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We construct generators for modules of vector-valued Picard modular forms on a unitary group of type (2, 1) over the Eisenstein integers. We also calculate eigenvalues of Hecke operators acting on cusp forms.
Modular forms are functions with an enormous amount of symmetry that play a central role in number theory, connecting it with analysis and geometry. They have played a prominent role in mathematics since the 19th century and their study continues to flourish today. Modular forms formed the inspiration for Langlands' conjectures and play an important role in the description of the cohomology of varieties defined over number fields. This collection of up-to-date articles originated from the conference 'Modular Forms' held on the Island of Schiermonnikoog in the Netherlands. A broad range of topics is covered including Hilbert and Siegel modular forms, Weil representations, Tannakian categories and Torelli's theorem. This book is a good source for all researchers and graduate students working on modular forms or related areas of number theory and algebraic geometry.
This volume grew out of a very succesful conference on Modular Forms that was held in October 2006 on the Dutch island of Schiermonnikoog and that was organised with financial support from the Foundation Compositio Mathematica. For some of the participants the journey to the island was a long one, but once on the island this was soon forgotten, and we look back at a very pleasant conference in beautiful surroundings. We thank the Foundation Compositio Mathematica for making this conference possible.
The present volume contains, in addition to an introduction by the editors, sixteen refereed papers, not necessarily related to lectures at the conference. We thank all authors and all referees for their contributions.
By using the Grothendieck–Riemann–Roch theorem we derive cycle relations modulo algebraic equivalence in the Jacobian of a curve. The relations generalize the relations found by Colombo and van Geemen and are analogous to but simpler than the relations recently found by Herbaut. In an appendix by Zagier, it is shown that these sets of relations are equivalent.