Computations with Modular Forms

This volume contains original research articles, survey articles and lecture notes related to the Computations with Modular Forms 2011 Summer School and Conference, held at the University of Heidelberg. A key theme of the Conference and Summer School was the interplay between theory, algorithms and experiment.

The 14 papers offer readers both, instructional courses on the latest algorithms for computing modular and automorphic forms, as well as original research articles reporting on the latest developments in the field.

The three Summer School lectures provide an introduction to modern algorithms together with some theoretical background for computations of and with modular forms, including computing cohomology of arithmetic groups, algebraic automorphic forms, and overconvergent modular symbols.

The 11 Conference papers cover a wide range of themes related to computations with modular forms, including lattice methods for algebraic modular forms on classical groups, a generalization of the Maeda conjecture, an efficient algorithm for special values of p-adic Rankin triple product L-functions, arithmetic aspects and experimental data of Bianchi groups, a theoretical study of the real Jacobian of modular curves, results on computing weight one modular forms, and more.

Unites lecture notes and research articles; lecture notes prepare the reader for most research articles

Part I Summer School: Lectures on Computing Cohomology of Arithmetic Groups by P.E. Gunnells.- Computing with Algebraic Automorphic Forms by D. Loeffler.- Overconvergent Modular Symbols by R. Pollack.- Part II Conference and Research Contributions: Congruence Subgroups, Cusps and Manin Symbols over Number Fields by J. E. Cremona and M. T. Aranés.- Computing Weight One Modular Forms over C and Fp by K. Buzzard.- Lattice Methods for Algebraic Modular Forms on Classical Groups by M. Greenberg and J. Voight.- Efficient Computation of Rankin p-Adic L-Functions by A.G.B. Lauder.- Formes Modulaires Modulo 2 et Composantes Réelles de Jacobiennes Modulaires by L.Merel.- Universal Hecke L-Series Associated with Cuspidal Eigenforms over Imaginary Quadratic Fields by A. Mohamed.- On Higher Congruences Between Cusp Forms and Eisenstein Series by B. Naskrecki.- Arithmetic Aspects of Bianchi Groups by M.H.Sengün.- A Possible Generalization of Maeda's Conjecture by P. Tsaknias.- Computing Power Series Expansions of Modular Forms by J. Voight and J. Willis.- Computing Modular Forms for GL2 over Certain Number Fields by D. Yasaki.