edgarcosta/controlledreduction

An implementation of the controlled reduction method for computing the Hasse-Weil zeta functions of smooth projective hypersurfaces over finite fields

controlled-reduction

PyPI package

For a PyPI package that integrates most of controlled-reduction into SageMath, see pycontrolledreduction.

Abstract

An implementation of the controlled reduction method for computing the Hasse-Weil zeta functions of smooth projective hypersurfaces over finite fields. Explicitly, by computing a p-adic approximation of Frobenius action on p-adic cohomology (Monsky-Washnizter) with sufficient precision and then lifting its characteristic polynomial to the integers. An overview of the method can be found in:

• "Computing zeta functions of nondegenerate projective hypersurfaces over finite fields", (under preparation) by Edgar Costa and David Harvey.
• "Effective computations of Hasse-Weil zeta functions", by Edgar Costa

Dependencies

It majorly depends on:

• NTL: A Library for doing Number Theory
• FLINT: Fast Library for Number Theory

Which depend on:

• GMP: GNU Multiple Precision Arithmetic Library (for NTL and FLINT)
• MPIR: Multiple Precision Integers and Rationals (for FLINT)
• MPFR: GNU Multiple Precision Floating-Point Reliably (for FLINT)

However, SageMath comes with all this libraries.

Installation

There are 3 options:

Using SageMath to provide the dependencies

1. Figuring out where SageMath is installed. We recommend doing this and storing in an environmental variable by doing:

SAGE_ROOT=$(sage -c "print SAGE_ROOT")

Alternatively, in Sage do print SAGE_ROOT and on the unix terminal: SAGE_ROOT=<the line printed in Sage>

2. Download controlled reduction

git clone https://github.com/edgarcosta/controlledreduction.git

3. Change your working directory and run the configure file

cd controlledreduction && ./configure --with-ntl=$SAGE_ROOT/local

4. Compile everything by doing

make

5. Set up the variable $LD_LIBRARY_PATH, so the executables can find the libraries at run time. One can do this for the current terminal by doing:

export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:$SAGE_ROOT/local/lib

and for this line to be ran at the start of every session one can do

echo export LD_LIBRARY_PATH=$LD_LIBRARY_PATH:$SAGE_ROOT/local/lib >> ~/.bashrc

6. You can optionaly run some tests by doing

make check

or check out some of the examples

build/examples/K3_dwork

Manually specifying the path for the dependencies

1. make sure you have the dependencies (see the source of the script build_dependencies if you would prefer to build them manually), if they are installed in a non-standard path, be sure to set $LD_LIBRARY_PATH accordingly.

2. ./configure to generate the makefile.

   To link against the libaries by SageMath it should be sufficient to run ./configure --with-ntl=<SAGE_DIR>/local/.

   Run ./configure --help for more options.

3. make to build everything

4. make check to run some tests.