Modular Forms

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This can be avoided with an extra variable substitution, for instance. The meaning of the parameters is as follows:. The level N can be replaced by a vector of allowed levels. The result is a vector of newforms f matching the search criteria, sorted by increasing level then increasing D. In other words it is normalized so that its linear coefficient is 1. If the valuation v at infinity is fractional, return 0. If L has rational coefficients, this will be enough to recognize the modular form in mf :. If L has inexact complex coefficients, one can for instance compute an eigenbasis for mf and check whether one of the attached L-function is reasonably close to L.

In the example, we cheat by producing the L function from an eigenform in a known space, but the function does not use this information:. Types A 4 , S 4 , A 5 are represented by minus their cardinality , , or , and type D n is represented by its cardinality, the integer 2n:. We take T n at the same level as the one used in mf.

Modular forms of half-integral weight are supported, in which case n must be a perfect square, else T n will act as 0 the operator T p for p N is not supported yet :. When n is a large composite, resp.

Modular Forms: A Classical Approach

If it is a vector, vector of such matrices, usually faster than calling each one individually. Alternatively, it can be a modular form F or modular form space, in which case we use mfparams to define the space parameters. For given level and weight, it is advantageous to compute simultaneously spaces attached to different Galois orbits of characters, especially in weight 1.

In weight 1, only S 1 new , S 1 and E 1 support wildcards. The structure is entirely algebraic and does not depend on the current realbitprecision. If flag is set, also return 0 if the eta quotient is not holomorphic. See mffrometaquo.

Arithmetic of Modular forms and Elliptic Curves

Since it gives a bijection of Hecke modules, this matrix can be used to transport modular form data from the easily computed space of level N and weight 2k-1 to the more difficult space of level 4N and weight k: matrices of Hecke operators, new space, splitting into eigenspaces and eigenforms. If we want the eigenforms, we write instead:. These are the two eigenfunctions in the space mf , the first resp.

The function mfkohneneigenbasis does this directly. This last construction allows to replace a general modular form by a simpler linear combination of basis functions, which is often more efficient:. If you want the true level of F from this result, use mfconductor mfinit F ,F.

  • Computational Aspects of Modular Forms and Galois Representations.
  • Organizing committee!
  • Modular Forms: A Classical Approach.

If flag is 0, ordinary period polynomial. If it is 1 or -1, even or odd part of that polynomial.

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If gs is omitted it is understood to be equal to fs. Note that f and g can both be noncuspidal, in which case the program returns an error if the product is divergent. The output is a list of matching forms with that given level and weight. The forms are sorted by increasing D. Updates and corrections are available for the book; please see errata. According to the preface, the authors expect the main use of this book to be for advanced graduate students to learn about the classical theory of modular forms.

However, given the tremendous amount of detail provided, the book should also be useful as a reference for established researchers in the area. Further, it can undoubtedly be mined by instructors for a graduate course on modular forms. This book gives a beautiful introduction to the theory of modular forms, with a delicate balance of analytic and arithmetic perspectives. They finish by showing us the frontiers of modern research, covering topics generalizing the classical theory in a variety of directions. Throughout, the authors expertly weave fine details with broad perspective.

The target readership for this text is graduate students in number theory, though it will also be accessible to advanced undergraduates and will, no doubt, serve as a valuable reference for researchers for years to come. This marvelous book is a gift to the mathematical community and more specifically to anyone wanting to learn modular forms.

The authors take a classical view of the material offering extremely helpful explanations in a generous conversational manner and covering such an impressive range of this beautiful, deep, and important subject. This book is an almost encyclopedic textbook on modular forms.

There are already numerous and some excellent books on the subject. But none of the existing books by themselves contain this much and this detailed information.

The authors' knowledge of the subject matter and the experience in writing books are clearly reflected in the end product. I would not only be very happy to use this book as a textbook next time I teach a course on modular forms, but I am also looking forward to having a hard copy in my library as an extensive reference book. Modular forms are central to many different fields of mathematics and mathematical physics. Having a detailed and complete treatment of all aspects of the theory by two world experts is a very welcome addition to the literature.

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modular forms

Join our email list. Sign up. 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.

Mock modular forms

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.

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And the book includes a chapter that describes actual computations. User Account Log in Register Help.

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