Modular forms modulo p
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In mathematics, modular forms are functions on the Upper half-plane. Their study is part of complex analysis. However, once reduced modulo 2, modular forms create a completely different theory, which may be classified as Algebraic number theory.
Reduction of Modular Forms Modulo 2
Conditions to Reduce Modulo 2
Modular forms are analytic functions, so they admit a Fourier series. As modular forms also satisfy a certain kind of functional equation with respect to the group action of the modular group, this Fourier series may be expressed in terms of . So if is a modular form, then there are coefficients such that . To reduce modulo 2, consider the subspace of modular forms with coefficients of the -series being all integers (since complex numbers, in general, may not be reduced modulo 2). It is then possible to reduce all coefficients modulo 2, which will give a modular form modulo 2.
Basis for Modular Forms Modulo 2
Modular forms are generated by and [1]. It is the possible to normalize and to and , having integers coeffieints in their -series. This gives generators for modular forms, which may be reduced modulo 2. Note the Miller basis has some interesting properties [2]. Once reduced modulo 2, and are just . That is, a trivial reduction. To get a non-trivial reduction, mathematicians use the modular discriminant . It is introduced as a "priority" generator before and . Thus, modular forms are seen as polynomials of , and (over the complex in general, but seen over integers for reduction), once reduced modulo 2, they become just polynomials of over .
The Modular Discriminant Modulo 2
The modular discriminant is defined by an infinite product: The coefficients taht matches are usually denoted , and correspond to the Ramanujan tau function. Results from Kolberg[3] and Jean-Pierre Serre[4] allows to show that modulo 2, we have: i.e., the -series of modulo 2 consists of to powers of odd squares.
Hecke Operators Modulo 2
Hecke operators are commonly considered as the most important operators acting on modular forms. It is therefore justified to try to reduce them modulo 2.
The Hecke operators for a modular form are defined as follows[5]: with .
Hecke operators may be defined on the -series as follows[6]: if , then with .
Since modular forms were reduced using the -series, it makes sense to use the -series definition. The sum simplifies a lot for Hecke operators of primes (i.e. when is prime): there are only two summands. This is very nice for reduction modulo 2, as the formula simplifies a lot. With more than 2 summands, there would be many cancellations modulo 2, and the legitimacy of the process would be doubtable. Thus, Hecke operators modulo 2 are usually defined only for primes numbers.
With a modular form modulo 2 with -representation , the Hecke operator on is defined by where
It is important to note that Hecke operators modulo 2 have the interesting property of being nilpotent. Finding their order of nilpotency is a problem solved by Jean-Pierre Serre and Jean-Louis Nicolas in [7].
The Hecke Algebra Modulo 2
The Hecke algebra may also be reduced modulo 2. It is defined to be the algebra generated by Hecke operators modulo 2, over .
Following Serre and Nicolas's notations from [8]: , i.e. . Writing so that , define as the -subalgebra of given by and .
That is, if is a sub-vector-space of , we get .
Finlly, define the Hecke algebra as follows: as , one can restrict elements of to to obtain an element of . When considering the map as the restriction to , then is a homomorphism. As is either identity or zero, . Therefore, the following chain is obtained: . Then, define the Hecke algebra to be the projective limit of the above as . Explicitly, this means .
The main property of the Hecke algebra is that it is generated by as series of and [9]. That is: .
So for any prime , it is possible to find coefficients such that:
References
- ^ Stein, William (2007). Modular Forms, a Computational Approach. Theorem 2.17: Graduate Studies in Mathematics. ISBN 978-0-8218-3960-7.
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: CS1 maint: location (link) - ^ Stein, William (2007). Modular Forms, a Computational Approach. Lemma 2.20: Graduate Studies in Mathematics. ISBN 978-0-8218-3960-7.
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: CS1 maint: location (link) - ^ Kolberg, O. (1962). Congruences for Ramanujan's function . Arbok Univ. Bergen Mat.-Natur. Ser. p. 8. ISSN 0522-9189.
- ^ Serre, Jean-Pierre (1973). A course in arithmetic. Springer-Verlag, New York-Heidelberg. p. 96. ISBN 978-1-4684-9884-4.
- ^ Serre, Jean-Pierre (1973). A course in arithmetic. Springer-Verlag, New York-Heidelberg. p. 100. ISBN 978-1-4684-9884-4.
- ^ Serre, Jean-Pierre (1973). A course in arithmetic. Springer-Verlag, New York-Heidelberg. p. 100. ISBN 978-1-4684-9884-4.
- ^ Nicolas, Jean-Louis; Serre, Jean-Pierre (2012). "Formes modulaires modulo 2: l'ordre de nilpotence des opérateurs de Hecke". C. R. Math. Acad. Sci. Paris. 350: 343–348. doi:10.1016/j.crma.2012.03.013. ISSN 1631-073X. Retrieved 2020-03-23.
- ^ Nicolas, Jean-Louis; Serre, Jean-Pierre (2012). "Formes modulaires modulo 2: structure de l'algèbre de Hecke". C. R. Math. Acad. Sci. Paris. 350: 449--454. doi:10.1016/j.crma.2012.03.019. ISSN 1631-073X. Retrieved 2020-03-23.
- ^ Nicolas, Jean-Louis; Serre, Jean-Pierre (2012). "Formes modulaires modulo 2: structure de l'algèbre de Hecke". C. R. Math. Acad. Sci. Paris. 350: 449--454. doi:10.1016/j.crma.2012.03.019. ISSN 1631-073X. Retrieved 2020-03-23.