Twisted Fermat curves over
totally real fields

Let $p$ be a prime number, $F$ a totally real field such that $[F(\mu_{p}):F]=2$ and $[F:{\mathbb{Q}}]$ is odd. For $\delta\in F^{\times}$, let $[\ \delta\ ]$ denote its class in $F^{\times}/F^{\times p}$. In this paper, we show

Main Theorem There are infinitely many classes $[\ \delta\ ]\in F^{\times}/F^{\times p}$ such that the twisted affine Fermat curves

have no $F$-rational points.

Remark It is clear that if $[\ \delta\ ]=[\ \delta^{\prime}\ ]$, then $W_{\delta}$ is isomorphic to $W_{\delta^{\prime}}$ over $F$. For any $\delta\in F^{\times},$ $W_{\delta}/F$ has rational points locally everywhere. \Enddemo

To obtain this result, consider the smooth open affine curve:

and the morphism:

Let $C_{\delta}\rightarrow J_{\delta}$ be the Jacobian embedding of $C_{\delta}/F$ defined by the point $(0,0)$. We will show that:

1. 1.

   If $L(1,J_{\delta}/F)\neq 0$, then $J_{\delta}(F)$ is a finite group (cf. Theorem 2.1. of §2).

   The proof is based on Zhang’s extension of the Gross-Zagier formula to totally real fields and on Kolyvagin’s technique of Euler systems. One might use techniques of congruence of modular forms to remove the restriction that the degree $[F:{\mathbb{Q}}]$ is odd.

2. 2.

   There are infinitely many classes $[\ \delta\ ]$ such that $L(1,J_{\delta}/F)\neq 0$ (cf. Theorem 3.1. of §3; see also 2.2.4.).

   The proof is based on the theory of double Dirichlet series. The condition that $[F(\mu_{p}):F]=2$ is essential for the technique we use here.

Combining $(1)$ and $(2)$, one can see that the set

is infinite.

Proof of the Main Theorem assuming $(1)$ and $(2)$ For any $\delta\in F^{\times}$, consider the twisting isomorphism (defined over $F(\sqrt[p]{\delta})$):

Define $\eta_{\delta}:J_{\delta}\longrightarrow J_{1}$ to be the homomorphism associated to $\iota_{\delta}$.

Let $\Sigma_{\delta}$ denote the set $\iota_{\delta}\left(C_{\delta}(F)\right)$. It is easy to see that:

1. (i)

   $\Sigma_{\delta}=\Sigma_{\delta^{\prime}}$, if $[\ \delta\ ]=[\ \delta^{\prime}\ ]$,

2. (ii)

   $\Sigma_{\delta}\cap\Sigma_{\delta^{\prime}}=\{(0,0),(1,0)\}$, otherwise.

For any $\delta\in F^{\times}$ with $[\ \delta\ ]\in\Pi$, and $[\ \delta\ ]\neq 1$, the diagram

commutes.

Since the set

is finite by the Northcott theorem, the set $\displaystyle{\bigcup_{[\ \delta\ ]\in\Pi}\Sigma_{\delta}}$ is finite. Thus, for all but finitely many $[\ \delta\ ]\in\Pi\setminus\{[1]\}$, $\Sigma_{\delta}=\{(0,0),(1,0)\}$, and therefore $W_{\delta}$ has no $F$-rational points. ∎

Remark Our method is, in fact, effective: for any $[\ \delta\ ]\in F^{\times}/F^{\times p}$, let

Let $L^{\prime}$ be the Galois closure of $F(\mu_{p})$, and let $S$ be the set of places of $F$ above $2D_{L^{\prime}/{\mathbb{Q}}}$, where $D_{L^{\prime}/{\mathbb{Q}}}$ is the discriminant of $L^{\prime}/{\mathbb{Q}}$. If ${\mathrm{Supp}}^{(p)}\left([\ \delta\ ]\right)$ is not contained in $S$ and $L(1,J_{\delta})\neq 0$, then the twisted Fermat curve $W_{\delta}$ has no $F$-rational points (see Proposition 2.2).

