Maximal abelian extension of $X_0(p)$ unramified outside cusps

TL;DR

This paper constructs a new maximal abelian extension of the modular curve $X_0(p)$ unramified outside cusps, extending Mazur's work by using generalized Dedekind eta functions to produce a cyclic cover of degree $2N_p$.

Contribution

It introduces a new cyclic abelian covering of $X_0(p)$ unramified outside cusps, expanding the understanding of modular curve extensions beyond Mazur's established covers.

Findings

The maximal abelian cover $X_2'(p)$ is cyclic of degree $2N_p$.

Construction utilizes generalized Dedekind eta functions.

Extends Mazur's results on unramified abelian coverings.

Abstract

Let $p$ be a prime number. Mazur proved that a geometrically maximal unramified abelian covering of $X_0(p)$ over $\mathbb Q$ is given by the Shimura covering $X_2(p) \to X_0(p)$, that is, a unique subcovering of $X_1(p) \to X_0(p)$ of degree $N_p := (p-1)/\gcd(p-1, 12)$. In this short paper, we show that a geometrically maximal abelian covering $X_2'(p) \to X_0(p)$ of $X_0(p)$ over $\mathbb Q$ unramified outside cusps is cyclic of degree $2N_p$. The main ingredient for the construction of $X_2'(p)$ is the generalized Dedelind eta functions.

Equations

$$X_2^{\prime }(p)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(M^{\ast }{)}^G\cong (M_G{)}^{\ast }\text{and}(M^G{)}^{\ast }\cong (M^{\ast }{)}_G.$$

$${\pi }_1(S{)}^{\mathrm{ab}}\cong H^1(S,\mathbb{Q}/\mathbb{Z}{)}^{\ast }$$

$$M^{\mu }:=\{a\in M_{\mathrm{Tor}}\mid \sigma (a)={\chi }_{\mathrm{cyc}}(\sigma )a\text{for all}\sigma \in G_k\},$$

$$r_D:=\sum _{x\in D}[k(x):k]-1.$$

$$0\to {\pi }_1(Y{)}^{\mathrm{ab},\mathrm{geo}}\to {\pi }_1(Y{)}^{\mathrm{ab}}\to G_k^{\mathrm{ab}}\to 0.$$

$$k(Y{)}^{\mathrm{ur},\mathrm{ab}}$$

$$0\to {\mathbb{G}}_D\to \tilde{J}\to J\to 0,$$

$${\mathbb{G}}_D:=\mathrm{Coker}[{\mathbb{G}}_{m,k}\to \underset{x\in D}{⨁}{\mathrm{Res}}_{k(x)/k}{\mathbb{G}}_{m,k(x)}].$$

$$\tilde{J}(k)$$

$$\cong \ker (H^1(X,{\mathbb{G}}_{X,D})\to H^1(X,{\mathbb{G}}_{m,X})\stackrel{\mathrm{deg}}{\to }\mathbb{Z}).$$

$$M_{\mathrm{Tor}}:=(T\tilde{J}{)}^{\ast }(1).$$

$$L_D:=\mathrm{Hom}({\mathbb{G}}_D,{\mathbb{G}}_{m,k})$$

$$0\to J_{\mathrm{Tor}}\to M_{\mathrm{Tor}}\to L_D\otimes \mathbb{Q}/\mathbb{Z}\to 0.$$

$$0\to J_{\mathrm{Tor}}^{\mu }(-1)\to M_{\mathrm{Tor}}^{\mu }(-1)\to (L_D\otimes \mathbb{Q}/\mathbb{Z}{)}^{\mu }(-1)\to H^1(k,J_{\mathrm{Tor}}(-1)).$$

$$0\to J_{\mathrm{Tor}}\to M_{\mathrm{Tor}}\to \mathbb{Q}/\mathbb{Z}\to 0.$$

$$M_{\mathrm{Tor}}[n]=J[n]\oplus (\mathbb{Z}/n\mathbb{Z})$$

$$\sigma (\alpha ,b)=(\sigma (\alpha )+b(\sigma (\beta )-\beta ),b)$$

$${\pi }_1(Y{)}^{\mathrm{ab},\mathrm{geo}}\cong M_{\mathrm{Tor}}^{\mu }(-1{)}^{\ast }.$$

$$0$$

$$\to H^2(k,\mathbb{Q}/\mathbb{Z})\stackrel{\varphi }{\to }{H}^2(Y,\mathbb{Q}/\mathbb{Z}).$$

$$0\to ({\pi }_1(\overline{Y}{)}^{\mathrm{ab}}{)}_{G_k}\to {\pi }_1(Y{)}^{\mathrm{ab}}\to G_k^{\mathrm{ab}}\to 1,$$

$${\pi }_1(\overline{Y}{)}^{\mathrm{ab}}\cong T\tilde{J}$$

$$0\to j_{!}({\mu }_{n,Y})\to {\mathbb{G}}_{X,D}\stackrel{n}{\to }{\mathbb{G}}_{X,D}\to 0$$

$${\tilde{J}}_{\mathrm{Tor}}[n]\cong H_c^1(\overline{Y},{\mu }_n),{\tilde{J}}_{\mathrm{Tor}}\cong H_c^1(\overline{Y},\mathbb{Q}/\mathbb{Z}(1)),T\tilde{J}\cong H_c^1(\overline{Y},\hat{\mathbb{Z}}(1)).$$

