From quadratic polynomials and continued fractions to modular forms

Asymptotic bounds for cycle integrals of the j-function on Markov geodesics

“…There is a finite number of forms satisfying (5.1) with fixed discriminant. Such forms are commonly called 'simple' after Zagier [22] and play an important role in the theory of rational periods and period functions of modular forms (see [17], [24], [5], [1]). It is shown in [6] that all simple forms SL(2, Z) equivalent to a form [a, b, c] are obtained by applying iteratively the following continued fraction algorithm.…”

“…This algorithm is cyclic because the continued fraction of w is purely periodic; we denote by ℓ = ℓ w the length of the cycle, so that w (ℓ+1) = w (1) . The cycle w (1) , . .…”

“…As usual, we denote by j(z) the modular function for SL(2, Z) introduced by Klein as an invariant of elliptic curves. The group Γ Q preserves j(z) √ D Q(z,1) dz, and so we can consider the integral 1) dz and define the 'value' (also called cycle integral ) of j at w, as the complex number:…”

“…The definitions of S u and S v are different for case 1 and case 2. In case 1, the sum S u below compares the first terms w (1) , . .…”

Meromorphic analogues of modular forms generating the kernel of Shimura’s lift

“…Also the even parts of the Eichler integrals of f k,D for D > 0 have been studied because of their link with Diophantine approximation, reduction of binary quadratic forms, special values of zeta functions and Dedekind sums ( [Zag99], [Ben14]).…”