Acknowledgment We would like to thank D. Goldfeld, S. Friedberg,J. Hoffstein, H. Jacquet, V. A. Kolyvagin, L. Szpiro for their help and encouragement, and the referees for useful remarks and suggestions. In particular, we are grateful to S. Zhang, who suggested the problem to us, for many helpful conversations. The second author was partially supported by the Clay Mathematics Institute.

Fix $\delta\in F^{\times}\cap{\mathcal{O}}_{F}$ such that $(\delta,p)=1$. Let $\zeta=\zeta_{p}$ be a primitive $p$-th root of unity. The abelian variety $J_{\delta}$ is absolutely simple, of dimension $\displaystyle{g=\frac{p-1}{2}}$, and has complex multiplication by ${\mathbb{Z}}[\zeta]$ over the field $F(\mu_{p})$. In this section we show:

Notation. In this section, for an abelian group $M$, set $\widehat{M}=M\otimes_{\mathbb{Z}}\prod_{p}{\mathbb{Z}}_{p}$ where $p$ runs over all primes. For any ring $R$, let $R^{\times}$ denote the group of invertible elements. For any ideal ${\mathfrak{a}}$ of $F,$ denote the norm ${\mathrm{N}}_{F/{\mathbb{Q}}}({\mathfrak{a}})$ by ${\mathrm{N}}{\mathfrak{a}}$. Let ${\mathbb{A}}$ denote the adele ring of $F$, and ${\mathbb{A}}_{f}$ its finite part. Sometimes, we shall not distinguish a finite place from its corresponding prime ideal.

The Hilbert newform associated to $J_{\delta}$ We first recall some facts about $L$-functions of twisted Fermat curves over arbitrary number fields (see [14], [32]). Let $F$ be any number field, $L=F(\mu_{p}),$ $L_{0}={\mathbb{Q}}(\mu_{p}),$ and $F_{0}=L_{0}\cap F$.

For any place $w$ of $L$, denote by $w_{0}$ and $v$ its restrictions to ${\mathbb{Q}}(\mu_{p})$ and $F$, respectively. Let $\chi_{w_{0}}$ and $\chi_{w}$ be the $p$-th power residue symbols on $L_{0}^{\times}$ and $L^{\times},$ respectively, given by class field theory. Then $\chi_{w}=\chi_{w_{0}}\circ{\mathrm{N}}_{L/{\mathbb{Q}}(\mu_{p})}$. The Jacobi sum

is an integer in $L_{0}$ satisfying $j(\chi_{w},\chi_{w})=j(\chi_{w_{0}},\chi_{w_{0}})^{i_{w/w_{0}}}$ and the Stickelberger relation:

as an ideal in $L_{0}.$ Here, $i_{w/w_{0}}$ is the inertial degree for $w/w_{0}$, and $\sigma_{i}\in{\mathrm{Gal}}(L_{0}/{\mathbb{Q}})$ is the image of $i$ under the isomorphism $({\mathbb{Z}}/p{\mathbb{Z}})^{\times}\longrightarrow{\mathrm{Gal}}(L_{0}/{\mathbb{Q}})$.

Since $\delta\in{\mathcal{O}}_{F}$ is coprime to $p$, $C_{\delta}$ has good reduction at $w$ for any $w\nmid p\delta$. We know that the zeta-function of the reduction $\widetilde{C_{\delta}}$ of $C_{\delta}$ at a place $v$ of $F$ is

with

where $f_{v}$ is the order of ${\mathrm{N}}v$ modulo $p,$ and $\sigma$ runs over representatives in ${\mathrm{Gal}}({\mathbb{Q}}(\mu_{p})/{\mathbb{Q}})$ of ${\mathrm{Gal}}(F_{0}/{\mathbb{Q}})$. Then the number of points on $\tilde{J_{\delta}}$ (the reduction of $J_{\delta}$ at $v$) is $P_{v}(1)$.

