On the modular forms of weight 1/2 over algebraic number fields

doi:10.1016/j.jnt.2021.04.009

Journal of Number Theory

Volume 229, December 2021, Pages 364-385

https://doi.org/10.1016/j.jnt.2021.04.009 Get rights and content

Abstract

Serre and Stark succeeded in deciding a basis of the space of modular forms of weight 1/2 over the rational number field. Achimescu and Saha generalized their result to the case of modular forms of weight 1/2 over totally real algebraic number fields. Gove also solved this problem in the case of modular forms of weight 1/2 over imaginary quadratic fields.

In this paper, we determine an explicit basis of the space of modular forms of weight 1/2, level c and character ψ over algebraic number fields. We prove our assertion using their arguments and Shimura's transformation formula of theta series over algebraic number fields.

Introduction

In [8], Serre and Stark succeeded in determining a basis of the space of modular forms of weight 1/2, level N and character ψ over the rational number field in terms of certain theta series. This was an affirmative answer for a question given in [9]. Using the method of representation theory, Gelbart and Piatetski-Shapiro [2] proved that the space of cusp forms of weight 1/2 over algebraic number fields is spanned by theta series. However, this result does not give not only an explicit basis but also the explicit level of theta series. In [10], Shimura also verified that the space of modular forms of weight 1/2 over totally real number fields is spanned by theta series. But, no specific basis and its exact level was revealed in this article.

On the other hand, Gove [3] showed an explicit basis of the space of modular forms of weight 1/2, level N and character ψ over imaginary quadratic number fields of class number one. Moreover, Achimescu and Saha [1] gave an explicit basis of the space of such forms over totally real number fields of narrow class number one.

The purpose of this paper is to determine an explicit basis of the space $M (c, ψ)$ of modular forms of weight 1/2, level c and character ψ over arbitrary algebraic number fields of narrow class number one. We shall give an affirmative solution in this problem. Though the proof of our results is basically the same as that of [8], we need to overcome some technical difficulties which does not arise in the case of the rational number field. The overall structure of this article is as follows.

Section 1 is a preliminary section.

In section 2, we shall introduce modular forms of weight 1/2 and theta series and discuss a basic fact concerning the automorphic factor $h (γ, z)$ of the theta series attached to number fields. There, using [10] and [11], we determine an explicit expression of $h (γ, z)$ in a necessary form for our later arguments. It is crucial for our argument in section 4.

In section 3, we consider Hecke operators acting on the space $M (c, ψ)$ . In Proposition 3.3 and Theorem 3.4, we shall prove that almost all Fourier coefficients of Hecke eigen-forms of weight 1/2 of Hecke operators defined at almost all primes are explicitly determined. It is a key theorem for our study.

In section 4, we shall investigate about symmetric operators and some operators. Using the above explicit formula of $h (γ, z)$ , we prove several properties of these operators that are crucial to the theory of new forms for modular forms of weight 1/2.

In section 5, using results in section 4, we shall prove Theorem 5.7, which is the multiplicity one theorem of new forms of weight 1/2 with respect to Hecke operators.

In section 6, we obtain our main theorem. In Theorem 6.4, using Theorem 3.4, Theorem 5.7 and functional equations of L-functions associated with modular forms of weight 1/2, we deduce that a new form of weight 1/2 is a theta series whose level and character are explicitly determined. In [1, Theorem 6.2], Achimescu and Saha confirmed this structure theorem in the case of a new form of weight 1/2 and non-split level. Adapting more precise arguments related to zero points and poles of L-functions attached to modular forms of weight 1/2, we may derive Theorem 6.4 and relieve the assumption concerning with the level of modular forms of weight 1/2 in [1, Theorem 6.2]. Applying this theorem and theorems in section 5, we establish the structure theorem of the space $M (c, ψ)$ of modular forms of weight 1/2, level c and character ψ. In particular, in Theorems 6.5 and 6.6, we shall determine a basis of $M (c, ψ)$ .

Finally, we mention that our results give a generalization of those of [1], [3] and [8].

Section snippets

Notation and preliminaries

We denote by $Z$ , $Q$ , $R$ and $C$ the ring of rational integers, the rational number field, the real number field and the complex number field, respectively. For an associative ring R with identity element, we denote by $R^{\times}$ the group of all its invertible elements and by $M_{n} (R)$ the ring of $n \times n$ -matrices with entries in R. Let ${GL}_{n} (R^{'})$ (resp. ${SL}_{n} (R^{'})$ ) denote the general linear group (resp. the special linear group) of degree n over a commutative ring $R^{'}$ with identity element. For $x = (\begin{matrix} a & b \\ c & d \end{matrix}) \in M_{2} (R)$ , we put $a_{x} = a$

Modular forms of weight 1/2 and theta series

In this section, we recall the definition of modular forms of weight 1/2. We refer to [5], [6] and [7] for our notation of modular forms of weight 1/2 over an algebraic number field. We denote by $r_{1}$ (resp. $r_{2}$ ) the cardinality of $\tilde{r}$ (resp. $\tilde{c}$ ). Let $φ_{1}, \dots, φ_{r_{1}}$ be the embeddings of F into $R$ . Furthermore, let $φ_{r_{1} + 1}, \dots, φ_{r_{1} + r_{2}}$ be the embeddings of F into $C$ with $φ_{r_{1} + i} \neq \overline{φ_{r_{1} + j}} (1 \leq i, j \leq r_{2}) .$ For $α \in F$ , we put $α^{(i)} = φ_{i} (α) (1 \leq i \leq r_{1})$ and $α^{(r_{1} + i)} = φ_{r_{1} + i} (α) (1 \leq i \leq r_{2})$ . We denote by ${GL}_{2}^{+} (F)$ the subgroup of ${GL}_{2} (F)$ consisting