$$M_{\mathrm{Tor}}[n]\cong H^1(\overline{Y},{\mu }_n),M_{\mathrm{Tor}}\cong H^1(\overline{Y},\mathbb{Q}/\mathbb{Z}(1)),T{M}_{\mathrm{Tor}}\cong H^1(\overline{Y},\hat{\mathbb{Z}}(1)).$$

$$\Sigma (f):=\ker (f^{\ast }:J(\overline{k})\to J^{\prime }(\overline{k})),{\Sigma }_D(f):=\ker (f^{\ast }:M_{\mathrm{Tor}}\to M_{\mathrm{Tor}}^{\prime }).$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathrm{Gal}(f_X^{\mathrm{uab}})\cong \mathrm{Hom}(\Sigma (f),\mathbb{Q}/\mathbb{Z}(1)),$$

$$\mathrm{Gal}(f_Y^{\mathrm{uab}})\cong \mathrm{Hom}({\Sigma }_D(f),\mathbb{Q}/\mathbb{Z}(1)).$$

$$Y^{\prime }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

1. Introduction

1.1. Main result

Let $p$ be a prime. We consider the modular curves $X_{0}(p)$ and $X_{1}(p)$ as geometrically integral smooth proper curves over $\mathbb{Q}$. We choose a model of $X_{1}(p)$ over $\mathbb{Q}$ such that the cusp at infinity splits completely in a finite cyclic covering $f_{p}:X_{1}(p)\to X_{0}(p)$ of degree $(p-1)/2$. Note that $f_{p}$ is possibly ramified. We denote by $\underline{f}_{p}:X_{2}(p)\to X_{0}(p)$ its (unique) subcovering of degree $N_{p}:=(p-1)/\gcd(p-1,12)$, which is the maximal unramified subcovering of $f_{p}$, called the Shimura covering. We recall an important result due to Mazur.

Theorem 1.1.1** ([3, Theorem 2]).**

The Shimura covering $\underline{f}_{p}:X_{2}(p)\to X_{0}(p)$ is geometrically maximal over $\mathbb{Q}$.

Here, we say a finite unramified abelian covering $Y^{\prime}\to Y$ of smooth geometrically integral curves over $\mathbb{Q}$ is geometrically maximal over $\mathbb{Q}$ if any finite unramified abelian covering of $Y$ is isomorphic to a subcovering of $Y^{\prime}\times_{\operatorname{Spec}\mathbb{Q}}\operatorname{Spec}\mathbb{Q}^{\operatorname{ab}}\to Y$, where $\mathbb{Q}^{\operatorname{ab}}$ is a maximal abelian extension of $\mathbb{Q}$ (see Definition 2.2.2).

Let $Y_{0}(p)\subset X_{0}(p)$ be an open subscheme such that $X_{0}(p)\setminus Y_{0}(p)$ consists of all (two) cusps. In this short note, we shall construct a geometrically maximal unramified covering of $Y_{0}(p)$ and prove the following result.

Theorem 1.1.2**.**

There exists a cyclic covering $\underline{f}_{p}^{\prime}:X_{2}^{\prime}(p)\to X_{0}(p)$ of degree $2N_{p}$ such that

1. (1)

$\underline{f}_{p}^{\prime}|_{Y_{2}^{\prime}(p)}:Y_{2}^{\prime}(p)\to Y_{0}(p)$* is a geometrically maximal unramified covering of $Y_{0}(p)$ over $\mathbb{Q}$, where $Y_{2}^{\prime}(p):={\underline{f}_{p}^{\prime}}^{-1}(Y_{0}(p))$; and* 2. (2)

$\underline{f}_{p}^{\prime}$* factors as $\underline{f}_{p}^{\prime}=\underline{f}_{p}\circ\pi_{p}$, where $\pi_{p}:X_{2}^{\prime}(p)\to X_{2}(p)$ is a degree two covering that is unramified outside cusps.*

[TABLE]

We shall prepare some general facts on abelian coverings of smooth curves in §2. We then prove Theorem 1.1.2 in §3. The function field $\mathbb{Q}(X_{2}^{\prime}(p))$ of $X_{2}^{\prime}(p)$ will be obtained as a quadratic extension of $\mathbb{Q}(X_{2}(p))$ generated by a square root of an explicitly constructed rational function in $\mathbb{Q}(X_{2}(p))$. A key ingredient for this construction is the generalized Dedekind eta functions, which we recall in §3.2.

1.2. Notations and conventions

Let $G$ be a profinite group. We write $G^{\operatorname{ab}}$ for the quotient of $G$ by the closure of its commutator subgroup. For a $G$-module $M$, we denote by $M^{G}\subset M$ and $M\twoheadrightarrow M_{G}$ its $G$-invariant subgroup and $G$-coinvariant quotient, respectively. When $M$ is locally compact, we write $M^{*}$ for the Pontryagin dual of $M$. In practice, we shall only consider the cases where $M$ is either a discrete torsion group or a compact free $\hat{\mathbb{Z}}$-module of finite rank, hence $M$ is a $\hat{\mathbb{Z}}$-module and $M^{*}=\operatorname{Hom}(M,\mathbb{Q}/\mathbb{Z})$, where $\operatorname{Hom}$ means the group of $\hat{\mathbb{Z}}$-linear maps. Note also that we have canonical isomorphisms

[TABLE]

Indeed, both sides of the first isomorphism are identified with the group of continuous $G$-equivariant homomorphisms $M\to\mathbb{Q}/\mathbb{Z}$. The second follows from the first by replacing $M$ by $M^{*}$ and $M^{**}\cong M$.