Now we give a bound on torsion points of $J_{\delta}(F)$. Let $F^{\prime}$ be the Galois closure of $F/{\mathbb{Q}}$, and assume that $F\cap L_{0}=F^{\prime}\cap L_{0}.$ This assumption is satisfied if $F$ is as in the main theorem, or $F$ is Galois over ${\mathbb{Q}}$. Let $L^{\prime}=F^{\prime}(\mu_{p}),$ and let $q\nmid 2D_{L^{\prime}/{\mathbb{Q}}}$ be a prime. Let $\ell$ be a prime for which there exists a place $w^{\prime}|\ell$ of $L^{\prime}$ such that ${\mathrm{Frob}}_{L_{0}/F_{0}}(w^{\prime}|_{L_{0}})$ is a generator of ${\mathrm{Gal}}(L_{0}/F_{0})$, ${\mathrm{Frob}}_{F^{\prime}/F_{0}}(w^{\prime}|_{F^{\prime}})=1$ and ${\mathrm{Frob}}_{{\mathbb{Q}}(\mu_{q})/{\mathbb{Q}}}(w^{\prime}|_{{\mathbb{Q}}(\mu_{q})})=1$. Then, $\ell\equiv 1\mod q$. Let $v,$ $w$ and $w_{0}$ be the places of $F,$ $L$ and $L_{0}$, respectively, below $w^{\prime}$. Then, $v$ is inert in $L/F$ and $i_{w/w_{0}}=1.$ We have

Since $v$ is inert in $L/F$ and $\delta\in F^{\times},$ we have $\chi_{w}(\delta^{2})=1.$ Using the Stickelberger relation and the fact that $j(\chi_{w_{0}},\chi_{w_{0}})\equiv 1\mod(1-\zeta_{p})^{2}$, one can show that$j(\chi_{w},\chi_{w})=-\ell^{f},$ for $f=\frac{p-1}{2[F_{0}:{\mathbb{Q}}]}.$ Then, $P_{v}(1)=(1+\ell^{f})^{[F_{0}:{\mathbb{Q}}]}\equiv 2^{[F_{0}:{\mathbb{Q}}]}\mod q.$ Consequently, there are no $q$-torsion points in $J_{\delta}(F)$.

Similarly, for the case $q|2D_{L^{\prime}/{\mathbb{Q}}}$, let $c_{q}\geq 1$ be the smallest positive integer such that there is a $\sigma\in{\mathrm{Gal}}(L^{\prime}(\mu_{q^{c_{q}}})/{\mathbb{Q}})$ for which $\sigma|_{L}$ is a generator of ${\mathrm{Gal}}(L/F)$, $\sigma|_{F^{\prime}}=1$, and the restriction of $\sigma$ to ${\mathrm{Gal}}({\mathbb{Q}}(\mu_{q^{c_{q}}})/{\mathbb{Q}})$ has order greater than $f=\frac{p-1}{2[F_{0}:{\mathbb{Q}}]}.$ Then, $P_{v}(1)\equiv\hskip-9.5pt/\ 0\mod q^{c_{q}[F_{0}:{\mathbb{Q}}]}$. Let $M$ be definedby $M:=\prod_{q|2D_{L^{\prime}/{\mathbb{Q}}}}q^{c_{q}[F_{0}:{\mathbb{Q}}]}.$ It follows that $J_{\delta}(F)_{\mathrm{tor}}\subset J_{\delta}[M],$ the subgroup of $M$-torsion points of $J_{\delta}(\overline{F})$.

Let $F$ be a totally real field as in the main theorem. We have:

Let $F$ be as in the introduction. Then $F_{0}={\mathbb{Q}}(\mu_{p})^{+}$ is the maximal totally real subfield of $L_{0}={\mathbb{Q}}(\mu_{p}).$ By the reciprocity law, one can see that $w\mapsto\chi_{w}(\delta^{2})$ defines a Hecke character, which we denote by $\chi_{[\delta^{2}]}.$ It depends only on the class of $\delta^{2}$ and has conductor above $\delta.$ By Weil [32], the map $w\mapsto j(\chi_{w},\chi_{w}){\mathrm{N}}_{L/{\mathbb{Q}}}w^{-\frac{1}{2}}$ also defines a Hecke character on $L,$ denoted by $\psi,$ which has conductor above $p.$ Thus, we have a (unitary) Hecke character on $L$,

which is not of the form $\phi\circ{\mathrm{N}}_{L/F},$ for any Hecke character $\phi$ over $F.$ Then, there exists a unique holomorphic Hilbert newform $f/F$ of pure weight $2$ with trivial central character such that,

for all places $v$ of $F.$ Actually, the field over ${\mathbb{Q}}$ generated by the Hecke eigenvalues attached to $f$ is $F_{0}={\mathbb{Q}}(\mu_{p})^{+},$ and for the CM abelian variety $J_{\delta},$ we have

Note that $L(s,J_{\delta})$ only depends on the class $[\ \delta\ ]$ of $\delta,$ and the above equality holds for any local factor.