Hecke operators of modular forms of weight 1/2

In this section, we recall several properties of Hecke operators of modular forms of weight 1/2. We can embed $D$ into $G$ via $γ \mapsto [γ, h (γ, z)]$ . For $γ = w_{1} t w_{2}$ with $w_{1}, w_{2} \in D$ and $t = (\begin{matrix} a^{- 1} & 0 \\ 0 & a \end{matrix})$ with some $a \in o - {0}$ , we define the function $J_{Ξ} (γ, z) = h (w_{1} w_{2}, z) .$ Let f be an element of $M (c, ψ)$ . For each totally positive prime element $π \in o$ , define a Hecke operator $T_{π^{2}}$ on $M (c, ψ)$ by $f | T_{π^{2}} = N {(π)}^{- 3 / 2} \overline{ψ_{\infty} (π)} (\sum_{b \in o / π^{2}} f | [α_{b}, J_{Ξ} (α_{b}, z)] + \overline{ψ_{c} (π)} \sum_{h \in {(o / π)}^{\times}} f | [β_{h}, J_{Ξ} (β_{h}, z)] + \overline{ψ_{c} (π^{2})} f | [α^{'}, J_{Ξ} (α^{'}, z)]),$ where $α_{b} = (\begin{matrix} π^{- 1} & 2 b / (δ π) \\ 0 & π \end{matrix}), β_{h} = (\begin{matrix} 1 & 2 h / (δ π) \\ 0 & 1 \end{matrix}) and α^{'} = (\begin{matrix} π & 0 \end{matrix}$

Symmetry operators and some operators

For any totally positive element $m \in o$ , we define the shift operator $V (m)$ by $V (m) = N {(m)}^{- 1 / 4} [(\begin{matrix} m & 0 \\ 0 & 1 \end{matrix}), N {(m)}^{- 1 / 4}] .$ We fix a totally positive generator c of c. Define the symmetry operator $W (c)$ by $W (c) = [W_{0}, J_{Ξ} (W_{0}, z)] \cdot [V_{0} (c), N {(c)}^{- 1 / 4}],$ where $W_{0} = (\begin{matrix} 0 & - 2 δ^{- 1} \\ 2^{- 1} δ & 0 \end{matrix})$ and $V_{0} (c) = (\begin{matrix} c & 0 \\ 0 & 1 \end{matrix})$ . Then we obtain $f | W (c) (z) = {(\prod_{i = 1}^{r_{1}} - \sqrt{- 1} z_{i})}^{- 1 / 2} \prod_{i = 1}^{r_{2}} {| z_{r_{1} + i} |}^{- 1} N {(2^{- 2} δ^{2} c)}^{- 1 / 4} f (W_{0} V_{0} (z)) .$ Observe that $W (c)$ is an involution, that is, $f | W (c) | W (c) = f$ . We define the conjugate operator H by $(f | H) (z) = \overline{f (- \overline{z})} .$ From (2.23) and (2.24), we can prove

New forms of modular forms of weight 1/2

Let $f \in M (c, ψ)$ be an eigenform of all but finitely many $T_{π^{2}}$ . We call f an old form if there is a totally positive prime element $π \in o$ dividing $c / 4$ such that one of the following holds: $\begin{matrix} (i) & r (ψ) divides c / π and f \in M (c / π, ψ) . \\ (ii) & r (ψ ϵ_{π}) divides c / π and f = g | V (π) with g \in M (c / π, ψ ϵ_{π}) . \end{matrix}$ We denote by $M^{0} (c, ψ)$ the subspace of $M (c, ψ)$ spanned by old forms of $M (c, ψ)$ . If $f \in M (c, ψ)$ is an eigenform of all but finitely many $T_{π^{2}}$ and f does not belong to $M^{0} (c, ψ)$ , we call f a new form of level c. According to the same argument as that

L-series associated with modular forms of weight 1/2 and the proof of the main theorem

Let $f (z) = \sum_{ξ \in o} a (ξ) e [(ξ / 2) \cdot z]$ be an element of $M (c, ψ)$ .

For $ϵ \in U$ , we consider $(\begin{matrix} ϵ & 0 \\ 0 & ϵ^{- 1} \end{matrix})$ . From (2.15), we can define the L-series $L (s, f)$ by $L (s, f) = \sum_{α \in o^{+} / U^{2}} a (α) N {(α)}^{- s},$ where $o^{+} = {α \in o | α ≫ 0}$ . By the same method as that of [1], we may confirm the following theorem.

Theorem 6.1

Suppose that $f \in M (c, ψ)$ and assume that c is a square of an ideal in $o$ . Then $L (s, f)$ can be analytically continued to an entire function with the exception of a simple pole at $s = 1 / 2$ if f is not a cusp form. Moreover, if we set $Λ (s, f) = {({(2 π)}^{- s})}^{r_{1}} ({(2 π)}^{- 2 s}$

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General SectionOn the modular forms of weight 1/2 over algebraic number fields

Abstract

Introduction

Section snippets

Notation and preliminaries

Modular forms of weight 1/2 and theta series

Hecke operators of modular forms of weight 1/2

Symmetry operators and some operators

New forms of modular forms of weight 1/2

L-series associated with modular forms of weight 1/2 and the proof of the main theorem

J. Number Theory

J. Number Theory

Distinguished representations and modular forms of half-integral weight

Invent. Math.

Modular forms of weight 1/2 over class number 1 imaginary quadratic number fields

Acta Arith.

Lectures on the Theory of Algebraic Numbers

On the Fourier coefficients of Maass wave forms of half integral weight over an imaginary quadratic field

J. Reine Angew. Math.

General Section
On the modular forms of weight 1/2 over algebraic number fields