Let $S$ be a scheme which is separated and of finite type over a field. We write $\mathbb{G}_{m,S}$ for the multiplicative group scheme over $S$, and $\mu_{n,S}:=\ker(n:\mathbb{G}_{m,S}\to\mathbb{G}_{m,S})$. When $S=\operatorname{Spec}R$ is affine, we write $G_{\operatorname{Spec}R}=G_{R}$. For an étale sheaf $F$ on $S$, we write $H^{*}(S,F)$ (resp. $H^{*}_{c}(S,F)$) for the étale cohomology (resp. the étale cohomology with compact support). A $G_{k}$-module $M$ is identified with an étale sheaf on $\operatorname{Spec}k$ and we write $H^{i}(\operatorname{Spec}k,M)=H^{i}(k,M)$. Suppose now $S$ is connected. We write $\pi_{1}(S)^{\operatorname{ab}}:=\pi_{1}(S,x)^{\operatorname{ab}}$, where $\pi_{1}(S,x)$ is the étale fundamental group of $S$ with respect to some geometric point $x\to S$ (on which $\pi_{1}(S,x)^{\operatorname{ab}}$ does not depend up to unique isomorphism). In particular, we have a canonical isomorphism

[TABLE]

Let $k$ be a field of characteristic zero. We take an algebraic closure $\overline{k}$ and put $G_{k}:=\operatorname{Gal}(\overline{k}/k)$. We write $\mathbb{Z}/n\mathbb{Z}(r)=\mu_{n,k}(\overline{k})^{\otimes r}$ if $r\geq 0$, and $\mathbb{Z}/n\mathbb{Z}(r)=\operatorname{Hom}(\mathbb{Z}/n\mathbb{Z}(-r),\mathbb{Z}/n\mathbb{Z})$ if $r<0$. Let $M$ be a $G_{k}$-module. We write $M[n]:=\{x\in M\mid nx=0\}$. We define the (twisted) Tate module by $TM(r):=\operatornamewithlimits{\varprojlim}_{n}M[n]\otimes\mathbb{Z}/n\mathbb{Z}(r)$. If $M$ is torsion, then we define $M(r):=M\otimes_{\hat{\mathbb{Z}}}\hat{\mathbb{Z}}(r)$, where $\hat{\mathbb{Z}}(r):=\operatornamewithlimits{\varprojlim}_{n}\mathbb{Z}/n\mathbb{Z}(r)$. For general $M$, we define its maximal $\mu$-type subgroup by

[TABLE]

where $\chi_{\operatorname{cyc}}:G_{k}\to\hat{\mathbb{Z}}^{\times}$ is the cyclotomic character. (In other words, $M^{\mu}(-1)=M_{\operatorname{Tor}}(-1)^{G_{k}}$.)

A commutative algebraic group $A$ over $k$ defines a $G_{k}$-module $A(\overline{k})$, and hence an étale sheaf on $\operatorname{Spec}k$. In this case, to ease the notation we often write $A$ for $A(\overline{k})$, e.g. $A[n]=A(\overline{k})[n],TA(r)=TA(\overline{k})(r)$ and $A^{\mu}=A(\overline{k})^{\mu}$.

2. Abelian coverings of smooth curves

In this section, we collect basic facts about abelian fundamental group of a smooth curve.

2.1. Setting

Let $X$ be a smooth proper geometrically integral curve over a field $k$ of characteristic zero. Let $D$ be an effective reduced divisor on $X$ and set $Y:=X\setminus D$. We write $g_{X}$ for the genus of $X$, and set

[TABLE]

We suppose that $X$ admits a degree one divisor. Note that this implies the existence of a degree one divisor $E$ supported on $Y$ by the approximation lemma (see e.g. [6, p. 12]). Indeed, let $E_{1}=\sum n_{x}x$ be a degree one divisor on $X$. For each $x\in|D|\cap|E_{1}|$ (resp. $x\in|D|\setminus|E_{1}|$), let $a_{x}\in k(X)$ be such that $\operatorname{ord}_{x}(a_{x})=n_{x}$ (resp. $a_{x}=1$). By applying the cited result to the family $(a_{x})_{x\in|D|}$, we obtain $f\in k(X)$ such that $\operatorname{ord}_{x}(f)=n_{x}$ for all $x\in|D|\cap|E_{1}|$ and $\operatorname{ord}_{x}(f)=0$ for all $x\in|D|\setminus|E_{1}|$. Then $E:=E_{1}-{\operatorname{div}}(f)$ is a degree one divisor supported on $Y$.

2.2. Geometrically maximal unramified abelian covering

Let $k^{\operatorname{ab}}$ be the maximal abelian extension of $k$. We have $G_{k}^{\operatorname{ab}}=\operatorname{Gal}(k^{\operatorname{ab}}/k)$ and $\pi_{1}(Y)^{\operatorname{ab}}=\operatorname{Gal}(k(Y)^{{\operatorname{ur}},{\operatorname{ab}}}/k(Y))$, where $k(Y)^{{\operatorname{ur}},{\operatorname{ab}}}$ is the maximal unramified abelian extension of the function field $k(Y)$ of $Y$. Denote by $\pi_{1}(Y)^{{\operatorname{ab}},{\operatorname{geo}}}$ the kernel of the canonical map $\pi_{1}(Y)^{\operatorname{ab}}\to G_{k}^{\operatorname{ab}}$. We have an exact sequence

[TABLE]

Remark 2.2.1*.*

It is shown in [1, Theorem 1] that $\pi_{1}(Y)^{{\operatorname{ab}},{\operatorname{geo}}}$ is finite if $k$ is finitely generated over its prime subfield.

