Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Algebra --- Hecke operators --- Trace formulas. --- Opérateurs de Hecke --- Formules de trace --- Hecke operators. --- 51 <082.1> --- Mathematics--Series --- Opérateurs de Hecke --- Trace formulas --- Formulas, Trace --- Automorphic forms --- Discontinuous groups --- Representations of groups --- Operators, Hecke --- Forms, Modular --- Operator theory

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

In 1938, at the Institute for Advanced Study, E Hecke gave a series of lectures on his theory of correspondence between modular forms and Dirichlet series. Since then, the Hecke correspondence has remained an active feature of number theory and, indeed, it is more important today than it was in 1936 when Hecke published his original papers. This book is an amplified and up-to-date version of the former author's lectures at the University of Illinois at Urbana-Champaign, based on Hecke's notes. Providing many details omitted from Hecke's notes, it includes various new and important developments

Dirichlet series. --- Forms (Mathematics) --- Modular functions. --- Hecke operators. --- Operators, Hecke --- Forms, Modular --- Operator theory --- Functions, Modular --- Elliptic functions --- Group theory --- Number theory --- Quantics --- Algebra --- Mathematics --- Series, Dirichlet --- Series

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Local Newforms for GSp(4) describes a theory of new- and oldforms for representations of GSp(4) over a non-archimedean local field. This theory considers vectors fixed by the paramodular groups, and singles out certain vectors that encode canonical information, such as L-factors and epsilon-factors, through their Hecke and Atkin-Lehner eigenvalues. While there are analogies to the GL(2) case, this theory is novel and unanticipated by the existing framework of conjectures. An appendix includes extensive tables about the results and the representation theory of GSp(4).

Representations of groups. --- Automorphic forms. --- Hecke operators. --- Représentations de groupes --- Formes automorphes --- Opérateurs de Hecke --- Representations of groups --- Automorphic forms --- Hecke operators --- Mathematical Theory --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Operators, Hecke --- Group representation (Mathematics) --- Groups, Representation theory of --- Mathematics. --- Algebra. --- Topological groups. --- Lie groups. --- Number theory. --- Number Theory. --- Topological Groups, Lie Groups. --- Number study --- Numbers, Theory of --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Topological --- Continuous groups --- Mathematical analysis --- Math --- Science --- Forms, Modular --- Operator theory --- Automorphic functions --- Forms (Mathematics) --- Group theory --- Topological Groups.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

"Modular forms are tremendously important in various areas of mathematics, from number theory and algebraic geometry to combinatorics and lattices. Their Fourier coefficients, with Ramanujan's tau-function as a typical example, have deep arithmetic significance. Prior to this book, the fastest known algorithms for computing these Fourier coefficients took exponential time, except in some special cases. The case of elliptic curves (Schoof's algorithm) was at the birth of elliptic curve cryptography around 1985. This book gives an algorithm for computing coefficients of modular forms of level one in polynomial time. For example, Ramanujan's tau of a prime number P can be computed in time bounded by a fixed power of the logarithm of P. Such fast computation of Fourier coefficients is itself based on the main result of the book: the computation, in polynomial time, of Galois representations over finite fields attached to modular forms by the Langlands program. Because these Galois representations typically have a nonsolvable image, this result is a major step forward from explicit class field theory, and it could be described as the start of the explicit Langlands program. The computation of the Galois representations uses their realization, following Shimura and Deligne, in the torsion subgroup of Jacobian varieties of modular curves. The main challenge is then to perform the necessary computations in time polynomial in the dimension of these highly nonlinear algebraic varieties. Exact computations involving systems of polynomial equations in many variables take exponential time. This is avoided by numerical approximations with a precision that suffices to derive exact results from them. Bounds for the required precision--in other words, bounds for the height of the rational numbers that describe the Galois representation to be computed--are obtained from Arakelov theory. Two types of approximations are treated: one using complex uniformization and another one using geometry over finite fields. The book begins with a concise and concrete introduction that makes its accessible to readers without an extensive background in arithmetic geometry. And the book includes a chapter that describes actual computations"-- "This book represents a major step forward from explicit class field theory, and it could be described as the start of the 'explicit Langlands program'"--

Galois modules (Algebra) --- Class field theory. --- Algebraic number theory --- Galois module structure (Algebra) --- Galois's modules (Algebra) --- Modules (Algebra) --- Arakelov invariants. --- Arakelov theory. --- Fourier coefficients. --- Galois representation. --- Galois representations. --- Green functions. --- Hecke operators. --- Jacobians. --- Langlands program. --- Las Vegas algorithm. --- Lehmer. --- Peter Bruin. --- Ramanujan's tau function. --- Ramanujan's tau-function. --- Ramanujan's tau. --- Riemann surfaces. --- Schoof's algorithm. --- Turing machines. --- algorithms. --- arithmetic geometry. --- arithmetic surfaces. --- bounding heights. --- bounds. --- coefficients. --- complex roots. --- computation. --- computing algorithms. --- computing coefficients. --- cusp forms. --- cuspidal divisor. --- eigenforms. --- finite fields. --- height functions. --- inequality. --- lattices. --- minimal polynomial. --- modular curves. --- modular forms. --- modular representation. --- modular representations. --- modular symbols. --- nonvanishing conjecture. --- p-adic methods. --- plane curves. --- polynomial time algorithm. --- polynomial time algoriths. --- polynomial time. --- polynomials. --- power series. --- probabilistic polynomial time. --- random divisors. --- residual representation. --- square root. --- square-free levels. --- tale cohomology. --- torsion divisors. --- torsion.