A nonvanishing result Let $\pi$ be the automorphic representation associated to $f,$ and let $N$ be its conductor. Let $S_{0}$ be any finite set of places of $F,$ including all infinite places and the places dividing $N.$ Choose a quadratic Hecke character $\xi$ corresponding to a totally imaginary quadratic extension of $F,$ unramified at $N,$ where $\xi(N)\cdot(-1)^{g}=-1$ (since $F$ is of odd degree, we have $(-1)^{g}=-1$); i.e., the epsilon factor of $L(s,\pi\otimes\xi)$ is $-1.$ Let ${\mathcal{D}}(\xi;S_{0})$ denote the set of quadratic characters $\chi$ of $F^{\times}/{\mathbb{A}}_{F}^{\times},$ for which $\chi_{v}=\xi_{v},$ for all $v\in S_{0}.$ With the above notation and assumptions, by a theorem of Friedberg and Hoffstein [11], there exist infinitely many quadratic characters $\chi\in{\mathcal{D}}(\xi;S_{0})$ such that $L(s,\pi\otimes\chi)$ has a simple zero at the center $s=1/2.$

Choose such a $\chi,$ and let $K$ be the totally imaginary quadratic extension of $F$ associated to it. The conductor of $\chi$ is coprime to $N,$ and the $L$-function $L(s,f/K)=L(s-1/2,\pi)L(s-1/2,\pi\otimes\chi)$ has a simple zero at $s=1.$ Let $d$ denote the discriminant of $K/F.$

Zhang’s formula \SubsubsecThe $(N,K)$-type Shimura curves Let ${\mathcal{O}}$ be the subalgebra of ${\mathbb{C}}$ over ${\mathbb{Z}}$ generated by the eigenvalues of $f$ under the Hecke operators. In our case, ${\mathcal{O}}={\mathbb{Z}}[\zeta+\zeta^{-1}]$ is the ring of integers of $F_{0}.$ In [33] (see also [5], [6]), Zhang constructs a Shimura curve $X$ of $(N,K)$-type, and proves that there exists a unique abelian subvariety $A$ of the Jacobian ${\mathrm{Jac}}(X)$ of dimension $[{\mathcal{O}}:{\mathbb{Z}}]=g,$ such that

for all places $v$ of $F.$ By the construction of $f,$ it follows that $L_{v}(s,A/F)=L_{v}(s,J_{\delta}/F)$ for all places $v$ of $F.$ Therefore, by the isogeny conjecture proved by Faltings, $A$ is isogenous to $J_{\delta}$ over $F.$ In particular, the complex multiplication by ${\mathcal{O}}\subset{\mathbb{Q}}(\mu_{p})^{+}$ on $A$ is defined over $F.$

Now, let us recall the constructions of $X$ and $A.$

The $L$-function of $\pi\otimes\chi$ satisfies the functional equation

where $\Sigma=\Sigma(N,K)$ is the following set of places of $F:$

Since the sign of the functional equation is $-1,$ by our choice of $K,$ the cardinality of $\Sigma$ is odd. Let $\tau$ be any real place of $F.$ Then, we have:

1. 1.

   Up to isomorphism, there exists a unique quaternion algebra $B$ such that $B$ is ramified at exactly the places in $\Sigma\backslash\{\tau\}$;

2. 2.

