Arithmetic Geometry, Number Theory, and Computation

Foreword
Contents
A Robust Implementation for Solving the S-Unit Equation and Several Applications
1 Introduction
1.1 Overview
2 Notation
2.1 S-Units in Number Fields
2.2 Absolute Values and Completions
2.3 Height Functions
2.4 p-Adic Logarithms
2.5 Solutions to the S-Unit Equation
3 The Bounds of Baker-Wüstholz and Yu
3.1 Statement of Yu's Bound
3.2 The Constants Ω and Ω'
3.3 The Constant C1*
3.4 A Remark About Implementation
3.5 Bound of Baker-Wüstholz
3.6 Obtaining the Initial Bound
4 Initial Exponent Bounds
4.1 An Upper Bound at the Extremal Place
4.2 Case I: p Is Finite
4.3 Case II: p Is Infinite
5 LLL Reduction
5.1 Finite Places
5.2 Complex Places
5.3 Real Places
5.4 Implementation
6 Further Reducing the Search Space: Sieving
6.1 Setup for the Sieve
6.2 Execution of the Sieve
7 Experimental Observations and Computational Choices
7.1 Sieving vs. Simple Exhaustion
7.2 Finite Place vs. Infinite Place Bounds
8 Applications
8.1 Asymptotic Fermat
8.2 Cubic Ramanujan-Nagell Equations
References
Computing Classical Modular Forms for Arbitrary Congruence Subgroups
1 Introduction
1.1 Motivation
1.2 Main Results
1.3 Recovering Known Results
1.4 New Results
1.5 Related Literature
1.6 Organization
1.7 Acknowledgements
2 Setup and Notation
2.1 Congruence Subgroups
2.2 Modular Forms
3 Explicit Computation of
3.1 Modular Symbols
3.2 Manin Symbols
3.3 Modular Symbols with Character
3.4 Cuspidal Modular Symbols
3.5 Efficient Computation of the Boundary Map
3.6 Pairing Modular Symbols and Modular Forms
3.6.1 The Action of Complex Conjugation
4 Hecke Operators
4.1 Hecke Operators on Mk()
4.2 Naive Computation of Tα
4.3 Faster Implementation of the Hecke Operator Tα
4.4 The Hecke Operators Tn
4.4.1 Adelic Definition
4.4.2 Classical Definition
4.5 Efficient Implementation of the Hecke Operators Tn, n det(G)
5 Degeneracy Maps
5.1 Petersson Inner Product
5.2 Degeneracy Maps
5.3 Degeneracy Maps and Hecke Operators
6
6.1 Constructing Dual Vector Spaces
6.2 Decomposition of Sk()
6.3 Computing the Zeta Functions
7 Applications
7.1 Classification of 2-Adic Images of Galois Representations Associated to Elliptic Curves over Q
7.2 Modular Curves of Prime-Power Level with Infinitely Many Rational Points
7.3 Efficient Computation of q-Eigenforms for Xns(N) and Xns+(N)
7.4 Decomposition of the Jacobian of Xns+(p)
7.5 Computation of q-Eigenforms for XG(p) When G Is Exceptional
7.6 Smooth Plane Models for Modular Curves
References
Square Root Time Coleman Integration on Superelliptic Curves
1 Introduction
1.1 Acknowledgements
2 Set-Up and Notation
3 Coleman Integration
3.1 Local Coordinates on Superelliptic Curves
3.2 The Action of q-Power Frobenius
4 Reductions in Cohomology
4.1 Vertical Reduction
4.2 Horizontal Reduction
5 Runtime and Precision
6 Implementation
6.1 Examples
6.1.1 A Picard Curve
6.2 An Elliptic Curve over a Quartic Field
6.3 Implementation Details
6.3.1 A Comparison of Computer Algebra Systems
6.4 Timings
References
Computing Classical Modular Forms
1 Introduction
1.1 Motivation
1.2 Organization
1.3 Acknowledgments
2 History
3 Characters
3.1 Definitions
3.2 Conrey Labels
3.3 Orbit Labels
4 Computing Modular Forms
4.1 Setup
4.2 Galois Digression
4.3 Dimensions
4.4 Eisenstein Series
4.5 Decomposition of Newspaces into Hecke Orbits
4.6 Hecke Eigenvalues
4.7 L-Functions
5 Algorithms
5.1 Modular Symbols
5.2 Trace Formula
5.3 Definite Methods
5.4 Other Methods
