In mathematics, modular forms are particular complex analytic functions on the upper half-plane of interest in complex analysis and number theory. When reduced modulo a prime p, there is an analogous theory to the classical theory of complex modular forms and the p-adic theory of modular forms.
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 q = e 2 π i z {\displaystyle q=e^{2\pi iz}} . So if f {\displaystyle f} is a modular form, then there are coefficients c ( n ) {\displaystyle c(n)} such that f ( z ) = ∑ n ∈ N c ( n ) q n {\displaystyle f(z)=\sum _{n\in \mathbb {N} }c(n)q^{n}} . To reduce modulo 2, consider the subspace of modular forms with coefficients of the q {\displaystyle q} -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 G 4 {\displaystyle G_{4}} and G 6 {\displaystyle G_{6}} . It is then possible to normalize G 4 {\displaystyle G_{4}} and G 6 {\displaystyle G_{6}} to E 4 {\displaystyle E_{4}} and E 6 {\displaystyle E_{6}} , having integers coefficients in their q {\displaystyle q} -series. This gives generators for modular forms, which may be reduced modulo 2. Note the Miller basis has some interesting properties: once reduced modulo 2, E 4 {\displaystyle E_{4}} and E 6 {\displaystyle E_{6}} are just 1 {\displaystyle 1} ; that is, a trivial reduction. To get a non-trivial reduction, one must use the modular discriminant Δ {\displaystyle \Delta } . Thus, modular forms are seen as polynomials of E 4 {\displaystyle E_{4}} , E 6 {\displaystyle E_{6}} and Δ {\displaystyle \Delta } (over the complex C {\displaystyle \mathbb {C} } in general, but seen over integers Z {\displaystyle \mathbb {Z} } for reduction), once reduced modulo 2, they become just polynomials of Δ {\displaystyle \Delta } over F 2 {\displaystyle \mathbb {F} _{2}} .
The modular discriminant modulo 2 The modular discriminant is defined by an infinite product,
Δ ( q ) = q ∏ n = 1 ∞ ( 1 − q n ) 24 = ∑ n = 1 ∞ τ ( n ) q n , {\displaystyle \Delta (q)=q\prod _{n=1}^{\infty }(1-q^{n})^{24}=\sum _{n=1}^{\infty }\tau (n)q^{n},}
where τ ( n ) {\displaystyle \tau (n)} is the Ramanujan tau function. Results from Kolberg and Jean-Pierre Serre demonstrate that, modulo 2, we have
Δ ( q ) ≡ ∑ m = 0 ∞ q ( 2 m + 1 ) 2 mod 2 {\displaystyle \Delta (q)\equiv \sum _{m=0}^{\infty }q^{(2m+1)^{2}}{\bmod {2}}} i.e., the q {\displaystyle q} -series of Δ {\displaystyle \Delta } modulo 2 consists of q {\displaystyle q} to powers of odd squares.
Hecke operators modulo 2 The action of the Hecke operators is fundamental to understanding the structure of spaces of modular forms. It is therefore justified to try to reduce them modulo 2. The Hecke operators for a modular form f {\displaystyle f} are defined as follows:
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