Definition 2.2.2**.**

We say a finite unramified abelian covering $Y^{\prime}\to Y$ is geometrically maximal over $k$ if $Y^{\prime}$ is geometrically integral over $k$ and if the composition $\pi_{1}(Y)^{{\operatorname{ab}},{\operatorname{geo}}}\hookrightarrow\pi_{1}(Y)^{{\operatorname{ab}}}\twoheadrightarrow\operatorname{Gal}(Y^{\prime}/Y)$ is an isomorphism:

[TABLE]

(Equivalently, any finite unramified abelian covering $Y^{\prime\prime}\to Y$ is a subcovering of $Y^{\prime}\times_{\operatorname{Spec}k}\operatorname{Spec}k^{\operatorname{ab}}\to Y$. See also Remark 2.6.5 and Proposition 2.6.6 below.)

When $Y$ has a $k$-rational point $x$, a maximal abelian unramified covering of $Y$ in which $x$ splits completely yields a geometrically maximal covering. There is, however, no such a description if no $k$-rational point is available. Given $Y$, we are interested in finding a geometrically maximal unramified abelian covering.

2.3. Generalized Jacobian

Let ${\widetilde{J}}:=\operatorname{Jac}(X,D)$ be the generalized Jacobian of $X$ with modulus $D$ in the sense of Rosenlicht-Serre [5], which is a semi-abelian variety over $k$. It fits in an exact sequence

[TABLE]

where $J=\operatorname{Jac}(X)$ is the Jacobian variety of $X$, and

[TABLE]

We also recall that, if we put $\mathbb{G}_{X,D}:=\ker(\mathbb{G}_{m,X}\to\mathbb{G}_{m,D})$, there are isomorphisms

[TABLE]

2.4. A Galois module $M_{\operatorname{Tor}}$

We define a divisible torsion $G_{k}$-module $M_{\operatorname{Tor}}$ by

[TABLE]

(See §1.2 for the notations for $T{\widetilde{J}},~{}(-)^{*}$ and $(-)(1)$). Although this is not the torsion subgroup of a group $M$, the notation $M_{\operatorname{Tor}}$ may be justified by the fact that $M_{\operatorname{Tor}}$ agrees with the group of torsion points of the dual $1$-motive $M$ of ${\widetilde{J}}$ in the sense of [7, Chapter V, §3]. (We will not use this fact.) Let

[TABLE]

be the character group of $\mathbb{G}_{D}$ from (2.3.2). This is thus a free abelian group of rank $r_{D}$ (see (2.1.1)), equipped with a continuous $G_{k}$-action. Recall that $g_{X}$ is the genus of $X$.

Lemma 2.4.1**.**

There is an exact sequence of $G_{k}$-modules

[TABLE]

In particular, $M_{\operatorname{Tor}}\cong(\mathbb{Q}/\mathbb{Z})^{\oplus(g_{X}+r_{D})}$ as abelian groups.

Proof.

The first statement follows from (2.3.1) and an isomorphism $J_{\operatorname{Tor}}\cong(TJ)^{*}(1)$ deduced by the Weil pairing $TJ\times J_{\operatorname{Tor}}\to\mathbb{Q}/\mathbb{Z}(1)$. The last statement follows form (2.4.3). ∎

By taking the Tate twist $(-1)$ and the long exact sequence attached to (2.4.3) (see also (1.2.3)), we get an exact sequence that will be used later

[TABLE]

Example 2.4.2*.*

Suppose that $D=P+Q$ consists of two $k$-rational points. Then we have an exact sequence

[TABLE]

Explicitely, the $G_{k}$-module $M_{\operatorname{Tor}}=\operatornamewithlimits{\varinjlim}_{n}M_{\operatorname{Tor}}[n]$ can be described as follows. We have

[TABLE]

as an abelian group, and $G_{k}$-action is given by

[TABLE]

where $\sigma\in G_{k},~{}\alpha\in J[n],~{}b\in\mathbb{Z}/n\mathbb{Z}$ and $\beta\in J(\overline{k})$ is a fixed element such that $n\beta$ is the class of $P-Q$ in $J(\overline{k})$ (see (2.3.3)).

2.5. Abelian fundamental group

The following result is essentially shown in [1], but it is embedded in a (long) proof, and hence we decide to include a proof to cut out the desired part.

Proposition 2.5.1**.**

The exact sequence (2.2.1) splits (non-canonically), and we have a canonical isomorphism (see §1.2 for the notations used in the right hand side)

[TABLE]

Proof.

