The PI and his research team are proposing a major new project to develop theory and organize methods for understanding and computing with L-functions and modular forms. Broadly speaking, they plan to chart the landscape of L-functions and modular forms in a systematic and concrete fashion. They will study these functions, develop algorithms for their computation, and test fundamental conjectures, including: the Generalized Riemann Hypothesis, the Birch and Swinnerton-Dyer conjecture, the Bloch-Kato conjecture, the correlation conjectures of Montgomery and of Rudnick-Sarnak, the density conjectures of Katz-Sarnak, automorphy of the Hasse-Weil zeta functions, and the Selberg eigenvalue conjecture. They plan to carry out a systematic study, theoretically, algorithmically, and experimentally of degree 1, 2, 3, 4 L-functions and their associated modular forms, including classical modular forms, Maass forms for GL(2), GL(3), GL(4), Siegel modular forms, and Hilbert modular forms. They will also investigate symmetric square and cube L-functions, Rankin-Selberg convolution L-functions, the Hasse-Weil L-functions of algebraic varieties, Artin L-functions associated to 3- and 4-dimensional Galois representations, and, less systematically, look at a few high degree L-functions associated to higher symmetric powers and higher dimensional Galois representations.
L-functions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. Virtually all branches of number theory have been touched by L-functions and modular forms. Besides containing deep information concerning the distribution of prime numbers and the structure of elliptic curves, they feature prominently in Andrew Wiles' solution of the famous 350-year-old Fermat's Last Theorem, and in the twentieth century classification of congruent numbers, a problem first posed by Arab mathematicians one thousand years. In spite of their central importance, mathematicians have only scratched the surface of these crucial and powerful functions. The PI and his research team are undertaking a major new project to systematically tabulate and study these functions. Their work will fall into four categories: theoretical, algorithmic, experimental, and data gathering. The theoretical work will be stimulated by their goal of charting the world of L-functions and modular forms. Their experimental work will involve testing many key conjectures concerning these functions. The project will produce a large amount of training, with plans for three graduate student schools, an undergraduate research experience, and support for a score of postdocs and graduate students who will assist in research. It will result in the creation of a vast amount of data about a wide range of modular forms and L-functions, which will far surpass in range and depth anything computed before in this area. The data will be organized in a freely available online data archive, along with the actual programs that were used to generate these tables. By providing these tables and tools online, the researchers will guarantee that the usefulness of this project will extend far beyond the circle of researchers on this FRG. The archive will be a rich source of examples and tools for researchers working on L-functions and modular forms for years to come, and will allow for future updates and expansion.
Our project `L-functions and Modular Forms' was concerned with developing theory and organizing methods for understanding and computing with L-functions and modular forms. L-functions and modular forms underlie much of twentieth century number theory and are connected to the practical applications of number theory in cryptography. Virtually all branches of number theory have been touched by L-functions and modular forms. Besides containing deep information concerning the distribution of prime numbers and the structure of elliptic curves, they feature prominently in Andrew Wiles' solution of the famous 350-year-old Fermat's Last Theorem, and in the twentieth century classification of congruent numbers, a problem first posed by Arab mathematicians one thousand years. In spite of their central importance, mathematicians have only scratched the surface of these crucial and powerful functions. Our work has resulted in the creation of a vast amount of data about a wide range of modular forms and L-functions, which surpasses in range and depth anything computed before in this area. The extensive tables that we produced provide numerical confirmation of a number of central conjectures in number theory including the Generalized Riemann Hypothesis, Birch and Swinnerton Dyer Conjecture, conjectures for the value distribution of L-functions, and conjectures concerning special values of L-functions. The research that we carried out in order to perform these record breaking computations has led to the development of exciting new algorithms and their implementations for computing with L-functions and modular forms. In order to disseminate our project to the public, we created an online data archive and interactive web pages for L- functions, modular forms, and related arithmetic and algebraic objects, at www.lmfdb.org. It represents the concerted effort of many of the project's researchers, postdocs, students, and colleagues. The LMFDB seeks to be a modern handbook for objects of research in number theory and related areas. It provides formulas, tables, graphs, and data, with an emphasis on exhibiting concrete examples and illustrating the connections between related objects. Browse and search pages provide easy access for both experts and non-experts. Every object has its own 'homepage,' which has a permanent and mathematically meaningful URL.
