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FREE LECTURES ON MODULAR FORMS 1ST EDITION PDF Joseph J Lehner | 9780486821405 | | | | | Lectures on Modular Forms - Robert C. Gunning - Google книги In mathematicsa modular form is a complex analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular groupand also satisfying a growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topologysphere packingand string theory. Instead, modular functions are meromorphic that is, they are almost holomorphic except for a set of isolated points. Modular form theory is a special case of the more general theory of automorphic formsand therefore can now be seen as just the most concrete part of a rich theory of discrete groups. Modular forms can also be interpreted as sections of a specific line bundles on modular varieties. The dimensions of these spaces of modular forms can be computed using the Riemann—Roch theorem [2]. A Lectures on Modular Forms 1st edition form of weight k for the modular group. A modular form can equivalently be defined as a function F from the set of lattices in C to the set of complex numbers which satisfies certain conditions:. The simplest examples from this point of view are the Eisenstein series. Then E k is a modular form of weight k. An even unimodular lattice L in R n is a lattice generated by n Lectures on Modular Forms 1st edition forming the columns of a matrix of determinant 1 and satisfying the condition that the square of the length of each vector in L is an even integer. The so-called theta function. It is not so easy to construct even unimodular lattices, but here is one way: Let n be an integer divisible by 8 and consider all vectors v in R n such that 2 v has integer coordinates, either all even or all odd, and such that the Lectures on Modular Forms 1st edition of the coordinates of v is an even integer. We call this lattice L n. Because there is only one modular form of weight 8 up to scalar multiplication. John Milnor observed that the dimensional tori obtained by dividing R 16 by these two lattices are consequently examples of compact Riemannian manifolds which are isospectral but not isometric see Hearing the shape of a drum. The Dedekind eta function is defined as. The presence of 24 is related to the fact that the Leech lattice has 24 dimensions. This was confirmed by the work of Eichler, Shimura, Kuga, Ihara, and Pierre Deligne as a result of Deligne's proof of the Weil conjectureswhich were shown to imply Ramanujan's conjecture. The second and third examples give some hint of the connection between modular forms and classical questions in number theory, such as representation of integers Lectures on Modular Forms 1st edition quadratic forms and the partition function. The crucial conceptual link between modular forms and number theory is furnished by the theory of Hecke operatorswhich also gives the link between the theory of modular forms and representation theory. When the weight k is zero, it can be shown using Liouville's theorem that the only modular forms are constant functions. However, relaxing the requirement that f be Lectures on Modular Forms 1st edition leads to the notion of modular functions. This is also referred to as the q -expansion of f. Thus, a modular function can also be regarded as a meromorphic function on the set of isomorphism Lectures on Modular Forms 1st edition of elliptic curves. For example, the j-invariant j z of an elliptic curve, regarded as a function on the set of all elliptic curves, is a modular function. More conceptually, modular functions can be thought of as functions on the moduli space of isomorphism classes of complex elliptic curves. A modular unit is a modular function whose poles and zeroes are confined to the cusps. The functional equation, i. Let G be a subgroup of SL 2, Z that is of finite index. Typically it is not compact, but can be compactified by adding a finite number of points called cusps. These are points at the boundary of Hi. What is more, it can be endowed with the structure of a Riemann surfacewhich allows one to speak of holo- and meromorphic functions. Important examples are, for any positive integer Neither one of the congruence subgroups. A modular form for G of weight k is a function on H satisfying the above functional equation for all matrices in Gthat is holomorphic on H and at all cusps of G. Again, modular forms that vanish at all cusps are called cusp forms for G. The C -vector spaces of modular and cusp forms of weight k are denoted M k G and S k Grespectively. For example, the spaces M k G and S k G are finite-dimensional, and their dimensions can be computed thanks to the Riemann-Roch theorem in terms of the geometry of the G -action on H. The modular functions constitute the field of functions of the Riemann surface, and hence form a field of transcendence degree one over C. Unfortunately, the only such functions are constants. If we allow denominators rational functions instead of polynomialswe can let F be the ratio of two homogeneous polynomials of the same degree. The solutions are then the homogeneous polynomials of degree k. One might ask, since the homogeneous polynomials are not really functions on P Vwhat are they, geometrically speaking? The algebro-geometric answer is that they are sections of a sheaf one could also say a line bundle in this case. The situation with modular forms is precisely analogous. Modular forms can also be profitably approached from this geometric direction, as sections of Lectures on Modular Forms 1st edition bundles on the moduli space of elliptic curves. Rings of modular forms of congruence subgroups of SL 2, Z are finitely generated due to a result of Pierre Deligne and Michael Rapoport. Such rings of modular forms are generated in weight at most 6 and the relations are generated in weight at most 12 when the congruence subgroup has nonzero odd weight modular forms, and the corresponding bounds are 5 and 10 when there are no nonzero odd weight modular forms. More generally, there are formulas for bounds on the weights of generators of the ring of modular forms and its relations for arbitrary Fuchsian groups. If f is meromorphic but not holomorphic at the cusp, it is called a non-entire modular form. The other forms are called old forms. A cusp form is a modular form with a zero constant coefficient in its Fourier series. It is called a cusp form because the form vanishes at all cusps. Maass forms are real-analytic eigenfunctions of the Laplacian but need not be holomorphic. The holomorphic parts of certain weak Maass wave forms turn out to be essentially Ramanujan's mock theta functions. Groups which are not subgroups of SL 2, Z can be considered. Siegel modular forms Lectures on Modular Forms 1st edition associated to larger symplectic groups in the same way in which classical modular forms are associated to SL 2, R ; in other words, they are related to abelian varieties in the same sense that Lectures on Modular Forms 1st edition modular forms which are sometimes called elliptic modular forms to emphasize the point are related to elliptic curves. Jacobi forms are a mixture of modular forms and elliptic functions. Examples of such functions are very classical - the Jacobi theta functions and the Fourier coefficients of Siegel modular forms of genus two - but it is a relatively recent observation that the Jacobi forms have an arithmetic theory very analogous to the usual theory of modular forms. Automorphic forms extend the notion of modular forms to general Lie groups. Modular integrals of weight k are meromorphic functions on the upper half plane of moderate growth at infinity which fail to be modular of weight Lectures on Modular Forms 1st edition by a rational function. The theory of modular forms was developed in four periods: first in connection with the theory of elliptic functionsin the first part of the nineteenth century; then by Felix Klein and others towards the end of the nineteenth century as the automorphic form concept became understood for one variable ; then by Erich Hecke from about ; and then in the s, as the needs of number theory and the formulation of the modularity theorem in particular made it clear that modular forms are deeply implicated. From Wikipedia, the free encyclopedia. Analytic function on the upper half-plane with a certain behavior under the modular group. A distinct use of this term appears in relation to Haar measure. Main article: Ring of modular forms. Main Lectures on Modular Forms 1st edition Atkin—Lehner theory. Main article: Cusp form. This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. October Learn how and when to remove this template message. Archived PDF from the Lectures on Modular Forms 1st edition on 1 August Archived PDF from the original on 31 July Topics in algebraic curves. Five points determine a conic Projective line Rational normal curve Riemann sphere Twisted cubic. Elliptic function Elliptic integral Fundamental pair of periods Modular form. Counting points on elliptic curves Division polynomials Hasse's theorem on elliptic curves Mazur's torsion theorem Modular elliptic curve Modularity theorem Mordell—Weil theorem Nagell—Lutz theorem Supersingular elliptic curve Schoof's algorithm Schoof—Elkies—Atkin algorithm.
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