Set $\overline{Y}:=Y\times_{\operatorname{Spec}k}\operatorname{Spec}\overline{k}$. The spectral sequence $E_{2}^{i,j}=H^{i}(k,H^{j}(\overline{Y},\mathbb{Q}/\mathbb{Z}))\Rightarrow H^{i+j}(Y,\mathbb{Q}/\mathbb{Z})$ yields an exact sequence

[TABLE]

Recall that by assumption we have a degree one divisor $E=\sum n_{x}x$ supported on $Y$ (see §2.1). It gives rise to a section $\sum n_{x}N_{k(x)/k}\circ i_{x}^{*}$ of $\phi$, where $i_{x}^{*}:H^{2}(Y,\mathbb{Q}/\mathbb{Z})\to H^{2}(k(x),\mathbb{Q}/\mathbb{Z})$ is the pull-back along the closed immersion $i_{x}:\operatorname{Spec}k(x)=x\to Y$ and $N_{k(x)/k}:H^{2}(k(x),\mathbb{Q}/\mathbb{Z})\to H^{2}(k,\mathbb{Q}/\mathbb{Z})$ is the norm (corestriction) map. This shows the injectivity of $\phi$. By taking the dual sequence, we obtain a short exact sequence (see (1.2.1))

[TABLE]

which splits again by the presence of $E$. We have shown $(\pi_{1}(\overline{Y})^{\operatorname{ab}})_{G_{k}}\cong\pi_{1}(Y)^{{\operatorname{ab}},{\operatorname{geo}}}$. We also have $(T{\widetilde{J}})_{G_{k}}\cong\operatorname{Hom}(M_{\operatorname{Tor}}^{\mu}(-1),\mathbb{Q}/\mathbb{Z})$ by the definition (2.4.1).

Hence it remains to show an isomorphism of $G_{k}$-modules

[TABLE]

from which the proposition follows by taking $G_{k}$-coinvariant quotients. By the definitions of $\mathbb{G}_{X,D}$ from (2.3.4), there exists an exact sequence of étale sheaves

[TABLE]

for any $n$, where $j:Y\to X$ is the open immersion so that $j_{!}(\mu_{n,Y})\cong\ker(\mu_{n,X}\to\mu_{n,D})$. It follows that

[TABLE]

The last isomorphism shows $T{\widetilde{J}}\cong H^{1}(\overline{Y},\mathbb{Q}/\mathbb{Z})^{*}$ by the Poincaré duality. Now (2.5.1) follows from (1.2.2). ∎

Remark 2.5.2*.*

By applying the functor $A\mapsto A^{*}(1)$ to (2.5.2), we obtain isomorphisms

[TABLE]

2.6. Covering of $Y$

Let $X^{\prime}$ be another proper smooth geometrically integral curve over $k$ admitting a degree one divisor, and let $f:X^{\prime}\to X$ be a finite $k$-morphism. Put $Y^{\prime}:=f^{-1}(Y)$ and $D^{\prime}:=X^{\prime}\setminus Y^{\prime}$. (We regard $D^{\prime}$ as a reduced effective divisor on $X^{\prime}$.) Set $J^{\prime}:=\operatorname{Jac}(X^{\prime}),{\widetilde{J}}^{\prime}:=\operatorname{Jac}(X^{\prime},D^{\prime})$ and $M_{\operatorname{Tor}}^{\prime}:=\operatorname{Hom}(T{\widetilde{J}}^{\prime},\mathbb{Q}/\mathbb{Z}(1))$. We have the pull-back maps $f^{*}:J\to J^{\prime},{\widetilde{J}}\to{\widetilde{J}}^{\prime},M_{\operatorname{Tor}}\to M_{\operatorname{Tor}}^{\prime}$ and the push-forward maps $f_{*}:J^{\prime}\to J,{\widetilde{J}}^{\prime}\to{\widetilde{J}},M_{\operatorname{Tor}}^{\prime}\to M_{\operatorname{Tor}}$.

Definition 2.6.1**.**

We define

[TABLE]

Note that $\Sigma(f)\subset J[d]$ with $d:=\operatorname{deg}(f)$, because $f_{*}\circ f^{*}=d$. It follows that $\Sigma(f)$ is finite and hence $\Sigma(f)\subset J_{\operatorname{Tor}}$. Similarly, $\Sigma_{D}(f)\subset M_{\operatorname{Tor}}^{\prime}[d]$ is finite too. The canonical maps $J_{\operatorname{Tor}}\to M_{\operatorname{Tor}}$ and $J_{\operatorname{Tor}}^{\prime}\to M_{\operatorname{Tor}}^{\prime}$ from (2.4.3) induce $\Sigma(f)\to\Sigma_{D}(f)$.

Lemma 2.6.2**.**

1. (1)

The map $\Sigma(f)\to\Sigma_{D}(f)$ is injective. 2. (2)

Suppose that $\mathrm{gcd}\{e(x^{\prime}/x)\mid x^{\prime}\in f^{-1}(x)\}=1$ for all $x\in D$, where $e(x^{\prime}/x)$ denotes the ramification index. Then $\Sigma(f)\to\Sigma_{D}(f)$ is an isomorphism.

Proof.

Let $L_{D}$ and $L_{D^{\prime}}$ be the character groups from (2.4.2). By Lemma 2.4.1, we have a commutative diagram with exact rows:

[TABLE]

This shows (1). For (2), it suffices to show the injectivity of the right vertical map under the stated assumption. We have a commutative diagram with exact rows:

[TABLE]

Here $(x,x^{\prime})$-component of the middle vertical map is given by [math] if $f(x^{\prime})\not=x$ and by $e(x^{\prime}/x)$ if $f(x^{\prime})=x$. The lemma follows. ∎

Example 2.6.3*.*

Let $N\in\mathbb{Z}_{>0}$ and consider the canonical map $f_{N}:X_{1}(N)\to X_{0}(N)$, with respect to divisors consisting of all cusps. If $N$ is square free, then $f_{N}$ is unramified at all cusps, and hence $\Sigma(f_{N})\cong\Sigma_{D}(f_{N})$, where $D$ is the sum of all cusps on $X_{0}(N)$.

