A continuity of cycle integrals of modular functions
- PDF / 283,313 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 14 Downloads / 196 Views
A continuity of cycle integrals of modular functions Yuya Murakami1 Received: 11 November 2019 / Accepted: 27 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we study a continuity of the “values” of modular functions at the real quadratic numbers which are defined in terms of their cycle integrals along the associated closed geodesics. Our main theorem reveals a more finer structure of the continuity of these values with respect to continued fraction expansions and it turns out that it is different from the continuity with respect to Euclidean topology. Keywords Elliptic modular functions · Cycle integrals · j-Invariant · Values at real quadratic numbers · Continued fractions Mathematics Subject Classification 11F03
1 Introduction √ Let H := {z = x + y −1 ∈ C | x, y ∈ R, y > 0} be the upper half plane and f : H → C be a holomorphic modular function with respect to SL2 (Z). Kaneko [4] introduced the “value” of f at any real quadratic number w as follows. For such a w and its non-trivial Galois conjugate w we define 1 1 dz. ηw := − z − w z−w ∗∗ Let SL2 (Z)w be the stabilizer of w and γw = be the unique element of SL2 (Z)w cd n such that SL2 (Z)w = {±γw | n ∈ Z}, εw := j(γw , w) := cw + d > 1. For z 0 ∈ H we define γw z 0 γw z 0 f ηw , 1(w) := ηw . f (w) := z0
B 1
z0
Yuya Murakami [email protected] Mathematical Institute, Tohoku University, 6-3, Aoba, Aramaki, Aoba-Ku, Sendai 980-8578, Japan
123
Y. Murakami
The integral is independent of z 0 since f ◦γw = f , γw∗ ηw = ηw and f is holomorphic. We set f (w) :=
f (w) . 1(w)
This value has a representation as a cycle integral 1 f ds f (w) = length(SL2 (Z)w \Sw ) SL2 (Z)w \Sw where ds := y −1 d x 2 + dy 2 , Sw is the geodesic in H from w to w and length(SL2 (Z)w \Sw ) :=
SL2 (Z)w \Sw
ds = 1(w).
The values f (w) and f (w) have been studied in the last decade. Duke, Imamo¯glu and Tóth [3] proved that a generating function of traces of these values is a mock modular form of weight 1/2 for 0 (4). This work is a real quadratic analog of Zagier’s work on traces of singular moduli in [7]. Masri [5] studied equidistribution of traces of these values. Päpcke [6] showed the interesting formula lim j1
n→∞
n+
√ n2 − 4 = 0. 2
We can regard it as a certain continuity of f (w). We remark that the real quadratic numbers in the left-hand side have the continued fraction expansions n+
√ n2 − 4 =n+ 2
1
.
1
n+
n+
1 .. .
Kaneko [4] conjectured another continuity of f (w) when w is a Markov quadratic number and this conjecture is proved by Bengoechea and Imamo¯glu [2]. In this paper, we pursue more finer structure in the continuity with respect to the continued fraction expansions. Our main result is to show that the function f (w) has a continuity with respect to how many cyclic parts appear in a continued fraction expansion of w. Each real number can be represented by the unique continued fraction 1
[k1 , k2 , k3 , . . . ] := k1 + k2 +
1 k3 +
123
1 .. .
Data Loading...
