BSD via Operator-Fredholm Framework - Computational Tools

Overview

Computational verification of key components in the operator-Fredholm approach to BSD conjecture.

This repo is demo pack for "Birch–Swinnerton–Dyer via an Operator–Fredholm Method" research by miruka.

Main document loccated at https://zenodo.org/records/17201882

Installation

Prerequisites

• Windows with WSL (Windows Subsystem for Linux)
• SageMath 9.5 or later installed in WSL
• Python 3.8+ in WSL
• NumPy

Setup for WSL

# Open WSL (from PowerShell or CMD):
wsl -d Ubuntu-22.04  # or your WSL distribution name

# Navigate to project directory
cd /mnt/'your dir'  # adjust path as needed

# Make scripts executable
chmod +x run_all.sh

# Optional: Create Python virtual environment
python3 -m venv .venv
source .venv/bin/activate

# Install Python dependencies
pip install -r requirements.txt

Usage

Quick Start - Run All Demos

From WSL

# Open WSL and navigate to project
wsl -d Ubuntu-22.04
cd /mnt/'your dir'

# Run all demos for curve 37a1 (rank 1)
./run_all.sh 37a1

# Run for curve 11a1 (rank 0)
./run_all.sh 11a1

# Run with custom log directory
./run_all.sh 37a1 my_logs

From PowerShell (one-liner)

# Run directly from Windows PowerShell
wsl -d Ubuntu-22.04 -e bash -c "cd /mnt/'your dir'
 && ./run_all.sh 37a1"

Individual Script Usage (in WSL)

1. Extract curve data and Fourier coefficients

sage src/01_curve_data.sage <curve_label> [B=5000]

# Examples:
sage src/01_curve_data.sage 37a1        # Default B=5000
sage src/01_curve_data.sage 11a1 10000  # Compute a_p for p < 10000

Output: data/a_ell_<curve>_B<B>.csv

2. Principal part computation

sage src/02_principal_part_demo.sage <curve> [eps=1e-3] [B=30000] [eta=1.2]

# Examples:
sage src/02_principal_part_demo.sage 37a1                    # All defaults
sage src/02_principal_part_demo.sage 37a1 1e-4              # Smaller eps
sage src/02_principal_part_demo.sage 37a1 1e-3 50000 1.5    # Higher precision

Output: data/principal_part_<curve>_<timestamp>.json

3. Gap polynomial computation

sage src/03_gap_poly_demo.sage <curve> [Lmax=29] [Mmax=2]

# Examples:
sage src/03_gap_poly_demo.sage 37a1          # Default parameters
sage src/03_gap_poly_demo.sage 37a1 37 3     # More features

Output: data/gap_poly_<curve>_<timestamp>.json

4. Renormalization matrix

python3 src/04_renorm_matrix_demo.py <conductor>

# Examples:
python3 src/04_renorm_matrix_demo.py 37    # For curve 37a1
python3 src/04_renorm_matrix_demo.py 11    # For curve 11a1

Output: data/renorm_matrix_<timestamp>.npz

5. BSD components and kappa

sage src/05_kappa_leading_coeff.sage <curve>

# Examples:
sage src/05_kappa_leading_coeff.sage 37a1
sage src/05_kappa_leading_coeff.sage 11a1

Output: data/bsd_components_<curve>_<timestamp>.json

View Results

# In WSL
python3 src/view_results.py 37a1

# View results for default curve (37a1)
python3 src/view_results.py

# Save summary to file
python3 src/view_results.py 37a1 > results_summary.txt

The view_results.py script:

• Finds the latest saved data files for a curve
• Displays all computed values in a readable format
• Shows results from all 5 scripts in one place
• Does NOT save anything - only reads and displays

Output Files

The project creates two types of output:

1. Data Files (in data/ directory)

Structured data files with all numerical results:

• JSON files: Human-readable, contains all computed values, parameters, and results
• NPZ files: NumPy format for matrices and arrays
• CSV files: Simple tabular data (Fourier coefficients)

2. Console Logs (in logs/ directory)

Text files capturing console output from each script run:

• Shows what each script prints during execution
• Useful for debugging and documentation
• Created by run_all.sh using tee command

Example:

• data/principal_part_37a1_20250924_201201.json - Actual computation results
• logs/02_principal_part_37a1.txt - Console output from running the script

See OUTPUT_FORMAT.md for detailed description of data file formats.

Example Curves

Rank 0 curves (good for testing)

• 11a1: conductor 11, rank 0, L(E,1) ≠ 0
• 14a1: conductor 14, rank 0
• 15a1: conductor 15, rank 0

Rank 1 curves

• 37a1: conductor 37, rank 1, minimal example
• 43a1: conductor 43, rank 1
• 53a1: conductor 53, rank 1

Higher rank curves (slower computation)

• 389a1: conductor 389, rank 2
• 5077a1: conductor 5077, rank 3

Notes

• Demos are illustrative; they validate the shape of the method
• Computation time increases with conductor and rank
• For production use, increase parameters B, X, Lmax for better precision
• All timestamps are in local time

What each script computes

01_curve_data.sage

• Fourier coefficients a_p for p < B not dividing N
• Tamagawa numbers at bad primes
• Torsion subgroup structure
• Rank (via Sage's built-in methods)

02_principal_part_demo.sage

• Windowed Hecke operator t_X(s) with C^1 cutoff
• Principal part matching: log det_fin(I - A_X(s)) vs log L(E,s)
• Slope verification for rank
• Individual term contributions

03_gap_poly_demo.sage

• Hecke separator polynomial P(T) with P(f)=1
• Feature vectors for all newforms at level N
• Spectral gap δ = 1 - max|P(g)| for g≠f
• Polynomial coefficients via constrained least squares

04_renorm_matrix_demo.py

• R×R Vandermonde-style matrix (R = 1 + #multiplicative primes)
• Matrix conditioning and determinant
• Eigenvalue computation
• Handles archimedean and multiplicative normalization

05_kappa_leading_coeff.sage

• Complete BSD formula components
• L^(r)(E,1)/r! via Dokchitser's algorithm
• Regulator, periods, Tamagawa numbers
• Generator points and heights (when available)

Troubleshooting

WSL Issues

WSL not installed

Install WSL from PowerShell (as Administrator):

wsl --install -d Ubuntu-22.04

SageMath not found in WSL

Install SageMath in WSL:

sudo apt update
sudo apt install sagemath

Permission denied when running scripts

Make scripts executable:

chmod +x run_all.sh

Import errors in Python scripts

Activate the virtual environment in WSL:

source .venv/bin/activate

Memory issues

For large conductors, reduce parameters:

• Lower B in script 02
• Lower Lmax in script 03