ZVHS: Conjectures
5.2. The Zilber–Pink conjecture for ZVHS: Results
5.3. On the algebraicity of the Hodge locus
5.4. On the typical Hodge locus in level one and two
6. Arithmetic aspects
6.1. Field of definition of special subvarieties
6.2. Absolute Hodge locus
References
C. Li: Canonical Kähler metrics and stability
1. Canonical Kähler metrics on algebraic varieties
1.1. Constant scalar curvature Kähler metrics
1.2. Kähler–Einstein metrics and weighted Kähler–Ricci soliton
1.3. Kähler–Einstein metrics on log Fano pairs
1.4. Ricci-flat Kähler cone metrics
1.5. Analytic criteria for the existence
2. Stability of algebraic varieties and non-Archimedean geometry
2.1. K-stability and non-Archimedean geometry
2.2. Non-Archimedean pluripotential theory
2.3. Stability of Fano varieties
2.4. Normalized volume and local stability theory of klt singularities
3. Archimedean (complex analytic) theory vs. non-Archimedean theory
3.1. Correspondence between Archimedean and non-Archimedean objects
3.2. Yau–Tian–Donaldson conjecture for general polarized manifolds
3.3. YTD conjecture for Fano varieties
3.3.1. Non-Archimedean approach
3.3.2. Other approaches
References
A. Pixton: The double ramification cycle formula
1. Introduction
2. Tautological classes
2.1. Preliminaries
2.2. Stable graphs
2.3. Compact type
3. Previous formulas and results
4. Motivation for the formula
4.1. Expanding Hain's formula
4.2. Cohomological field theory axioms
4.3. Givental's R-matrix action
4.4. Divergent averages
4.5. Interpolating finite rank CohFTs
4.6. Geometric interpretation from (k/r)-spin structures
5. The double ramification cycle formula
References
Yu. Prokhorov: Effective results in the three-dimensional minimal model program
1. Singularities
2. Minimal model program
3. General elephant
4. Divisorial contractions to a point
5. Del Pezzo fibrations
6. Extremal curve germs
6.1. Construction of germs by deformations
7. Extremal curve germs: general elephant
7.1. Local method
7.2. Extension from S∈|-2K_X|
7.3. Global method
8. Semistable germs
9. Exceptional curve germs
10. Q-Fano threefolds
References
O. Wittenberg: Some aspects of rational points and rational curves
1. Introduction
2. Solvable groups and the Grunwald problem in inverse Galois theory
2.1. Homogeneous spaces
2.2. Geometry
2.3. Descent
2.4. Fibration
2.5. An application to Massey products
3. Rational curves on real algebraic varieties
3.1. A few questions
3.2. Tight approximation
3.3. Descent
3.4. Fibration
3.5. Homogeneous spaces
3.6. Further comments
4. Function fields of curves over p-adic fields
4.1. Some motivation: rational curves over number fields
4.2. Rational curves on varieties over p-adic fields
4.3. Quadrics and other homogeneous spaces
4.4. Reciprocity obstructions
4.5. Sufficiency of the reciprocity obstruction
4.6. Further questions
4.7. Other fields
References
List of contributors
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