Lemma 2.6.4**.**

Let $f^{\operatorname{uab}}_{X}:X^{\operatorname{uab}}_{f}\to X$ (resp. $f^{\operatorname{uab}}_{Y}:Y^{\operatorname{uab}}_{f}\to Y$) be the maximal unramified abelian subcovering of $f:X^{\prime}\to X$ (resp. $f:Y^{\prime}\to Y$). Then we have

[TABLE]

In particular, $\Sigma(f)\subset J_{\operatorname{Tor}}^{\mu}$ and $\Sigma_{D}(f)\subset M_{\operatorname{Tor}}^{\mu}$ (see (1.2.3)).

Proof.

Since the first statement is a special case of second (for $D=\emptyset$), we only prove the latter. For a $k$-scheme $V$, we write $\overline{V}:=V\times_{\operatorname{Spec}k}\operatorname{Spec}\overline{k}$. Let $\overline{f}^{\operatorname{uab}}_{Y}:\overline{Y}^{\operatorname{uab}}_{f}\to\overline{Y}$ be the maximal unramified abelian subcovering of the base change $\overline{f}:\overline{Y}^{\prime}\to\overline{Y}$ of $f$. Let $\alpha:\overline{V}\to\overline{Y}^{\prime}$ be a Galois closure of $\overline{f}$. We have a commutative diagram

[TABLE]

The left two squares are Cartesian since $Y,Y^{\prime}$ (and hence $Y^{\operatorname{uab}}_{f}$ too) are geometrically integral over $k$. It follows that $\operatorname{Gal}(f_{Y}^{\operatorname{uab}})\cong\operatorname{Gal}(\overline{f}_{Y}^{\operatorname{uab}})$. Using (2.5.3) and (1.2.2), we get

[TABLE]

The last statement follows since $G_{k}$ acts on $\operatorname{Gal}(f^{\operatorname{uab}}_{Y})$ trivially. ∎

Note that this lemma implies $\Sigma(f)=\Sigma(f_{X}^{\operatorname{uab}})$ and $\Sigma_{D}(f)=\Sigma_{D}(f_{Y}^{\operatorname{uab}})$.

Remark 2.6.5*.*

Suppose that $f:X^{\prime}\to X$ (resp. $f:Y^{\prime}\to Y$) is a finite abelian unramified covering. Then $f$ is geometrically maximal (see Definition 2.2.2) if and only if $\Sigma(f)=J_{\operatorname{Tor}}^{\mu}$ (resp. $\Sigma_{D}(f)=M_{\operatorname{Tor}}^{\mu}$).

Proposition 2.6.6**.**

Suppose that $k$ is finitely generated over its prime subfield. Given $Y$, there exists a geometrically maximal abelian unramified covering $f:Y^{\prime}\to Y$.

Proof.

This follows from Proposition 2.5.1 and Remark 2.2.1. ∎

3. Modular curves

In this section, we are going to prove Theorem 1.1.2.

3.1. First reduction

Let $p$ be a prime and set $k=\mathbb{Q},~{}X=X_{0}(p),~{}Y=Y_{0}(p)$. Recall that $D=X\setminus Y$ consists of two $\mathbb{Q}$-rational points (i.e. [math] and $\infty$ cusps). It follows that $L_{D}=\mathbb{Z}$ with trivial $G_{\mathbb{Q}}$-action, and hence $(L_{D}\otimes\mathbb{Q}/\mathbb{Z})^{\mu}(-1)\cong\mathbb{Z}/2\mathbb{Z}$ (see Example 2.4.2). By (2.4.4), we get an exact sequcne

[TABLE]

Since $J_{\operatorname{Tor}}^{\mu}(-1)$ is a cyclic group of order $N_{p}$ by Theorem 1.1.1, the order of $M_{\operatorname{Tor}}^{\mu}(-1)$ is either $N_{p}$ or $2N_{p}$. In view of Proposition 2.5.1 and Lemma 2.6.4, this implies that if $\underline{f}_{p}^{\prime}:X_{2}^{\prime}(p)\to X_{0}(p)$ is a cyclic covering of degree $2N_{p}$ such that $X_{2}^{\prime}(p)$ is geometrically integral and such that the condition (2) in Theorem 1.1.2 holds, then it automatically satisfies (1) as well. We are going to construct such $\underline{f}_{p}^{\prime}$. If $p=2$ or $3$, we may take $\underline{f}_{p}^{\prime}$ to be the map $\mathbb{G}_{m,k}\to\mathbb{G}_{m,k},~{}x\mapsto x^{2}$ under the identification $Y_{0}(p)\cong\mathbb{G}_{m,k}=\mathbb{P}^{1}_{k}\setminus\{0,\infty\}$. Below we assume $p\geq 5$. We need a preparation.

3.2. Generalized Dedekind eta functions

Let $N$ be a positive integer. For an integer $g$ not congruent to [math] modulo $N$, define the generalized Dedekind eta function $E_{g}$ by

[TABLE]

where $B(x)=x^{2}-x+1/6$ is the second Bernoulli polynomial (see [4]). Up to scalars, these functions are also called Siegel functions (see [2]). We have the following properties of $E_{g}$.