   There exist embeddings $\rho:K\hookrightarrow B$ over $F.$

From now on, we fix an embedding $\rho:K\rightarrow B$ over $F.$

Let $G$ denote the algebraic group over $F,$ which is an inner form of ${\mathrm{PGL}}_{2}$ with $G(F)\cong B^{\times}/F^{\times}.$ The group $G(F_{\tau})\cong{\mathrm{PGL}}_{2}({\mathbb{R}})$ acts on ${\mathcal{H}}^{\pm}={\mathbb{C}}\setminus{\mathbb{R}}.$ Now, for any open compact subgroup $U$ of $G({\mathbb{A}}_{f}),$ we have an analytic space

where $G(F)_{+}$ denotes the subgroup of elements in $G(F)$ with positive determinant via $\tau.$

Shimura has shown that $S_{U}({\mathbb{C}})$ is the set of complex points of an algebraic curve $S_{U},$ which descends canonically to $F$ (as a subfield of ${\mathbb{C}}$ via $\tau$). The curve $S_{U}$ over $F$ is independent of the choice of $\tau.$

There exists an order $R_{0}$ of $B$ containing ${\mathcal{O}}_{K}$ with reduced discriminant $N.$ One can choose $R_{0}$ as follows. Let ${\mathcal{O}}_{B}$ be a maximal order of $B$ containing ${\mathcal{O}}_{K},$ and let ${\mathcal{N}}$ be an ideal of ${\mathcal{O}}_{K}$ such that

where ${\mathrm{disc}}_{B/F}$ is the reduced discriminant of ${\mathcal{O}}_{B}$ over ${\mathcal{O}}_{F}.$ Then, we take

Take $U=\prod_{v}R^{\times}_{v}/{\mathcal{O}}_{v}^{\times}.$ The corresponding Shimura curve $X:=S_{U}$ is compact.

Let $\xi\in\mathrm{Pic}(X)\otimes{\mathbb{Q}}$ be the unique class whose degree is $1$ on each connected component and such that,

for all integral ideals $m$ of ${\mathcal{O}}_{F}$ coprime to $Nd.$ Here, the ${\mathrm{T}}_{m}$ are the Hecke operators.

Gross-Zagier-Zhang formula Now, we define the basic class in ${\mathrm{Jac}}(X)(K)\otimes{\mathbb{Q}},$ where ${\mathrm{Jac}}(X)$ is the connected component of $\mathrm{Pic}(X),$ from the CM-points on the curve $X.$ The CM points corresponding to $K$ on $X$ form a set:

where $h_{0}\in{\mathcal{H}}^{+}$ is the unique fixed point of the torus $T(F)=K^{\times}/F^{\times}.$

For a CM point $z=[g]\in{\mathcal{C}},$ represented by $g\in G({\mathbb{A}}_{f}),$ let

Then, ${\mathrm{End}}(z):=\Phi_{g}^{-1}(\widehat{R_{0}})$ is an order of $K,$ say ${\mathcal{O}}_{n}={\mathcal{O}}_{F}+n{\mathcal{O}}_{K},$ for a (unique) ideal $n$ of $F.$ The ideal $n,$ called the conductor of $z,$ is independent of the choice of the representative $g.$ By Shimura’s theory, every CM point of conductor $n$ is defined over the abelian extension $H_{n}^{\prime}$ of $K$ corresponding to $K^{\times}\setminus\widehat{K}^{\times}/\widehat{F}^{\times}\widehat{{\mathcal{O}}}_{n}^{\times}$ via class field theory.

Let $P_{1}$ be a CM point in $X$ of conductor $1,$ which is defined over $H_{1}^{\prime},$ the abelian extension of $K$ corresponding to $K^{\times}\setminus\widehat{K}^{\times}/\widehat{F}^{\times}\widehat{{\mathcal{O}}}_{K}^{\times}.$ The divisor $P={\mathrm{Gal}}(H_{1}^{\prime}/K)\cdot P_{1}$ together with the Hodge class defines a class

where $\deg P$ is the multi-degree of $P$ on the geometric components. Let $x_{f}$ be the $f$-typical component of $x.$ In [34], Zhang generalized the Gross-Zagier formula to the totally real field case, by proving that

where $\|f\|^{2}$ is computed on the invariant measure on

induced by $dxdy/y^{2}$ on ${\mathcal{H}}^{g},$ and where

and $\|x_{f}\|^{2}$ is the Neron-Tate pairing of $x_{f}$ with itself.

