On a class of elliptic functions associated with even Dirichlet characters
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On a class of elliptic functions associated with even Dirichlet characters Dandan Chen1,2 · Rong Chen1,2 Received: 9 June 2019 / Accepted: 27 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We construct a class of companion elliptic functions associated with even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function ℘ (z|τ ) as a blueprint, we will derive their representations in terms of qseries and partial fractions. We also explore the significance of the coefficients of their power series expansions and establish the modular properties under the actions of the arithmetic groups 0 (N ) and 1 (N ). Keywords Elliptic function · Theta function · Eisenstein series Mathematics Subject Classification 33E05 · 14H42 · 11M36
1 Introduction We will consider a pair of companion elliptic functions generated from the twisting of the logarithmic derivative of the Jacobi theta function θ1 (z|τ ) by even Dirichlet characters over certain subgroups of the period lattice. We first mention some familiar properties for Dirichlet characters and the Jacobi theta functions which can be found in standard literature.
The authors were supported in part by the National Natural Science Foundation of China (Grant No. 11971173) and ECNU Short-term Overseas Research Scholarship for Graduate Students (Grant Nos. 201811280046 and 201811280047).
B
Rong Chen [email protected] Dandan Chen [email protected]
1
School of Mathematical Sciences, East China Normal University, Shanghai, People’s Republic of China
2
Department of Mathematics, University of Florida, Gainesville, FL 32601, USA
123
D. Chen, R. Chen
Let N be a positive integer and χ be a Dirichlet character modulo N , which is extended to the set of integers Z. For all integers m and n, it satisfies the following properties: (1) (2) (3) (4)
χ (1) = 1, χ (n + N ) = χ (n), χ (mn) = χ (m)χ (n), χ (n) = 0 if gcd(n, N ) > 1.
We say χ is even if χ (−1) = 1 and odd if χ (−1) = −1. Let N be a positive integer which is divisible by N . For any character χ modulo N , we can form a character χ modulo N as follows:
χ (k) =
χ (k) if gcd(k, N ) = 1, 0 if gcd(k, N ) > 1.
We say that χ is induced by the character χ . Let χ be a character modulo N . If there is a proper divisor d of N and a character modulo d which induces χ , then the character χ is said to be non-primitive, otherwise it is called primitive. Define the Gauss sum gn (χ ) =
N −1
χ (k)e2iπ nk/N .
k=1
From [4, p. 334], if χ is primitive, then gn (χ ) = χ (n)g1 (χ ). Definition 1.1 (Cf. [12, p. 166]) Jacobi theta functions θ j for j = 1, 2, 3, 4 are defined as θ1 (z|τ ) = 2q 1/8 θ2 (z|τ ) = 2q 1/8
∞ n=0 ∞
(−1)n q n(n+1)/2 sin(2n + 1)z, θ3 (z|τ ) = 1 + 2
∞
2 q n /2 cos 2nz,
n=1
q n(n+1)/2 cos(2n + 1)z, θ4 (z|τ ) = 1 + 2
n=0
∞
2 (−1)n q n /2 cos 2nz,
n=1
where q = exp(2πiτ ) with τ > 0. The infinite product representations of theta functions are given by the following proposition. Proposition 1
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