Proposition 3.2.1** ([8, Corollaries 2 and 3 and Lemma 2]).**

1. (1)

We have

[TABLE] 2. (2)

Let $\gamma=\begin{pmatrix}a&b\\ cN&d\end{pmatrix}\in\Gamma_{0}(N)$. We have, for $c=0$,

[TABLE]

and, for $c\neq 0$,

[TABLE]

where

[TABLE] 3. (3)

Suppose that $\prod_{g}E_{g}^{e_{g}}$ is a product of generalized Dedekind eta functions satisfying

[TABLE]

Then $\prod_{g}E_{g}^{e_{g}}$ is a modular function on $\Gamma_{1}(N)$. Moreover, if $N$ is odd, then the conditions can be reduced to

[TABLE] 4. (4)

Given a matrix

[TABLE]

the Fourier expansion of $E_{g}(\gamma\tau)$ starts from $\zeta q^{\delta}+(\text{\rm higher powers})$, where $\zeta$ is a root of unity and

[TABLE]

where $P_{2}(x)=\{x\}^{2}-\{x\}+1/6$ is the second Bernoulli function, $\{x\}$ denotes the fraction part of $x$, and $(a,b):=\gcd(a,b)$.

3.3. The modular curve $X_{2}^{\prime}(p)$

Let $p$ be a prime with $p\geq 5$ and $g$ be an odd generator of $(\mathbb{Z}/p\mathbb{Z})^{\times}$. To ease the notation, set

[TABLE]

Let $\Gamma_{2}(p)$ be the group generated by $\Gamma_{1}(p)$ and any matrix of the form $\left(\begin{smallmatrix}g^{k}&\ast\\ p&\ast\end{smallmatrix}\right)$ and $X_{2}(p)$ be the corresponding modular curve. By [3, §II, 2], $X_{2}(p)\to X_{0}(p)$ is the maximal unramified subcover of $X_{1}(p)\to X_{0}(p)$. We have

[TABLE]

For an integer $h$ not congruent to [math] modulo $p$, using (3.2.1), we set

[TABLE]

Lemma 3.3.1**.**

The functions $F_{h}$ have the following properties.

1. (1)

$F_{h}=F_{-h}=F_{p+h}$. 2. (2)

$F_{g^{k}h}=\begin{cases}-F_{h},&\text{if }p\equiv 1\mod 4,\\ F_{h},&\text{if }p\equiv 3\mod 4.\end{cases}$**

Proof.

The fact that $F_{h}=F_{-h}$ follows immediately from (3.2.2) in Proposition 3.2.1 since $F_{h}$ is a product of $6$ generalized Dedekind eta functions. By the same property of the generalized Dedekind eta functions, we have

[TABLE]

Since $g$ is assumed to be odd, we find that $F_{p+h}=F_{h}$. We next prove Part (2).

By the definition of $F_{h}$, we have

[TABLE]

Now $g^{k\ell}=g^{(p-1)/2}\equiv-1\mod p$. Hence from (3.2.2) in Proposition 3.2.1 again, we find that

[TABLE]

It follows that

[TABLE]

This completes the proof. ∎

Let $\psi$ be the character of order $2$ of $\mathrm{SL}(2,\mathbb{Z})$ defined by

[TABLE]

for $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\mathrm{SL}(2,\mathbb{Z})$. In addition, if $\ell$ is even, i.e., if $p\equiv 1\mod 4$, there is a unique character $\chi$ of order $2$ of $\Gamma_{2}(p)$ with $\Gamma_{1}(p)\subset\operatorname{ker}\chi$. Explicitly, for $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{2}(p)$, let $n$ be an integer such that $a\equiv g^{nk}\mod p$. Then $\chi(\gamma)$ is equal to $(-1)^{n}$. Note that if $p\equiv 5\mod 8$, then $k$ is odd and $\chi$ is simply the character $\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\mapsto\left(\frac{d}{p}\right)$ of nebentype.

Let $\Gamma_{2}^{\prime}(p)$ be the kernel of the character

[TABLE]

on $\Gamma_{2}(p)$.

Lemma 3.3.2**.**

The group $\Gamma_{2}^{\prime}(p)$ is a normal subgroup of $\Gamma_{0}(p)$ and $\Gamma_{0}(p)/\Gamma_{2}^{\prime}(p)$ is cyclic of order $2k$.

Proof.

Let $\rho$ be a character on $\Gamma_{0}(p)$ of order $(p-1)/2$ with kernel $\Gamma_{1}(p)$. By definition, $\Gamma_{2}^{\prime}(p)$ is the kernel of the character $\rho^{\ell}\psi$ (resp. $\rho^{\ell/2}\chi$) if $\ell$ is odd (resp. even) on $\Gamma_{0}(p)$ of order $2k$. ∎

Combined with this lemma, Theorem 1.1.2 follows from the following result:

Proposition 3.3.3**.**

The modular curve $X_{2}^{\prime}(p)$ associated to $\Gamma_{2}^{\prime}(p)$ admits a geometrically integral model over $\mathbb{Q}$ and the covering $\underline{f}_{p}^{\prime}:X_{2}^{\prime}(p)\to X_{2}(p)$ ramifies precisely at each cusp.

We first prove the proposition assuming $p\not\equiv 11\mod 12$.

Lemma 3.3.4**.**

Assume that $\ell\neq 1$, i.e., that $p\not\equiv 11\mod 12$. Let $h$ be an integer not congruent to [math] modulo $p$.