The equivalence of nonvanishing of $L$-factors For any $\sigma:F\hookrightarrow{\mathbb{C}},$ it is known by a result of Shimura that $L(1,f/F)\neq 0$ is equivalent to $L(1,f^{\sigma}/F)\break\neq 0.$ One can also show this using Zhang’s formula above. To see this, assume $L(1,f/F)\neq 0.$ Then, $\|x_{f}\|\neq 0,$ and therefore, $\|x_{f^{\sigma}}\|\neq 0.$ It follows that $L^{\prime}(1,f^{\sigma}/K)\neq 0.$ Since $L(1,f/F)\neq 0,$ the $L$-function $L(s,f^{\sigma}/F)$ has a positive sign in its functional equation. Thus, $L(1,f^{\sigma}/F)\neq 0.$ In fact, to obtain our main theorem, we do not need this equivalence, but we may see that Theorem 3.1 is equivalent to statement (2) in the introduction.

The Euler system of CM points We now assume that $L(1,\chi_{[\delta^{2}]}\psi)\neq 0,$ or equivalently, $L(1,f/F)\neq 0.$ Then by the equivalence of nonvanishing of $L(1,f^{\sigma})$ for all embeddings $\sigma:F\hookrightarrow{\mathbb{C}},$ we have that $L(1,J_{\delta}/F)\neq 0.$ By Zhang’s formula, we also know that $\|x_{f}\|\neq 0.$

Let ${\mathcal{N}}$ be the set of square-free integral ideals of $F$ whose prime divisors are inert in $K$ and coprime to $Nd.$ For any $n\in{\mathcal{N}},$ define

Let $u_{n}$ denote the cardinality of $(\widehat{{\mathcal{O}}}_{n}^{\times}\cap K^{\times}\widehat{F}^{\times})/\widehat{{\mathcal{O}}}_{F}^{\times}.$ Then, $H_{\ell}/H_{1}$ is a cyclic extension of degree $t(\ell)=\frac{{\mathrm{N}}(\ell)+1}{u_{1}/u_{\ell}}.$

For each $n\in{\mathcal{N}},$ let $P_{n}$ be a CM point of order $n$ such that $P_{n}$ is contained in ${\mathrm{T}}_{\ell}P_{m}$ if $n=m\ell\in{\mathcal{N}}$ and $\ell$ is a prime ideal of $F.$ Let $y_{n}={\mathrm{Tr}}_{H_{n}^{\prime}/H_{n}}\pi(P_{n})\in A(H_{n}),$ where $\pi$ is a morphism from $X$ to ${\mathrm{Jac}}(X)$ defined by a multiple of the Hodge class.

The points $\{y_{n}\}_{n\in{\mathcal{N}}}$ form an Euler system (see [29, Prop. 7.5], or [33, Lemma 7.2.2]) so that, for any $n=m\ell\in{\mathcal{N}}$ with $\ell$ a prime ideal of $F,$

1. 1.

   $\displaystyle{{u_{n}}^{-1}\sum_{\sigma\in{\mathrm{Gal}}(H_{n}/H_{m})}y_{n}^{\sigma}={u_{m}}^{-1}a_{\ell}y_{m}}$;

2. 2.

   For any prime ideal $\lambda_{m}$ of $H_{m}$ above $\ell,$ and for $\lambda_{n}$ the unique prime above $\lambda_{m},$

   $${\mathrm{Frob}}_{\lambda_{m}}y_{m}\equiv y_{n}\mod\lambda_{n};$$

3. 3.

   The class $x_{f}$ is equal to $y_{K}:={\mathrm{tr}}_{H_{1}/K}y_{1}$ in $\big{(}A(K)\otimes{\mathbb{Q}}\big{)}\big{/}{\mathbb{Q}}^{\times}.$

Theorem 2.1 follows with the nontrivial Euler system by Kolyvagin’s standard argument (see [21], [23], [13], and [33, Th. A]).

up

Let $r=4$ or an odd prime, and let $L=F(\zeta_{r}),$ with $[L:F]=2.$ Let $\psi$ be a unitary Hecke character of $L.$ In this section, we show:

Let $\rho$ be a unitary Hecke character of $F.$ The purpose of this section is to construct a perfect double Dirichlet series $Z(s,w;\psi;\rho)$ similar to an Asai-Flicker-Patterson type Rankin-Selberg convolution, which possesses meromorphic continuation to ${\mathbb{C}}^{2}$ and functional equations. Then, Theorem $3.1$ will follow from the analytic properties of $Z(s,w;\psi;\rho)$ (when $r=4$, see [7]). To do this, it is necessary to recall the Fisher-Friedberg symbol in [9].

