Modular Arithmetic Geometry

Kevin Buzzard (Imperial College) made available his code for computing modular forms of weight one. The Magma implementation was developed using this as a starting point.

Lassina Dembélé (Warwick) wrote part of the code implementing his algorithm for computing Hilbert modular forms.

Enrique González-Jiménez (Madrid) contributed a package to compute curves over ℚ, of genus at least 2, which are images of X0(N) and X1(N) for a given level N.

Matthew Greenberg (Calgary) and John Voight (Vermont) developed and implemented an algorithm for computing Hilbert modular forms using Shimura curves.

A new implementation (V2.19) of Brandt modules associated to definite quaternion orders, over ℤ and over function fields \F{q}[t], has been developed by Markus Kirschmer (Aachen) and Steve Donnelly (Magma).

David Kohel (Singapore–NUS, Sydney) has provided implementations of division polynomials and isogeny structures for Brandt modules and modular curves. Jointly with William Stein (Harvard), he implemented the module of supersingular points.

Allan Lauder (Oxford) has contributed code for computing the characteristic polynomial of a Hecke operator acting on spaces of overconvergent modular forms.

An improved version of code due to Alan Lauder (Oxford) for computing the characteristic series of the Atkin–Lehner operator Up on p-adic modular forms was included in Magma V2.20 (2013).

Magma routines for constructing building blocks of modular abelian varieties were contributed by Jordi Quer (Cataluna).

A package for computing with modular symbols (known as HECKE) has been developed by William Stein (Harvard). William has also provided much of the package for modular forms.

In 2003–2004, William Stein (Harvard) developed extensive machinery for computing with modular abelian varieties within Magma.

A package for computing with congruence subgroups of the group PSL(2,ℝ) has been developed by Helena Verrill (LSU).

John Voight (Vermont) produced the package for Shimura curves and arithmetic Fuchsian groups.

Dan Yasaki (UNC) developed a Magma package for Bianchi modular forms over imaginary quadratic fields which was distributed in Magma V2.16 (2009). A new faster version developed by Dan was released in Magma 2.20 (2013).

Jeremy Le Borgne (Rennes) contributed his package for working with mod p Galois representations.

Code for constructing Artin representations of the Galois group of the absolute extension of a number field was developed by Tim Dokchitser (Cambridge).

Machinery which provides a uniform method for working with Galois representations over p-adic fields was implemented by Tim Dokchitser (Bristol).

Jared Weinstein (UCLA) wrote the package on admissible representations of GL2(ℚp).

Tim Dokchitser (Cambridge) has implemented efficient computation of many kinds of L-functions, including those attached to Dirichlet characters, number fields, Artin representations, elliptic curves and hyperelliptic curves. Vladmir Dokchitser (Cambridge) has contributed theoretical ideas.

Anton Mellit (Bonn) has contributed code for computing symmetric powers and tensor products of L-functions.

Bartosz Nasrecki (Bristol) has contributed a package for computing associated schemes in product projective spaces. This finds application in the study of hypergeometric motives.

A package for computing with Jacobi sum motives has been developed by Mark Watkins (Magma) in conjunction with David Roberts (University of Minnesota Morris) and Fernando Rodriguez-Villegas (ICTP, Trieste).