Modular group

Modular group

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Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016

Here we let τ vary in the complex upper half plane, the complex numbers with positive imaginary part. By construction we have ΘΓ(τ + 1) = ΘΓ(τ). Recall that unimodular lattices are in analogy with self-dual codes. In the case of self-dual codes, the MacWilliams formula is an involutory operation, which leaves the weight enumerator invariant. The same principle applies in the setting of lattices. The classical Poisson summation formula plays the role of the MacWilliams formula and we obtain an identity relating ΘΓ(τ) and ΘΓ(−1/τ). All in all one obtains an action of a group, which in a sense leaves the theta series invariant. This group is the modular groupSL(2, ℤ), the group of all 2 × 2 matrices with integer entries and determinant ±1. Invariants under this kind of action of the modular group (or of subgroups of finite index) are known as elliptic modular forms. This is a classical and highly developed topic of number theory and complex analysis. The upshot is that modular forms form vector spaces of finite dimension. These dimensions are known and bases are known.

Dynamics of a family of continued fraction maps

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Published in Dynamical Systems, 2018

Muhammed Uludağ, Hakan Ayral

Consider the left action of on the set of functions on R defined by Set T(x) ≔ x + 1 and U(x) ≔ 1/x. Then and one may write Alternatively, Note that the sums inside the right-hand side are taken inside the group algebra of the modular group, i.e. inside the algebra

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Dynamics of a family of continued fraction maps