6 Two Technical Ingredients
6.1 Eichler–Selberg Trace Formula for Newforms
6.2 Certifying Generalized Eigenvalues
7 A Sample of the Implementations
7.1 Comparison of Methods
7.2 A Trace Formula Implementation with Complex Coefficients
8 Issues: Computational, Theoretical, and Practical
8.1 Analytic Conductor
8.2 Sturm Bound
8.3 Atkin–Lehner Operators and Eigenvalues
8.4 Self-Duality
8.5 Efficiently Recognizing Irreducibility
8.6 Trace Form
8.7 Presenting Coefficients Using LLL-Reduction
8.8 Presenting Coefficients Using a Sparse Cyclotomic Representation
8.9 Hecke Kernels
9 Computing L-Functions Rigorously
9.1 Embedded Modular Forms
9.2 Computations
9.3 Imprimitive L-Function
9.4 Verifying the Analytic Rank
9.5 Chowla's Conjecture
10 An Overview of the Computation
10.1 Data Extent
10.2 Statistics
10.3 Data Reliability
10.4 Interesting, Extreme Behavior and Examples from the Literature
10.5 Pictures
10.6 Features
11 Twisting
11.1 Definitions
11.2 Detecting Inner Twists
11.3 Computing Inner Twists
12 Weight One
12.1 Computational Observations
12.2 Classifying the Projective Image
12.3 Computing the Projective Field
12.4 Computing the Artin Image, the Artin Field, and the Associated Artin Representation
12.5 Interesting and Extreme Behavior
References
Elliptic Curves with Good Reduction Outside of the First Six Primes
1 Introduction
1.1 Summary of the Database
1.2 Distribution of Quantities
1.3 Comparison with Cremona's Database
2 Computation
2.1 Computation Method
2.2 Computing Mordell–Weil Bases
2.2.1 Standard Techniques
2.2.2 12-Descent
2.2.3 Remaining Curves
2.3 Completeness of the Data
2.4 An S-Integral Weak Hall Conjecture and the abc Conjecture
3 Attainability of Maximal Conductor by Curves in M(S)
4 Applications
4.1 Solving S-Unit Equations
4.2 Other Diophantine Problems
4.3 n-Congruences Between Elliptic Curves
References
Efficient Computation of BSD Invariants in Genus 2
1 Introduction
2 Tamagawa Numbers
2.1 Computation of a Regular Model in Magma
2.2 Outline of Algorithm to Compute the Galois Action
2.3 Problem Arising During the Algorithm
2.4 Construction of a Suitable Extension
2.5 Implementation
3 Real Periods
3.1 Computation of the Order of Vanishing of a Function
3.2 How to Avoid Difficult Computations in Characteristic 0
3.2.1 Working over Z/pnZ Instead of over Z
3.2.2 Optimising the Ideals for Fast Computation
3.2.3 Avoiding Function Field Computations
3.3 How to Avoid Checking p-to-the-g Differentials
3.4 Possible Ideas for Improvements
3.5 Implementation
4 Other BSD Invariants
References
Restrictions on Weil Polynomials of Jacobians of Hyperelliptic Curves
1 Introduction
2 Weil Polynomials Mod 2
3 Asymptotics
4 Point Counts
4.1 Restrictions on Parity of Point Counts
4.2 Restrictions on Point Counts Modulo Powers of 2
4.3 Reproving Theorem 2.8
5 Experimental Data
References
Zen and the Art of Database Maintenance
1 Introduction
2 Using Databases in Mathematics
2.1 The LMFDB Project
3 Nuts and Bolts
3.1 Back End
3.2 Front End
3.3 Hosting
4 MongoDB vs PostgreSQL
4.1 Comparing Two Database Systems
4.2 Transition
5 Benefits
5.1 Abstraction
5.2 New Features
References
Effective Obstruction to Lifting Tate Classes from PositiveCharacteristic
1 Introduction
1.1 Lifting Algebraic Cycles from Finite Characteristic
1.2 Tate Classes As a Substitute for Algebraic Classes
1.3 A Note on Using Finite Approximations
1.4 A Limitation and the Need for Integral Structure