1. (1)

For $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{0}(p)$, we have $F_{h}(\gamma\tau)=\psi(\gamma)F_{ah}(\tau)$. 2. (2)

Let $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{2}(p)$. Then

[TABLE] 3. (3)

For any $h$, $F_{h}^{2}$ is a modular function on $X_{2}(p)$ defined over $\mathbb{Q}$. 4. (4)

The order of $F_{h}^{2}$ at any cusp of $X_{2}(p)$ is odd, and is zero elsewhere.

Proof.

Assume that $\ell\neq 1$, i.e., that $p\not\equiv 11\mod 12$. Let $\gamma=\left(\begin{smallmatrix}a&b\\ cp&d\end{smallmatrix}\right)$ be a matrix in $\Gamma_{0}(p)$. By Part (2) of Proposition 3.2.1, for any $j=0,\ldots,\ell-1$,

[TABLE]

Notice that $\varepsilon(a,bp,c,d)$ is a $12$th root of unity independent of $h$ and $jk$. Thus,

[TABLE]

where

[TABLE]

We check directly from the definition of $\varepsilon$ that

[TABLE]

Since $\ell$ is assumed to be greater than $1$, we have

[TABLE]

Also, $S$ is even since either $6/\ell$ is even or $\sum_{j=0}^{\ell-1}g^{2jk}$ is even. We conclude that $(2p)|S$ and

[TABLE]

We next prove Part (2). Assume that $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{2}(p)$. By Part (1), $F_{h}(\gamma\tau)=\psi(\gamma)F_{ah}(\tau)$. Let $n$ be a nonnegative integer such that $a\equiv g^{nk}\mod p$. By Lemma 3.3.1, we have

[TABLE]

This yields the formula in Part (2). The first statement of Part (3) is an immediate consequence of Part (2), and the second statement follows immediately from the definition (3.3.1).

We now prove Part (4). The statement for non-cusp points are obvious from the definition (3.3.1). Let $a/c$, $(a,c)=1$, be a cusp of $X_{2}(p)$. Consider first the case $p\nmid c$. Let $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)$ be a matrix in $\mathrm{SL}(2,\mathbb{Z})$. By Part (4) of Proposition 3.2.1, the Fourier expansion of $E_{g^{jk}h}$ starts from the term $q^{1/12p}$ for any $j$. Since such a cusp has width $p$ and $F_{h}^{2}$ is the product of exactly $12$ $E_{g^{jk}h}$, the order of $F_{h}$ at $a/c$ is $1$.

Now consider the case $p|c$. Such a cusp has width $1$. By Part (4) of Proposition 3.2.1, the order of $F_{h}$ at $a/c$ is

[TABLE]

Observe that for any integer $x$, if we let $n=\lfloor x/p\rfloor$, then

[TABLE]

and hence

[TABLE]

It follows that

[TABLE]

That is, the order of $F_{h}^{2}$ at the cusp $a/c$ is odd. This completes the proof. ∎

It follows from the lemma that $\mathbb{Q}(X_{2}^{\prime}(p))$ is a quadratic extension of $\mathbb{Q}(X_{2}(p))$ generated by a square root of $F_{h}^{2}\in\mathbb{Q}(X_{2}(p))$, that $\mathbb{Q}$ is algebraically closed in $\mathbb{Q}(X_{2}^{\prime}(p))$, and that the covering $X_{2}^{\prime}(p)\to X_{2}(p)$ ramifies exactly at each cusp, showing Proposition 3.3.3 in this case.

To show the proposition for the case $p\equiv 11\mod 12$, we need a slightly different construction of modular functions. In this case, $\Gamma_{2}(p)=\Gamma_{1}(p)$. Let $h=(h_{1},h_{2},h_{3})$ be any triplet of integers not congruent to [math] modulo $p$ such that $h_{1}^{2}+h_{2}^{2}+h_{3}^{2}\equiv 0\mod p$ and define $G_{h}=(E_{h_{1}}E_{h_{2}}E_{h_{3}})^{2}$. Then $G_{h}^{2}$ is a modular function on $X_{1}(p)$ by Proposition 3.2.1. Also, by the same computation as above, the order of $G_{h}^{2}$ at each cusp is odd. In addition, we can verify as above that for $\gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma_{1}(p)$,

[TABLE]

Therefore, the proposition holds for the case $p\equiv 11\mod 12$. ∎

Remark 3.3.5*.*

If $p\not\equiv 1\mod 8$, then

[TABLE]

is also a modular function on $\Gamma_{2}^{\prime}(p)$, but not on $\Gamma_{2}(p)$. Thus, the function field of $X_{2}^{\prime}(p)$ can be obtained by adjoining either $F_{h}(\tau)$ or $z(\tau)$. The relation betwen $F_{h}$ and $z$ is

[TABLE]

(In particular, if $k$ is odd, i.e., if $p\not\equiv 1\mod 8$, then $z(\gamma\tau)=\psi(\gamma)z(\tau)$ for $\gamma\in\Gamma_{2}(p)$.)

Acknowledgement. The first author would like to thank Masataka Chida and Fu-Tsun Wei for fruitful discussion. He is partially supported by JSPS KAKENHI Grant (18K03232). The second author was partially supported by Grant 106-2115-M-002-009-MY3 of the Ministry of Science and Technology, Republic of China (Taiwan).

The authors would like to thank the anonymous referee for the detailed comments.