The $r$-th power residue symbol Let $S^{\prime}$ be a finite set of non-archimedean places of $L$ containing all places dividing $r,$ and such that the ring of $S^{\prime}$-integers ${\mathcal{O}}_{L}^{S^{\prime}}$ has class number one. We shall also assume that $S^{\prime}$ is closed under conjugation and that $\psi$ and $\rho$ are both unramified outside $S^{\prime}.$

Let $S_{\infty}$ denote the set of all archimedean places of $L,$ and set $S=S^{\prime}\cup S_{\infty}.$ Let $I_{L}(S)$ (resp. ${\mathcal{I}}_{L}(S)$) denote the group of fractional ideals (resp. the set of all integral ideals) of ${\mathcal{O}}_{L}$ coprime to $S^{\prime}.$ In [9], Fisher and Friedberg have shown that the $r$-th order symbol $\chi_{n}$ can be extended to $I_{L}(S)$ i.e., $\chi_{\mathfrak{n}}(\mathfrak{m})$ is defined for $\mathfrak{m},$ $\mathfrak{n}\in I_{L}(S).$ Let us recall their construction.

For a non-archimedean place $v\in S^{\prime},$ let $\mathfrak{P}_{v}$ denote the corresponding ideal of $L.$ Define $\mathfrak{c}=\prod_{v\in S^{\prime}}\mathfrak{P}_{v}^{r_{v}}$ with $r_{v}=1$ if ${\mathrm{ord}}_{v}(r)=0$, and $r_{v}$ sufficiently large such that, for $a\in L_{v},$ ${\mathrm{ord}}_{v}(a-1)\geq r_{v}$ implies that $a\in(L^{\times}_{v})^{r}$. Let $P_{L}({\mathfrak{c}})\subset I_{L}(S)$ be the subgroup of principal ideals $(\alpha)$ with $\alpha\equiv 1\mod{\mathfrak{c}},$ and let $H_{\mathfrak{c}}=I_{L}(S)/P_{L}({\mathfrak{c}})$ be the ray class group modulo ${\mathfrak{c}}.$ Set $R_{\mathfrak{c}}=H_{\mathfrak{c}}\otimes{\mathbb{Z}}/r{\mathbb{Z}},$ and write the finite group $R_{\mathfrak{c}}$ as a direct product of cyclic groups. Choose a generator for each, and let ${\mathfrak{E}}_{0}$ be a set of ideals of ${\mathcal{O}}_{L},$ prime to $S,$ which represent these generators. For each ${\mathfrak{e}}_{0}\in{\mathfrak{E}}_{0},$ choose $m_{{\mathfrak{e}}_{0}}\in L^{\times}$ such that ${\mathfrak{e}}_{0}{\mathcal{O}}_{L}^{S^{\prime}}=m_{{\mathfrak{e}}_{0}}{\mathcal{O}}_{L}^{S^{\prime}}.$ Let ${\mathfrak{E}}$ be a full set of representatives for $R_{\mathfrak{c}}$ of the form $\prod_{{\mathfrak{e}}_{0}\in{\mathfrak{E}}_{0}}{\mathfrak{e}}_{0}^{\lambda_{{\mathfrak{e}}_{0}}}.$ Note that ${\mathfrak{e}}{\mathcal{O}}_{L}^{S^{\prime}}=m_{\mathfrak{e}}{\mathcal{O}}_{L}^{S^{\prime}}$ for all ${\mathfrak{e}}\in{\mathfrak{E}}.$ Without loss, we suppose that ${\mathcal{O}}_{L}^{S^{\prime}}\in{\mathfrak{E}}$ and $m_{{\mathcal{O}}_{L}^{S^{\prime}}}=1.$