1.5 Previous Approaches
1.6 Overview
2 Lifting Algebraic Cycles from Positive Characteristic
2.1 Passage from Number Fields to Finite Fields
2.1.1 Good Reduction
2.1.2 The Specialization Map on Subvarieties
2.1.3 A Hodge Filtration on Crystalline Cohomology
2.1.4 Cycle Class Maps
2.1.5 Dimensions of the Space of Algebraic Cycles Over Different Fields
2.1.6 The Obstruction Map
2.2 Finding the Image of the Chern Class Map via Tate's Conjecture
2.3 Enlarging the Base Field
2.4 Improving the Obstruction Using the Frobenius Decomposition
2.5 Bounds on the Characteristic Polynomial of the Frobenius
3 Computing in Crystalline Cohomology
3.1 Crystalline to de Rham Cohomology
3.2 Splitting the Cohomology of a Hypersurface
3.2.1 Splitting in Characteristic Zero
3.2.2 Splitting Over the Ring of Witt Vectors
3.3 Torsion-Free Obstruction Space
3.4 Primitive Cohomology After Griffiths
3.4.1 Approximating the Frobenius Matrix in Terms of the Griffiths Basis
4 Main Algorithm
4.1 Clarification of the Steps in the Algorithm
4.1.1 Pick a Good Prime
4.1.2 Compute a Griffiths Basis for Primitive Cohomology
4.1.3 Compute a Minimal Working Precision
4.1.4 Compute an N-digit Approximation of the Frobenius Matrix
4.1.5 Compute the Characteristic Polynomial of the Frobenius Matrix
4.1.6 Represent the Obstruction Map
4.1.7 Extract Cyclotomic Factors from the Characteristic Polynomial
4.1.8 Approximate the Space of Tate Classes
4.1.9 Approximate the Map pii
4.1.10 Bound Dimension of Li
4.1.11 Return the Upper Bound on the Middle Picard Rank
5 Examples
5.1 Jacobians of Plane Curves
5.2 Surfaces in Projective Space
5.2.1 K3 Surfaces
5.2.2 Quintic Surfaces
References
Conjecture: 100% of Elliptic Surfaces Over Q have Rank Zero
1 Introduction
2 Probabilistic Heuristics
3 Obstacles for Analytic Proofs
4 An Approach via Finite Fields
References
On Rational Bianchi Newforms and Abelian Surfaces with Quaternionic Multiplication
1 Introduction
2 Rational Bianchi Newforms with No Associated Elliptic Curve
3 Characters
4 A Classical Newform of Level 1225 and an Abelian Eightfold
5 Base Change to QQsqrtm7
6 Twisting
7 An Explicit Model for the Building Block
8 Other Rational Bianchi Newforms at Level (175)
9 Relation with the Paramodular Conjecture
References
A Database of Hilbert Modular Forms
1 The Algorithm: An Overview
2 Computing Systems of Hecke Eigenvalues
3 Data Computed
4 Comments on Data
References
Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB
1 Introduction
2 Background
2.1 Weil Numbers, Characteristic Polynomials of Frobenius, and Zeta Functions
2.2 Weil Polynomials
2.3 Weil Polynomials and Isogeny Classes of Abelian Varieties
2.4 Newton Polygons, p-rank and Ordinarity
2.5 Galois Groups
2.6 Frobenius Angle Rank
2.7 Bounds on Point Counts
3 Algorithms
3.1 Enumerating Weil Polynomials
3.2 Point Counts
3.3 Curve Point Counts
3.4 Base Change, Primitivity and Isogeny Twists
3.5 Endomorphism Algebras
3.6 Principal Polarizations
3.7 Jacobian Testing
3.8 Angle Rank
4 Statistics vs. Heuristics
4.1 The Number of Isogeny Classes
4.2 Galois Groups
4.3 Newton Polygons Data and p-rank Strata
4.4 Frobenius Angle Rank
4.5 Endomorphism Algebras
4.6 Isogeny Sato-Tate Distribution
4.7 Maximal and Minimal Point Counts
5 An Isogeny Class Scavenger Hunt
5.1 Some Basic Examples
5.2 Supersingular Curves
5.3 Ordinarity and Angle Ranks
5.4 Function Fields of Class Number One
5.5 Hypersymmetric Abelian Varieties
5.6 Isomorphic Endomorphism Algebras and Different p-ranks
5.7 Abelian Fourfolds as Jacobians
5.8 Distinguishing Isogeny Classes by Point Counts
6 Possible Generalizations and Bottlenecks
6.1 Bottlenecks
6.2 Jacobians
6.3 Isomorphism Classes
6.4 K3 Surfaces and Higher Weight
A Tables and Figures
References
Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves
1 Introduction
2 Background on Coleman Integration
3 Algorithm
4 Analysis and Examples
References
Curves with Sharp Chabauty-Coleman Bound
1 Introduction
1.1 The Method of Chabauty and Coleman and Its History
1.2 On the Computation of the Rank of the Jacobian of Hyperelliptic Curves
1.3 Introduction to the Descent Method
2 Examples of Sharp Curves of Genus Two
2.1 Two Known Sharp Curves in Genus Two
2.2 A Finite Family of Sharp Curves of Genus Two
2.3 An Example with Descent and a Sharp Curve
2.4 A Sharp Curve with the Smallest Possible Number of Points
2.5 Examples on Improving Lower Rank Bounds in Magma
3 Examples of Sharp Curves of Higher Genus
3.1 Concrete Examples of Sharp Curves of Genus Three, Four and Five
3.2 Bertrand's Postulate for Primes Modulo 8
3.3 Potentially Sharp Curves of Genus g≥2
References
Chabauty–Coleman Computations on Rank 1 Picard Curves
1 Introduction
2 The Chabauty–Coleman Method
2.1 Chabauty–Coleman
2.2 Explicit Coleman Integration After Balakrishnan and Tuitman
3 The Algorithm
3.1 With a Rational Point Whose Class in the Jacobian Is of Infinite Order
3.2 Precision Bounds
3.3 Beyond Rational Points
4 The Data
4.1 Examples
Appendix 1: Precision Heuristics
Appendix 2: Iterative Hensel Lifting
References
Linear Dependence Among Hecke Eigenvalues
1 Introduction
2 Linear Relations Among Algebraic Integers in a Given Number Field
3 Discriminants and Odlyzko Bound
4 Lattice Reduction and Orthogonality Defect
5 Kronecker's Theorem
6 The Proof of the Main Theorem and the Trivial Bound
7 Application: An Algorithm for Modular Forms
References
Congruent Number Triangles with the Same Hypotenuse
1 Introduction and Motivation
2 Algebraic Formulation
3 Description of Methodology
3.1 Enumeration Through Elliptic Curves
3.2 Hypotenuses, and Heights, Grow Exponentially
3.3 Two Strategies of Experimentation
3.4 Computing Rank and Generators
4 Experimental Results
4.1 Results for t ≤1000
4.2 Results for Curves of Rank 6 and 7
Appendix: Invariants of the Lowry-Duda Moduli Space
Double Cover Realization
Resolving the Bad Singularities
Canonical Class of X
X"0365X Is Simply Connected
The Remaining Invariants
References
Visualizing Modular Forms
1 Introduction
1.1 Motivation
1.2 Broad Overview of Complex Function Plotting
1.3 Paper Overview
2 Survey of Visualizations
2.1 Halfplane and Disk Models
2.2 Visualizations
2.2.1 Standard Domain Coloring
2.2.2 Magnitude Plot Without Color
2.2.3 Magnitude Plot with Periodic Linear Color
2.2.4 Magnitude Plot with Periodic Logarithmic Spacing
2.2.5 Pure Phase Plots
2.2.6 Phase Plots with Magnitude Contours
3 Choosing Color
3.1 Implementation Overview
3.2 Example Visualizations with Colormaps
References
A Prym Variety with Everywhere Good Reduction over Q(61)
1 Introduction
2 Computing the Period Matrix
3 Isolating the Building Block
4 Computing an Equation for the Cover
5 Simplifying, Twisting, and Verifying the Cover
References
The S-Integral Points on the Projective Line Minus Three Points via Finite Covers and Skolem's Method
1 Introduction
2 Review of the Skolem–Chabauty Method
3 Proof of Siegel's Theorem
References