Let ${\mathfrak{m}},{\mathfrak{n}}\in I_{L}(S)$ be coprime. Write ${\mathfrak{m}}=(m){\mathfrak{e}}{\mathfrak{g}}^{r}$ with ${\mathfrak{e}}\in{\mathfrak{E}},$ $m\in L^{\times}$, $m\equiv 1\mod{\mathfrak{c}}$ and ${\mathfrak{g}}\in I_{L}(S),$ $({\mathfrak{g}},{\mathfrak{n}})=1.$ Then the $r$-th power residue symbol $\left(\frac{mm_{\mathfrak{e}}}{{\mathfrak{n}}}\right)_{r}$ is defined. If ${\mathfrak{m}}=(m^{\prime}){\mathfrak{e}}^{\prime}{\mathfrak{g}}^{{}^{\prime}r}$ is another such decomposition, then ${\mathfrak{e}}^{\prime}={\mathfrak{e}}$ and $\left(\frac{m^{\prime}m_{{\mathfrak{e}}^{\prime}}}{{\mathfrak{n}}}\right)_{r}=\left(\frac{mm_{\mathfrak{e}}}{{\mathfrak{n}}}\right)_{r}.$

In view of this, the $r$-th power residue symbol $\left(\frac{{\mathfrak{m}}}{{\mathfrak{n}}}\right)_{r}$ is defined to be$\left(\frac{mm_{\mathfrak{e}}}{{\mathfrak{n}}}\right)_{r},$ and the character $\chi_{\mathfrak{m}}$ is defined by $\chi_{\mathfrak{m}}({\mathfrak{n}})=\left(\frac{{\mathfrak{m}}}{{\mathfrak{n}}}\right)_{r}.$ This extension of the $r$-th power residue symbol depends on the above choices. Let $S_{\mathfrak{m}}$ denote the support of the conductor of $\chi_{\mathfrak{m}}.$ It can be easily checked that if ${\mathfrak{m}}={\mathfrak{m}}^{\prime}{\mathfrak{a}}^{r}$, then $\chi_{\mathfrak{m}}({\mathfrak{n}})=\chi_{{\mathfrak{m}}^{\prime}}({\mathfrak{n}})$ whenever both are defined. This allows one to extend $\chi_{\mathfrak{m}}$ to a character of all ideals of $I_{L}(S\cup S_{\mathfrak{m}}).$

The extended symbol possesses a reciprocity law: if ${\mathfrak{m}},{\mathfrak{n}}\in I_{L}(S)$ are coprime, then $\alpha({\mathfrak{m}},{\mathfrak{n}})=\chi_{\mathfrak{m}}({\mathfrak{n}})\chi_{\mathfrak{n}}({\mathfrak{m}})^{-1}$ depends only on the images of ${\mathfrak{m}},{\mathfrak{n}}$ in $R_{\mathfrak{c}}.$

In our situation, we also need the following lemma:

If $[{\mathfrak{n}}]$ is in the kernel, i.e., ${\mathfrak{n}}=(\alpha)$ in $I_{L}(S)$ is a principal ideal with $\alpha\equiv 1\mod{\mathfrak{c}},$ then $\alpha/\overline{\alpha}$ is a root of unity with $\alpha/\overline{\alpha}\equiv 1\mod{\mathfrak{c}}.$ Now let $W$ be the set of roots of unity in $L$ which are $\equiv 1\mod{\mathfrak{c}}.$ Let $W_{0}$ be the subset of $W$ of elements of the form $u/\overline{u}$ for some unit $u$ in ${\mathcal{O}}_{L}$ and $u\equiv 1\mod{\mathfrak{c}}.$ It is clear that $W_{0}\supset W^{2}.$ Then, the map

is obviously injective; i.e., the order of the kernel of the natural map in this lemma is a power of $2.$ \Endproof

Since $r$ is odd, using the lemma, we may choose a suitable set ${\mathfrak{E}}_{0}$ of representatives since the beginning such that if ${\mathfrak{m}}\in I_{F}(S),$ then the decomposition ${\mathfrak{m}}=(m){\mathfrak{e}}{\mathfrak{g}}^{r}$ is such that $m\in F^{\times}$, ${\mathfrak{e}},{\mathfrak{g}}\in I_{F}(S).$

Using the symbol $\chi_{\mathfrak{n}},$ we shall construct a perfect double Dirichlet series $Z(s,w;\psi;\rho)$ (i.e., possessing meromorphic continuation to ${{\mathbb{C}}}^{2}$) of type: