2.1 Galois representations and the adelic level

Let $E/\mathbb {Q}$ be an elliptic curve. Associated with E, we consider the adelic Tate module, which is given by the inverse limit

where $E[n]$ denotes the n-torsion subgroup of $E(\overline {\mathbb {Q}})$ . Let $\widehat {\mathbb {Z}}$ denote the ring of profinite integers. It is well known that $T(E)$ is a free $\widehat {\mathbb {Z}}$ -module of rank $2$ . The absolute Galois group $\operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathbb {Q})$ acts naturally on $T(E)$ , giving rise to the adelic Galois representation of E,

$$\begin{align*}\rho_E \colon \operatorname{\mathrm{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \operatorname{\mathrm{Aut}}(T(E)). \end{align*}$$

Upon fixing a $\widehat {\mathbb {Z}}$ -basis for $T(E)$ , we consider $\rho _E$ as a map

$$\begin{align*}\rho_E \colon \operatorname{\mathrm{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \operatorname{\mathrm{GL}}_2(\widehat{\mathbb{Z}}). \end{align*}$$

Let $G_E$ denote the image of $\rho _E$ , which, because of the above choice of basis, is defined only up to conjugacy in $\operatorname {\mathrm {GL}}_2(\widehat {\mathbb {Z}})$ . With respect to the profinite topology on $\operatorname {\mathrm {GL}}_2(\widehat {\mathbb {Z}})$ , the subgroup $G_E$ is necessarily closed since $\rho _E$ is a continuous map.

We now state a foundational result of Serre, known as Serre’s open image theorem.

Suppose that $E/\mathbb {Q}$ is a non-CM elliptic curve. For each positive integer m, let $\pi _m$ be the natural reduction map

$$\begin{align*}\pi_m \colon \operatorname{\mathrm{GL}}_2(\widehat{\mathbb{Z}}) \longrightarrow \operatorname{\mathrm{GL}}_2(\mathbb{Z}/m\mathbb{Z}). \end{align*}$$

Let $G_E(m)$ be the image of the mod m Galois representation

$$\begin{align*}\rho_{E,m} \colon \operatorname{\mathrm{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \operatorname{\mathrm{GL}}_2(\mathbb{Z}/m\mathbb{Z}),\end{align*}$$

defined by the composition $\pi _m\circ \rho _E$ . It follows from Theorem 2.1 that there exists a positive integer m for which

(8) $$ \begin{align} G_E = \pi_m^{-1}(G_E(m)). \end{align} $$

One may observe that (8) is equivalent to the statement that for every $n \in \mathbb {N}$ ,

(9) $$ \begin{align} G_E(n) = \pi^{-1}(G_E(\gcd(n,m))), \end{align} $$

where $\pi \colon \operatorname {\mathrm {GL}}_2(\mathbb {Z}/n\mathbb {Z}) \to \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\gcd (n,m)\mathbb {Z})$ denotes the natural reduction map. The least positive integer m with this property is called the adelic level of E, and is denoted by $m_E$ . The constant $m_E$ accounts for both the nonsurjectivity of the $\ell $ -adic Galois representations of E as well as the entanglements between their images.

We now give a fundamental property of $m_E$ that we will use several times.

Lemma 2.2 Let $E/\mathbb {Q}$ be a non-CM elliptic curve of adelic level $m_E$ . For any $d_1,d_2 \in \mathbb {N}$ with $d_1 \mid m_E^\infty $ and $(d_2, m_E) = 1$ , we have

$$ \begin{align*}G_E(d_1d_2) \simeq G_E(d_1) \times \operatorname{\mathrm{GL}}_2(\mathbb{Z}/d_2\mathbb{Z})\end{align*} $$

via the map $\operatorname {\mathrm {GL}}_2(\mathbb {Z}/d_1d_2\mathbb {Z}) \to \operatorname {\mathrm {GL}}_2(\mathbb {Z}/d_1\mathbb {Z}) \times \operatorname {\mathrm {GL}}_2(\mathbb {Z}/d_2\mathbb {Z})$ .

Proof By the given conditions, we have $(d_1,d_2) = 1$ . Set $d^{\prime } = \gcd (d_1,m_E)$ . Let $\pi \colon \operatorname {\mathrm {GL}}_2(\mathbb {Z}/d_1d_2\mathbb {Z}) \to \operatorname {\mathrm {GL}}_2(\mathbb {Z}/d^{\prime }\mathbb {Z})$ and $\pi _1 \colon \operatorname {\mathrm {GL}}_2(\mathbb {Z}/d_1\mathbb {Z}) \to \operatorname {\mathrm {GL}}_2(\mathbb {Z}/d^{\prime }\mathbb {Z})$ be the natural reduction maps. By the Chinese remainder theorem, $\pi $ can be identified with

$$ \begin{align*}\pi_1 \times \text{triv} \colon \operatorname{\mathrm{GL}}_2(\mathbb{Z}/d_1\mathbb{Z}) \times \operatorname{\mathrm{GL}}_2(\mathbb{Z}/d_2\mathbb{Z}) \to \operatorname{\mathrm{GL}}_2(\mathbb{Z}/d^{\prime}\mathbb{Z}) \times \{1\}.\end{align*} $$

By (9), we have that

$$\begin{align*}G_E(d_1d_2) = \pi^{-1}(G_E(d^{\prime})) \simeq (\pi_1 \times \text{triv})^{-1}(G_E(d^{\prime})) = G_E(d_1) \times \operatorname{\mathrm{GL}}_2(\mathbb{Z}/d_2\mathbb{Z}).\\[-34pt] \end{align*}$$

We conclude this subsection by recalling Serre’s uniformity question.

Question 2.3 Does there exist an absolute constant c such that for each elliptic curve $E/\mathbb {Q}$ ,

$$ \begin{align*}G_E(\ell) = \operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})\end{align*} $$

holds for all rational primes $\ell> c$ ?

While Question 2.3 remains open, it is widely conjectured to be true with $c = 37$ [Reference Sutherland56, Reference Zywina63] and considerable partial progress has been made toward its resolution [Reference Balakrishnan, Netan Dogra, Müller, Tuitman and Vonk4, Reference Furio and Lombardo25, Reference Lemos40, Reference Mazur43, Reference Serre48, Reference Serre49].

2.2 Serre curves

In this subsection, we introduce the generic class of elliptic curves $E/\mathbb {Q}$ with maximal adelic Galois image $G_E$ , and provide an explicit description of $G_E$ for curves in this class.

Serre noted [Reference Serre48] that for an elliptic curve $E/\mathbb {Q}$ , the adelic Galois representation $\rho _E$ cannot be surjectiveFootnote ² , that is, the adelic level $m_E$ is never $1$ . We briefly give the argument here. If E has complex multiplication, then $[\operatorname {\mathrm {GL}}_2(\widehat {\mathbb {Z}}):G_E]$ is necessarily infinite [Reference Serre48], so we restrict our attention to the case that E is non-CM. Assume that E is defined by the factored Weierstrass equation

$$ \begin{align*}Y^2 = (X-e_1)(X-e_2)(X-e_3)\end{align*} $$

with $e_1,e_2,e_3 \in \overline {\mathbb {Q}}$ . Then, the $2$ -torsion of E is given by

$$\begin{align*}E[2] = \left\{\mathcal{O}, (e_1,0), (e_2,0), (e_3,0)\right\} \cong \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}.\end{align*}$$

Consequently, $\operatorname {\mathrm {Aut}}(E[2])$ can be identified with $S_3$ . The discriminant $\Delta _E$ of E is given by

(10) $$ \begin{align} \Delta_E = \left[(e_1-e_2) (e_2-e_3)(e_3-e_1) \right]^2. \end{align} $$

Let $\Delta '$ denote the squarefree part of $\Delta _E$ , i.e., the unique squarefree integer such that $\Delta _E/\Delta ^{\prime } \in (\mathbb {Q}^\times )^2$ . Note that the discriminant $\Delta _E$ depends on the Weierstrass model of E, but $\Delta ^{\prime }$ does not.

Let us first assume that $\Delta _E \not \in (\mathbb {Q}^\times )^2$ . Let $d_E$ be the conductor of $\mathbb {Q}(\sqrt {\Delta _E})$ , that is, the smallest positive integer such that $\mathbb {Q}(\sqrt {\Delta _E}) \subseteq \mathbb {Q}(\zeta _{d_E})$ . It is straightforward to check that

$$ \begin{align*}d_E = \begin{cases} |\Delta^{\prime}| & \text{ if } \Delta^{\prime} \equiv 1 \quad\pmod 4,\\ 4|\Delta^{\prime}| & \text{ otherwise}. \end{cases}\end{align*} $$

Let us define the quadratic character associated with $\mathbb {Q}(\sqrt {\Delta _E})$ as follows,

$$ \begin{align*}\chi_{\Delta_E} \colon \operatorname{\mathrm{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q}) \xrightarrow{\text{rest}.} \operatorname{\mathrm{Gal}}(\mathbb{Q}(\sqrt{\Delta_E})/\mathbb{Q}) \overset{\sim}{\longrightarrow} \{\pm 1\}.\end{align*} $$

Fix $\sigma \in \operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathbb {Q})$ . Viewing $\rho _{E,2}(\sigma ) \in G_E(2) \subseteq \operatorname {\mathrm {Aut}}(E[2]) \simeq S_3$ , by (10), we notice that

$$ \begin{align*}\chi_{\Delta_E}(\sigma) \left(\sqrt{\Delta_E}\right) = \epsilon(\rho_{E, 2}(\sigma))\left(\sqrt{\Delta_E}\right),\end{align*} $$

where $\epsilon \colon S_3 \to \{\pm 1\}$ denotes the signature map.Footnote ³ Hence, $\chi _{\Delta _E}(\sigma ) = \epsilon (\rho _{E,2}(\sigma ))$ .

On the other hand, we have that $\mathbb {Q}(\sqrt {\Delta _E}) \subseteq \mathbb {Q}(\zeta _{d_E})$ . Since $\operatorname {\mathrm {Gal}}(\mathbb {Q}(\zeta _{d_E})/\mathbb {Q}) \simeq (\mathbb {Z}/d_E\mathbb {Z})^\times $ , there exists a unique quadratic character $\alpha \colon \operatorname {\mathrm {Gal}}(\mathbb {Q}(\zeta _{d_E})/\mathbb {Q}) \to \{\pm 1\}$ for which $\chi _{\Delta _E}(\sigma ) = \alpha (\operatorname {det} \circ \rho _{E,{d_E}}(\sigma ))$ for any $\sigma \in \operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathbb {Q})$ . Therefore, we have

(11) $$ \begin{align} \epsilon(\rho_{E,2}(\sigma)) = \alpha(\operatorname{det} \circ \rho_{E,d_E}(\sigma)) \end{align} $$

for any $\sigma \in \operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathbb {Q})$ .

Let . Consider the subgroup

$$ \begin{align*}H_E(M_E) = \left\{ M \in \operatorname{\mathrm{GL}}_2(\mathbb{Z}/M_E\mathbb{Z}) : \epsilon(M_2) = \alpha(\operatorname{det} M_{d_E})\right\},\end{align*} $$

where $M_2$ and $M_{d_E}$ denote the reductions of M modulo $2$ and $d_E$ , respectively. Note that the index of $H_E(M_E)$ in $\operatorname {\mathrm {GL}}_2(\mathbb {Z}/M_E\mathbb {Z})$ is $2$ and that $G_E(M_E) \subseteq H_E(M_E)$ by (11). We define

(12)

where $\pi \colon \operatorname {\mathrm {GL}}_2(\widehat {\mathbb {Z}}) \to \operatorname {\mathrm {GL}}_2(\mathbb {Z}/M_E\mathbb {Z})$ is the natural reduction map. Then $H_E$ is an index $2$ subgroup of $\operatorname {\mathrm {GL}}_2(\widehat {\mathbb {Z}})$ that contains $G_E$ . We say that E is a Serre curve if $H_E = G_E$ , that is, $[\operatorname {\mathrm {GL}}_2(\widehat {\mathbb {Z}}):G_E] = 2$ .

In the above discussion, we supposed that $\Delta _E \not \in (\mathbb {Q}^\times )^2$ . We now consider the opposite case that $\Delta _E \in (\mathbb {Q}^\times )^2$ . Let $\mathbb {Q}(E[2]) = \mathbb {Q}(e_1,e_2,e_3)$ denote the $2$ -division field of E. Observe that $[\mathbb {Q}(E[2]):\mathbb {Q}]$ divides $3$ , and hence $[\operatorname {\mathrm {GL}}_2(\mathbb {Z}/2\mathbb {Z}) : G_E(2)]$ is divisible by $2$ . Thus, by [Reference Mayle and Rakvi42, Proposition 2.14], $[\operatorname {\mathrm {GL}}_2(\widehat {\mathbb {Z}}) : G_E] \geq 12$ , which follows by considering the index of the commutator of $G_E$ in $\operatorname {\mathrm {SL}}_2(\widehat {\mathbb {Z}})$ . In particular, E cannot be a Serre curve in this case.

Serre curves are useful for us for two key reasons. First, as mentioned in the introduction, Jones [Reference Jones32] showed that they are “generic” in the sense that the density of the subfamily of Serre curves among the family of all elliptic curves ordered by naive height is $1$ . Second, the adelic image $G_E$ of a Serre curve E can be explicitly described, as we will now discuss.

Proposition 2.4 Let $E/\mathbb {Q}$ be a Serre curve and write $\Delta ^{\prime }$ to denote the squarefree part of the discriminant of E. Then

(13) $$ \begin{align} m_E = \begin{cases} 2|\Delta^{\prime}| & \text{ if } \Delta^{\prime} \equiv 1 \quad\pmod 4,\\ 4|\Delta^{\prime}| & \text{ otherwise}. \end{cases} \end{align} $$

Furthermore, for any positive integer m,

$$ \begin{align*}G_E(m) = \begin{cases} \operatorname{\mathrm{GL}}_2(\mathbb{Z}/m\mathbb{Z}) & \text{ if } m_E \nmid m, \\ H_E(m) & \text{ if } m_E \mid m, \end{cases}\end{align*} $$

where $H_E(m)$ denotes the image of $H_E$ , defined in (12), under the reduction modulo m map.

In order to compute $C^{\mathcal {X}}_{E,n,k}$ , we need to know $G_E$ (meaning we must know the adelic level $m_E$ and the image of $G_E$ modulo $m_E$ ). For Serre curves, this is particularly tractable, and was exploited in the work of Jones [Reference Jones31]. We now give the description of $G_E$ for Serre curves.

First, we define $\chi _4 \colon (\mathbb {Z}/4\mathbb {Z})^\times \to \{\pm 1\}$ and $\chi _8 \colon (\mathbb {Z}/8\mathbb {Z})^\times \to \{\pm 1\}$ as follows:

$$\begin{align*}\chi_4(k) = \begin{cases} 1 & \text{ if }k\equiv 1\quad\pmod 4\\ -1 & \text{ if }k\equiv 3\quad\pmod 4 \end{cases}, \quad \chi_8(k)=\begin{cases} 1 & \text{ if }k\equiv 1, 7\quad\pmod 8\\ -1 & \text{ if }k\equiv 3, 5\quad\pmod 8 \end{cases}. \end{align*}$$

We define the character $\psi _m \colon \operatorname {\mathrm {GL}}_2(\mathbb {Z}/m\mathbb {Z}) \to \{\pm 1\}$ associated with E by

$$\begin{align*}\psi_m=\prod_{\ell^{\alpha}\parallel m}\psi_{\ell^{\alpha}}, \end{align*}$$

where $\psi _{\ell ^\alpha } \colon \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^\alpha \mathbb {Z}) \to \{\pm 1\}$ is defined for $M \in \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^\alpha \mathbb {Z})$ by

$$ \begin{align*} \psi_{\ell^\alpha}(M) = \begin{cases} \left( \frac{\operatorname{det} M_\ell}{\ell}\right) & \text{ if } \ell \text{ is odd},\\ \epsilon(M_2) & \text{ if } \ell= 2, \alpha \geq 1, \text{ and } \Delta^{\prime} \equiv 1 \quad\pmod 4, \\ \chi_4(\operatorname{det} M_4) \epsilon(M_2) & \text{ if } \ell = 2 , \alpha \geq 2,\text{ and } \Delta^{\prime} \equiv 3 \quad\pmod 4, \\ \chi_8(\operatorname{det} M_8) \epsilon(M_2) & \text{ if } \ell =2 , \alpha \geq 3,\text{ and } \Delta^{\prime} \equiv 2 \quad\pmod 8,\\ \chi_8(\operatorname{det} M_8) \chi_4(\operatorname{det} M_4) \epsilon(M_2) & \text{ if } \ell = 2 , \alpha \geq 3, \text{ and } \Delta^{\prime} \equiv 6 \quad\pmod 8, \\ 1 & \text{ otherwise}. \end{cases} \end{align*} $$

As noted in [Reference Jones31, p. 701], given $m_E\mid m$ , one may see that for $M \in \operatorname {\mathrm {GL}}_2(\mathbb {Z}/m\mathbb {Z})$ , we have

$$ \begin{align*}\epsilon(M_2) \left(\frac{\Delta^{\prime}}{\operatorname{det} M_{m_E}}\right) = \psi_m(M).\end{align*} $$

In particular, we have $H_E(m) = \ker \psi _m$ . Thus $G_E$ is the preimage of $\ker \psi _m$ in $\operatorname {\mathrm {GL}}_2(\widehat {\mathbb {Z}})$ .

2.3 Galois representations in the CM case

Having discussed Galois representations for non-CM elliptic curves, we now turn to the CM case. Suppose that E has CM by an order $\mathcal {O}$ in an imaginary quadratic field K. In this case, the absolute Galois group $\operatorname {\mathrm {Gal}}(\overline {K}/K)$ acts naturally on $T(E)$ , which is a one-dimensional $\widehat {\mathcal {O}}$ -module, where $\widehat {\mathcal {O}}$ denotes the profinite completion of $\mathcal {O}$ . Hence, we can construct the adelic Galois representation associated with E,

$$ \begin{align*}\rho_E \colon\operatorname{\mathrm{Gal}}(\overline{K}/K) \to \operatorname{\mathrm{Aut}}(T(E)) \simeq \operatorname{\mathrm{GL}}_1(\widehat{\mathcal{O}}) \simeq \widehat{\mathcal{O}}^\times.\end{align*} $$

Let $G_E$ denote the image of $\rho _E$ . We now state Serre’s open image theorem for CM elliptic curves.

For each positive integer m, consider the natural reduction map

$$ \begin{align*}\pi_m \colon \widehat{\mathcal{O}}^\times \to (\mathcal{O}/m\mathcal{O})^\times.\end{align*} $$

Let $G_E(m)$ denote the image of the modulo m Galois representation

$$\begin{align*}\rho_{E, m}: \operatorname{\mathrm{Gal}}(\overline{K}/K) \to (\mathcal{O}/m\mathcal{O})^\times \end{align*}$$

defined by the composition $\pi _m\circ \rho _E$ . It follows from Theorem 2.5 that

(14) $$ \begin{align} G_E = \pi_m^{-1}(G_E(m)) \end{align} $$

for some positive integer m. As in the non-CM case, (14) is equivalent to the statement that for every $n \in \mathbb {N}$ ,

(15) $$ \begin{align} G_E(n) = \pi^{-1}(G_E(\gcd(n,m))), \end{align} $$

where $\pi \colon (\mathcal {O}/n\mathcal {O})^{\times }\to (\mathcal {O}/\gcd (n, m)\mathcal {O})^{\times }$ is the natural reduction map.

In the CM case, we follow [Reference Jones31, p. 693] to define $m_E$ to be the smallest positive integer m such that (15) holds and for which

(16) $$ \begin{align} 4 \left( \prod_{\ell \text{ ramifies in }K} \ell \right) \text{ divides } m. \end{align} $$

One can prove the following using the same argument sketched in the proof of Lemma 2.2.

Lemma 2.6 Let $E/\mathbb {Q}$ be a CM elliptic curve of level $m_E$ . For any $d_1,d_2 \in \mathbb {N}$ with $d_1 \mid m_E^\infty $ and $(d_2, m_E) = 1$ , we have

$$ \begin{align*}G_E(d_1d_2) \simeq G_E(d_1) \times (\mathcal{O}/d_2\mathcal{O})^\times.\end{align*} $$

Lemmas 2.2 and 2.6 are used to express the constants $C^{\operatorname {\mathrm {cyc}}}_{E,n,k}$ and $C^{\operatorname {\mathrm {prime}}}_{E,n,k}$ as almost Euler products. It is worth noting that both lemmas hold even if $m_E$ is replaced by any positive multiple of it. Thus, the minimality condition in the definition of $m_E$ for both non-CM and CM curves is not required from a theoretical perspective for us. Nonetheless, the minimality of $m_E$ is useful for our computations as it allows us to extract more Euler factors.

Let $K/\mathbb {Q}$ be an imaginary quadratic field. We denote its ring of integers by $\mathcal {O}_K$ . Let $\mathcal {O}$ be an order of K. The index $f = [\mathcal {O}_K : \mathcal {O}]$ is necessarily finite and is called the conductor of $\mathcal {O}$ . Let $\chi _K$ be the Dirichlet character defined by

(17) $$ \begin{align} \chi_K(\ell) =\begin{cases} 0 & \text{ if } \ell \text{ ramifies in } K, \\ 1 & \text{ if } \ell \text{ splits in } K, \\ -1 & \text{ if } \ell \text{ is inert in } K. \end{cases} \end{align} $$

Let $d_K$ be the discriminant of K. One can check that

$$ \begin{align*}\chi_K(\ell) = \left(\frac{d_K}{\ell}\right)\end{align*} $$

for each odd prime $\ell $ . By [Reference Montgomery and Vaughan45, Theorem 9.13], we see that $\chi _K$ is a primitive quadratic character.

We now state a lemma on the size of the image of mod $\ell ^\alpha $ Galois representation of E for $\ell \nmid fm_E$ .

Lemma 2.7 Let $E/\mathbb {Q}$ be a CM elliptic curve. For $\ell \nmid fm_E$ , we have

$$ \begin{align*}|G_E(\ell^\alpha)| = \ell^{2(\alpha-1)}(\ell-1)(\ell-\chi_K(\ell)).\end{align*} $$

Proof Since $\mathcal {O}$ is an order of class number $1$ , we have

$$ \begin{align*}\mathcal{O}/ (\ell \mathcal{O}_K \cap \mathcal{O}) = \mathcal{O}/\ell \mathcal{O} \simeq \mathcal{O}_K / \ell \mathcal{O}_K\end{align*} $$

for any $\ell \nmid f$ . (See [Reference Cox21, Proposition 7.20].) By Lemma 2.6, we have $G_E(\ell ^\alpha ) \simeq \left (\mathcal {O}_K/\ell ^\alpha \mathcal {O}_K\right )^\times $ . Applying [Reference Bröker, Lauter and Sutherland13, Equation (4)], we obtain the desired results.

Moreover, we have the following uniformity result for CM elliptic curves over $\mathbb {Q}$ .

Proposition 2.8 There is an absolute constant C such that

$$\begin{align*}fm_E \leq C \end{align*}$$

holds for all CM elliptic curves $E / \mathbb {Q}$ .

4.1 On the cyclicity constant

We keep the notation from Section 2.1. In this subsection, we introduce the definition of the cyclicity constant $C_E^{\operatorname {\mathrm {cyc}}}$ , given by Serre, and its average counterpart $C^{\operatorname {\mathrm {cyc}}}$ . For coprime integers n and k, we introduce the cyclicity constant for primes in arithmetic progression $C_{E, n, k}^{\operatorname {\mathrm {cyc}}}$ , given by Akbal and Güloğlu, and its average counterpart $C^{\operatorname {\mathrm {cyc}}}_{n, k}$ .

First of all, Serre [Reference Serre51, pp. 465–468] defined the cyclicity constant $C^{\operatorname {\mathrm {cyc}}}_E$ to be

(18)

where $\mu (\cdot )$ denotes the Möbius function and $\mathbb {Q}(E[n])$ is the nth division field of E. He proved that, under GRH, $C^{\operatorname {\mathrm {cyc}}}_E$ is the density of primes of cyclic reduction for E; see Conjecture 1.1.

For a non-CM elliptic curve $E/\mathbb {Q}$ , Jones [Reference Jones31, p. 692] observed that (18) can be expressed as an almost Euler product involving the adelic level of E. Specifically, he showed that

(19) $$ \begin{align} C_E^{\operatorname{\mathrm{cyc}}} = \left(\sum_{d\mid m_E} \frac{\mu(d)}{[\mathbb{Q}(E[d]):\mathbb{Q}]}\right) \prod_{\ell \nmid m_E} \left( 1 - \frac{1}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right). \end{align} $$

The average counterpart of $C^{\operatorname {\mathrm {cyc}}}_E$ is

(20)

As mentioned in the introduction, Gekeler [Reference Gekeler26] demonstrated that $C^{\operatorname {\mathrm {cyc}}}$ represents the average cyclicity constant from the local viewpoint. Later, Banks and Shparlinski [Reference Banks and Shparlinski6] verified that the constant also describes the density of primes of cyclic reduction on average in the global sense. Furthermore, Jones [Reference Jones31] verified that the average of $C^{\operatorname {\mathrm {cyc}}}_E$ coincides with $C^{\operatorname {\mathrm {cyc}}}$ .

Let $\zeta _n$ denote a primitive nth root of unity, and let $\sigma _k \in \operatorname {\mathrm {Gal}}(\mathbb {Q}(\zeta _n)/\mathbb {Q})$ map $\zeta _n \mapsto \zeta _n^k$ . Define

Akbal and Güloğlu [Reference Akbal and Güloğlu1] defined the constant $C^{\operatorname {\mathrm {cyc}}}_{E,n,k}$ as follows,

(21)

They proved that this constant represents the density of primes $p \equiv k \ \pmod n$ of cyclic reduction for E, under GRH. Recently, Jones and the first author [Reference Jones and Lee33] demonstrated that for a non-CM elliptic curve $E/\mathbb {Q}$ , this density can be expressed as an almost Euler product as follows,

(22) $$ \begin{align} C^{\operatorname{\mathrm{cyc}}}_{E,n,k} = \left( \sum_{d\mid m_E} \frac{\mu(d)\gamma_{n,k}(\mathbb{Q}(E[d]))}{[\mathbb{Q}(E[d])\mathbb{Q}(\zeta_n) : \mathbb{Q}]} \right) \prod_{\substack{\ell \nmid m_E \\ \ell \mid (n,k-1)}} \left( 1 - \frac{\phi(\ell)}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right) \prod_{\substack{\ell \nmid nm_E}}\!\left( 1 - \frac{1}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right). \end{align} $$

Finally, the average counterpart of $C^{\operatorname {\mathrm {cyc}}}_{E,n,k}$ is given by

(23)

Observe that (23) coincides with (22) if $m_E$ is taken to be $1$ . While $m_E = 1$ is impossible for any given elliptic curve over $\mathbb {Q}$ , it is plausible to think that the role of $m_E$ is inconsequential when considered over the family of all elliptic curves ordered by height. Indeed, as mentioned in the introduction, the first author [Reference Lee38] demonstrated that $C^{\operatorname {\mathrm {cyc}}}_{n,k}$ represents the average density of primes $p \equiv k \ \pmod n$ of cyclic reduction for the family of elliptic curves ordered by height.

We now prove a proposition that serves as a reasonableness check for $C^{\operatorname {\mathrm {cyc}}}_{n,k}$ . While it can be derived from the main theorem of [Reference Lee38], we opt to include a self-contained proof to draw a parallel with the upcoming Proposition 4.6.

Proposition 4.1 For any positive integer n, we have

$$ \begin{align*}\sum_{\substack{1 \leq k \leq n \\ (n,k) = 1}}C^{\operatorname{\mathrm{cyc}}}_{n,k} = C^{\operatorname{\mathrm{cyc}}},\end{align*} $$

where $C^{\operatorname {\mathrm {cyc}}}$ and $C^{\operatorname {\mathrm {cyc}}}_{n,k}$ are defined in (20) and (23), respectively.

Proof For notational convenience, we define

$$ \begin{align*}f(\ell) = 1 - \frac{\phi(\ell)}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}.\end{align*} $$

It suffices to verify that

(24)

First, we prove that (24) holds for $n = p^a$ , a prime power. Observe that

$$ \begin{align*} F(p^a) &= \frac{1}{\phi(p^a)} \sum_{\substack{1\leq k\leq p^a\\ (k,p^a) = 1}} \prod_{\substack{\ell \mid p^a \\ k \equiv 1(\ell)}} f(\ell)\\ &= \frac{1}{\phi(p^a)} \left( p^{a-1} f(p) + p^{a-1}(p-2) \right) = 1 - \frac{1}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/p\mathbb{Z})|}. \end{align*} $$

Now, we prove that F is multiplicative. Let $p^a$ be a prime power and n be a positive integer coprime to p. Then

$$ \begin{align*} F(p^an) &= \frac{1}{\phi(p^an)}\sum_{\substack{1 \leq k \leq p^an \\ (k,p^an) = 1}} \prod_{\substack{\ell \mid p^an \\ k \equiv 1 (\ell)}} f(\ell) \\ &= \frac{1}{\phi(p^a)} \cdot \frac{1}{\phi(n)} \left[ \sum_{\substack{1 \leq k \leq p^a n \\ (k,pn) = 1 \\ k \equiv 1 (p)}} f(p) \prod_{\substack{\ell \mid n \\ k \equiv 1 (\ell)}} f(\ell) + \sum_{\substack{1 \leq k \leq p^a n \\ (k,pn) = 1 \\ k \not \equiv 1 (p)}} \prod_{\substack{\ell \mid n \\ k \equiv 1 (\ell)}}f(\ell) \right]\\ &= \frac{\left( p^{a-1} + p^{a-1}(p-2)\right)}{\phi(p^a)} \cdot \frac{1}{\phi(n)} \left[ \sum_{\substack{1 \leq k \leq n \\ (k,n) = 1 }}\prod_{\substack{\ell\mid n \\ k \equiv 1 (\ell)}} f(\ell)\right] = F(p^a) \cdot F(n). \end{align*} $$

This completes the proof.

4.2 On the Koblitz constant

We keep the notation from Section 2.1. Now we give the definition of the Koblitz constant $C_E^{\operatorname {\mathrm {prime}}}$ defined by Zywina and its average counterpart $C^{\operatorname {\mathrm {prime}}}$ given by Balog, Cojocaru, and David. Based on Zywina’s method, for coprime integers n and k, we propose the Koblitz constant $C_{E, n, k}^{\operatorname {\mathrm {prime}}}$ for primes in arithmetic progression and its average counterpart $C_{n, k}^{\operatorname {\mathrm {prime}}}$ .

Let $E/\mathbb {Q}$ be a non-CM elliptic curve of conductor $N_E$ and m be a positive integer. For $p \nmid mN_E$ , let $\operatorname {\mathrm {Frob}}_p$ be a Frobenius element at p in $\operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathbb {Q})$ (see [Reference Serre50, Chapter 2.1, I-6] for the definition of $\operatorname {\mathrm {Frob}}_p$ ). We have that

(25) $$ \begin{align} |\widetilde{E}_p(\mathbb{F}_p)| \equiv \operatorname{det}(I - \rho_{E,m}(\operatorname{\mathrm{Frob}}_p)) \quad\pmod m, \end{align} $$

by [Reference Silverman52, Chapter V. Theorem 2.3.1]. Thus, we see that an odd prime p is of Koblitz reduction if and only if the right-hand side of (25) is invertible modulo m, for every $m < |\widetilde {E}_p(\mathbb {F}_p)|$ such that $\gcd (p,m) = 1$ .Footnote ⁴ For such an integer m, we set

(26)

Define the ratio

The Koblitz constant, proposed by Zywina [Reference Zywina62], is defined by

(27)

where the limit is taken over all positive integers ordered by divisibility.

We start by proving some properties of $\delta _E^{\operatorname {\mathrm {prime}}}(\cdot )$ , which were originally remarked in [Reference Zywina62].

Proposition 4.2 Let $E/\mathbb {Q}$ be a non-CM elliptic curve of adelic level $m_E$ . Then $\delta _E^{\operatorname {\mathrm {prime}}}(\cdot )$ , as an arithmetic function, satisfies the following properties:

1. (1) for any positive integer m, $\delta _E^{\operatorname {\mathrm {prime}}}(m) = \delta _E^{\operatorname {\mathrm {prime}}}(\operatorname {\mathrm {rad}}(m))$ ;

2. (2) for any prime $\ell \nmid m_E$ and integer d coprime to $\ell $ , $\delta _E^{\operatorname {\mathrm {prime}}}(d \ell ) = \delta _E^{\operatorname {\mathrm {prime}}}(d) \cdot \delta _E^{\operatorname {\mathrm {prime}}}(\ell )$ .

Therefore, (27) can be expressed as follows,

(28) $$ \begin{align} C^{\operatorname{\mathrm{prime}}}_E = \frac{\delta_E^{\operatorname{\mathrm{prime}}}(\operatorname{\mathrm{rad}} (m_E))}{\prod_{\ell \mid m_E} (1-1/\ell)} \cdot \prod_{\ell \nmid m_E} \frac{\delta_{E}^{\operatorname{\mathrm{prime}}}(\ell)}{1-1/\ell}. \end{align} $$

Proof We first prove item (1). Let $r = \operatorname {\mathrm {rad}}(m)$ and $\varpi \colon G_E(m) \to G_E(r)$ be the usual reduction map. In particular, $\varpi $ is a surjective group homomorphism. We will show that

(29) $$ \begin{align} \varpi^{-1}\left(G_E(r) \cap \Psi^{\operatorname{\mathrm{prime}}}(r)\right) = G_E(m) \cap \Psi^{\operatorname{\mathrm{prime}}}(m). \end{align} $$

Let $M \in G_E(r) \cap \Psi ^{\operatorname {\mathrm {prime}}}(r)$ and $\widetilde {M} \in \varpi ^{-1}(M)$ . Recall that $\operatorname {det}(M-I)$ is invertible modulo r and that m is only supported by the prime factors of r. Thus, $\operatorname {det}(\widetilde {M}-I)$ is invertible modulo m and $\widetilde {M} \in G_E(m) \cap \Psi ^{\operatorname {\mathrm {prime}}}(m)$ . The other inclusion is obvious, and hence (29) is obtained. Therefore,

$$ \begin{align*}\delta_E^{\operatorname{\mathrm{prime}}}(m) = \frac{\left| G_E(m) \cap \Psi^{\operatorname{\mathrm{prime}}}(m)\right|}{\left|G_E(m)\right|} = \frac{\left|\varpi^{-1}\left(G_E(r) \cap \Psi^{\operatorname{\mathrm{prime}}}(r)\right)\right|}{\left|\varpi^{-1}\left(G_E(r)\right)\right|} = \delta_E^{\operatorname{\mathrm{prime}}}(r).\end{align*} $$

We now prove item (2). By Lemma 2.2, we have an isomorphism,

(30) $$ \begin{align} G_E(d\ell) \simeq G_E(d) \times \operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z}). \end{align} $$

It suffices to show that the isomorphism induces a bijection between the two sets

(31) $$ \begin{align} G_E(d\ell) \cap\Psi^{\operatorname{\mathrm{prime}}}(d\ell) \, \text{ and } \, \left(G_E(d) \cap \Psi^{\operatorname{\mathrm{prime}}}(d)\right) \times \left(\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z}) \cap \Psi^{\operatorname{\mathrm{prime}}}(\ell)\right). \end{align} $$

Take $M \in G_E(d\ell ) \cap \Psi ^{\operatorname {\mathrm {prime}}}(d\ell )$ . By a similar argument to the proof of (1), we have that $M_d \in G_E(d) \cap \Psi ^{\operatorname {\mathrm {prime}}}(d)$ and $M_\ell \in \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell \mathbb {Z}) \cap \Psi ^{\operatorname {\mathrm {prime}}}(\ell )$ . Now, let $M^{\prime } \in G_E(d) \cap \Psi ^{\operatorname {\mathrm {prime}}}(d)$ and $M^{\prime \prime } \in \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell \mathbb {Z}) \cap \Psi ^{\operatorname {\mathrm {prime}}}(\ell )$ . Viewing $(M^{\prime },M^{\prime \prime }) \in G_E(d) \times \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell \mathbb {Z})$ , there exists a unique element $M \in G_E(d\ell )$ with $M_d = M^{\prime }$ and $M_\ell = M^{\prime \prime }$ by (30). Since $\operatorname {det}(M^{\prime }-I)\in (\mathbb {Z}/d\mathbb {Z})^{\times }$ and $\operatorname {det}(M^{\prime \prime }-I)\in (\mathbb {Z}/\ell \mathbb {Z})^{\times }$ , we have $\operatorname {det}(M-I)\in (\mathbb {Z}/d\ell \mathbb {Z})^{\times }$ ; in particular, $M \in \Psi ^{\operatorname {\mathrm {prime}}}(d\ell )$ . Therefore, (31) is established.

Along with (30), we obtain

$$ \begin{align*} \delta_{E}^{\operatorname{\mathrm{prime}}}(d\ell) &= \frac{\left|G_E(d\ell)\cap \Psi^{\operatorname{\mathrm{prime}}}(d\ell)\right|}{\left|G_E(d\ell)\right|} \\ &= \frac{|G_E(d) \cap \Psi^{\operatorname{\mathrm{prime}}}(d)|}{|G_E(d)|} \cdot \frac{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z}) \cap \Psi^{\operatorname{\mathrm{prime}}}(\ell)|}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|} \\ &= \delta^{\operatorname{\mathrm{prime}}}_E(d) \cdot \delta^{\operatorname{\mathrm{prime}}}_E(\ell). \end{align*} $$

This completes the proof.

Remark 4.3 Suppose that $\ell \nmid m_E$ and $M \in \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell \mathbb {Z})$ . Note that $\operatorname {det}(M-I) \in (\mathbb {Z}/\ell \mathbb {Z})^\times $ if and only if $1$ is not an eigenvalue of M. One can check from Table 12.4 in [Reference Lang36, Chapter XVIII] that

$$ \begin{align*} \#\left\{ M \in \operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell \mathbb{Z}) : M \text{ has eigenvalues } 1 \text{ and } k \right\} = \begin{cases} \ell^2 + \ell & \text{ if } k \not \equiv 1 \quad\pmod \ell,\\ \ell^2 & \text{ if } k \equiv 1 \quad\pmod \ell. \end{cases} \end{align*} $$

Thus, we see that

(32) $$ \begin{align} \frac{\delta_E^{\operatorname{\mathrm{prime}}}(\ell)}{1-1/\ell} = \frac{\ell}{\ell-1} \cdot \left(1-\frac{(\ell-2)(\ell^2+\ell) + \ell^2}{(\ell^2-1)(\ell^2-\ell)}\right) = 1 - \frac{\ell^2-\ell-1}{(\ell-1)^3(\ell+1)} \sim 1 - \frac{1}{\ell^2} \hspace{0.5cm} \text{ as } \ell \to \infty, \end{align} $$

and hence the infinite product in (28) converges absolutely.

The average counterpart of $C^{\operatorname {\mathrm {prime}}}_E$ is given by

(33)

As mentioned earlier, Balog, Cojocaru, and David [Reference Balog, Cojocaru and David5] demonstrated that $C^{\operatorname {\mathrm {prime}}}$ represents the average Koblitz constant, while Jones [Reference Jones31] verified that the average of $C^{\operatorname {\mathrm {prime}}}_E$ coincides with $C^{\operatorname {\mathrm {prime}}}$ . Unlike for the cyclicity problem, the Koblitz problem has not yet been studied for primes in arithmetic progressions. We construct $C^{\operatorname {\mathrm {prime}}}_{E,n,k}$ in a parallel way to Zywina’s method and propose a candidate for the average constant $C^{\operatorname {\mathrm {prime}}}_{n,k}$ .

Let $E/\mathbb {Q}$ be a non-CM elliptic curve of conductor $N_E$ and m be a positive integer. For a prime $p \nmid nN_E$ , let $\operatorname {\mathrm {Frob}}_p$ be a Frobenius element lying above p in $\operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathbb {Q})$ . We have that

$$ \begin{align*}\operatorname{det}(\rho_{E,n}(\operatorname{\mathrm{Frob}}_p)) \equiv p \quad\pmod n.\end{align*} $$

Along with (25), let us consider the set

(34)

One may note that $\rho _{E,m}(\operatorname {\mathrm {Frob}}_p) \in G_E(m) \cap \Psi ^{\operatorname {\mathrm {prime}}}_{n,k}(m)$ if and only if $p \equiv k \pmod {\gcd (n, m)}$ and $|\widetilde {E}_p(\mathbb {F}_p)|$ is invertible $\mathbb {Z}/m\mathbb {Z}$ . For this reason, we consider the ratio

Building upon Zywina’s approach, we are led to define

(35)

where the limit is taken over all positive integers, ordered by divisibility.

Proposition 4.4 Let $E/\mathbb {Q}$ be a non-CM elliptic curve of adelic level $m_E$ and n be a positive integer. Let L be defined as in (7). Then, $\delta _{E,n,k}^{\operatorname {\mathrm {prime}}}(\cdot )$ , as an arithmetic function, satisfies the following properties:

1. (1) Let $L \mid L^{\prime } \mid L^\infty $ . Then, $\delta _{E,n,k}^{\operatorname {\mathrm {prime}}}(L) = \delta _{E,n,k}^{\operatorname {\mathrm {prime}}}(L^{\prime })$ ;

2. (2) Let $\ell ^\alpha $ be a prime power and d be a positive integer with $(\ell , Ld) = 1$ . Then, $\delta _{E,n,k}^{\operatorname {\mathrm {prime}}}(d\ell ^a) = \delta _{E,n,k}^{\operatorname {\mathrm {prime}}}(d) \cdot \delta _{E,n,k}^{\operatorname {\mathrm {prime}}}(\ell ^\alpha )$ .

3. (3) Let $\ell ^\alpha \parallel n$ and $(\ell , L) = 1$ . Then, for any $\beta> \alpha $ , $\delta ^{\operatorname {\mathrm {prime}}}_{E,n,k}(\ell ^\beta ) = \delta ^{\operatorname {\mathrm {prime}}}_{E,n,k}(\ell ^\alpha )$ . Further, if $\ell \nmid nL$ , we have $\delta _{E,n,k}^{\operatorname {\mathrm {prime}}}(\ell ^\beta ) = \delta ^{\operatorname {\mathrm {prime}}}_E(\ell )$ .

Therefore, (35) can be expressed as follows,

(36) $$ \begin{align} C^{\operatorname{\mathrm{prime}}}_{E,n,k} = \frac{\delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(L)}{\prod_{\ell \mid L}(1-1/\ell)} \cdot \prod_{\substack{\ell \nmid m_E \\ \ell^{\alpha} \parallel n}} \frac{\delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha})}{1-1/\ell} \cdot \prod_{\ell \nmid nm_E} \frac{\delta_E^{\operatorname{\mathrm{prime}}}(\ell)}{1-1/\ell}. \end{align} $$

and the infinite product converges absolutely.

Proof Let us prove item (1). Consider the natural reduction map $\varpi \colon G_E(L^{\prime }) \to G_E(L)$ , which is a surjective group homomorphism. We will show that

(37) $$ \begin{align} \varpi^{-1}\left(G_E(L) \cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(L)\right) = G_E(L^{\prime}) \cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(L^{\prime}). \end{align} $$

Let $M \in G_E(L) \cap \Psi ^{\operatorname {\mathrm {prime}}}_{n,k}(L)$ and $\widetilde {M} \in \varpi ^{-1}(M)$ . Recall that $\operatorname {det}(M-I)$ is invertible modulo L and that $L^{\prime }$ is only supported by the prime factors of L. Thus, ${\operatorname {det}(\widetilde {M}-I)}$ is invertible modulo $L^{\prime }$ . Since $\gcd (n,L) = \gcd (n,L^{\prime })$ , we also have $\operatorname {det} \widetilde {M} \equiv k \ \pmod {\gcd (n,L^{\prime })}$ . Thus, $\widetilde {M} \in G_E(L^{\prime }) \cap \Psi ^{\operatorname {\mathrm {prime}}}_{n,k}(L^{\prime })$ . The other inclusion is obvious, and hence (37) is obtained. Therefore, we have

$$ \begin{align*}\delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(L^{\prime}) = \frac{\left|G_E(L^{\prime}) \cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(L^{\prime})\right|}{|G_E(L^{\prime})|} = \frac{\left|\varpi^{-1}\left(G_E(L) \cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(L)\right)\right|}{|\varpi^{-1}(G_E(L))|} = \delta^{\operatorname{\mathrm{prime}}}_{E,n,k}(L).\end{align*} $$

Let us prove item (2). By Lemma 2.2, we have an isomorphism,

(38) $$ \begin{align} G_E(d\ell^\alpha) \simeq G_E(d) \times \operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell^\alpha\mathbb{Z}). \end{align} $$

It suffices to show that the isomorphism induces a map between the sets

(39) $$ \begin{align} G_E(d\ell^\alpha) \cap\Psi_{n,k}^{\operatorname{\mathrm{prime}}}(d\ell^\alpha) \text{ and } \left(G_E(d) \cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(d)\right) \times \left(\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell^\alpha\mathbb{Z}) \cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(\ell^\alpha)\right). \end{align} $$

Say $M \in G_E(d\ell ^\alpha ) \cap \Psi ^{\operatorname {\mathrm {prime}}}_{n,k}(d\ell ^\alpha )$ . By a similar argument to the proof of (1), one may see that $M_d \in G_E(d) \cap \Psi _{n,k}^{\operatorname {\mathrm {prime}}}(d)$ and $M_{\ell ^\alpha } \in \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^\alpha \mathbb {Z}) \cap \Psi _{n,k}^{\operatorname {\mathrm {prime}}}(\ell ^\alpha )$ . Now, let $M^{\prime } \in G_E(d) \cap \Psi ^{\operatorname {\mathrm {prime}}}(d)$ and $M^{\prime \prime } \in \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^\alpha \mathbb {Z}) \cap \Psi ^{\operatorname {\mathrm {prime}}}(\ell ^\alpha )$ . Viewing $(M^{\prime },M^{\prime \prime }) \in G_E(d) \times \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^\alpha \mathbb {Z})$ , there exists a unique element $M \in G_E(d\ell ^\alpha )$ with $M_d = M^{\prime }$ and $M_{\ell ^\alpha } = M^{\prime \prime }$ by (38). Note that since $\operatorname {det}(M^{\prime }-I)\in (\mathbb {Z}/d\mathbb {Z})^{\times }$ and $\operatorname {det}(M^{\prime \prime }-I)\in (\mathbb {Z}/\ell ^{\alpha }\mathbb {Z})^{\times }$ , we have $\operatorname {det}(M-I)\in (\mathbb {Z}/d\ell ^{\alpha }\mathbb {Z})^{\times }$ ; in particular, $M \in \Psi ^{\operatorname {\mathrm {prime}}}_{n,k}(d\ell ^\alpha )$ . Therefore, (39) is established.

Along with (38), we obtain

$$ \begin{align*} \delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(d\ell^\alpha) &= \frac{\left|G_E(d\ell^\alpha)\cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(d\ell^\alpha)\right|}{\left|G_E(d\ell^\alpha)\right|} \\ &= \frac{|G_E(d) \cap \Psi^{\operatorname{\mathrm{prime}}}(d)|}{|G_E(d)|} \cdot \frac{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell^\alpha\mathbb{Z}) \cap \Psi^{\operatorname{\mathrm{prime}}}(\ell^\alpha)|}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell^\alpha\mathbb{Z})|} = \delta^{\operatorname{\mathrm{prime}}}_{E,n,k}(d) \cdot \delta^{\operatorname{\mathrm{prime}}}_{E,n,k}(\ell^\alpha). \end{align*} $$

Finally, let us prove item (3). Since $\ell \nmid m_E$ , by Lemma 2.2, $G_E(\ell ^\alpha )$ and $G_E(\ell ^\beta )$ are the full groups, $\operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^\alpha \mathbb {Z})$ and $\operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^\beta \mathbb {Z})$ . Let $\varpi \colon \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^\beta \mathbb {Z}) \to \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^\alpha \mathbb {Z})$ be the natural reduction map which is a surjective group homomorphism. By a similar argument as in the proof of item (1), it suffices to check that

(40) $$ \begin{align} \varpi^{-1} \left( \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(\ell^{\alpha})\right) = \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(\ell^\beta). \end{align} $$

Take $M \in \Psi ^{\operatorname {\mathrm {prime}}}_{n,k}(\ell ^\alpha )$ and let $\widetilde {M} \in \varpi ^{-1}(M)$ . By the same reasoning in the proof of item (1), $\operatorname {det}(\widetilde {M}-I)$ is invertible modulo $\ell ^\beta $ . Since $\gcd (n,\ell ^\alpha ) = \gcd (n,\ell ^\beta ) = \ell ^\alpha $ , we also have $\operatorname {det} \widetilde {M} \equiv k \ \pmod {\ell ^\alpha }$ . The other inclusion is obvious, and hence (40) is obtained. Thus, we have

$$ \begin{align*}\delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(\ell^\beta) = \frac{\left|\Psi^{\operatorname{\mathrm{prime}}}_{n,k}(\ell^\beta)\right|}{\left|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell^\beta\mathbb{Z})\right|} = \frac{\left|\varpi^{-1}\left(\Psi^{\operatorname{\mathrm{prime}}}_{n,k}(\ell^\alpha)\right) \right|}{\left| \varpi^{-1}\left(\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell^\alpha\mathbb{Z})\right)\right|} = \delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(\ell^\alpha).\end{align*} $$

In case $\ell \nmid nL$ , let $\varpi \colon \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^\beta \mathbb {Z}) \to \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell \mathbb {Z})$ . It suffices to check

(41) $$ \begin{align} \varpi^{-1}\left(\Psi^{\operatorname{\mathrm{prime}}}(\ell)\right) = \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(\ell^\beta). \end{align} $$

Note that the condition $\operatorname {det} M \equiv k \ \pmod {\gcd (n,\ell )}$ is trivial, and hence $\Psi ^{\operatorname {\mathrm {prime}}}_{E,n,k}(\ell ) = \Psi ^{\operatorname {\mathrm {prime}}}_E(\ell )$ . Let $M \in \Psi ^{\operatorname {\mathrm {prime}}}_E(\ell )$ . Note that every lifting $\widetilde {M} \in \varpi ^{-1}(M)$ belongs to $\Psi ^{\operatorname {\mathrm {prime}}}_{E,n,k}(\ell ^\beta )$ . The other inclusion is obvious, and hence (41) is obtained. Thus, we have

$$ \begin{align*}\delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(\ell^\beta) = \frac{\left|\Psi^{\operatorname{\mathrm{prime}}}_{n,k}(\ell^\beta)\right|}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell^\beta\mathbb{Z})|} = \frac{\left|\varpi^{-1}\left(\Psi^{\operatorname{\mathrm{prime}}}(\ell)\right)\right|}{\left|\varpi^{-1}(\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z}))\right|} = \delta_E^{\operatorname{\mathrm{prime}}}(\ell).\end{align*} $$

By grouping the prime factors of M in (35) according to whether they divide L or not, we obtain (36). The absolute convergence of (36) follows from Remark 4.3.

The following lemma allows us to express $C_{E,n,k}^{\operatorname {\mathrm {prime}}}$ more explicitly.

Lemma 4.5 Suppose $\ell ^{\alpha } \parallel n$ and $\ell \nmid m_E$ . Then

$$ \begin{align*} \frac{\delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha})}{1-1/\ell} = \begin{cases} \frac{1}{\phi(\ell^{\alpha})} \left( 1 - \frac{\ell}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right) & \text{ if } k \equiv 1 \quad\pmod \ell, \\ \frac{1}{\phi(\ell^{\alpha})}\left( 1 - \frac{\ell^2+\ell}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right) & \text{ if } k \not \equiv 1 \quad\pmod \ell. \end{cases} \end{align*} $$

Proof By the assumption, we have $G_E(\ell ^{\alpha }) \simeq \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^{\alpha }\mathbb {Z})$ . Recall that

$$ \begin{align*} \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(\ell^{\alpha}) = \left\{M \in \operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell^{\alpha}\mathbb{Z}) : \operatorname{det}(M-I) \in (\mathbb{Z}/\ell^{\alpha}\mathbb{Z})^\times, \operatorname{det} M \equiv k \quad\pmod {\gcd(n,\ell^{\alpha})} \right\}, \end{align*} $$

whose cardinality was determined in Corollary 3.4. A brief calculation reveals the desired result.

Let $E/\mathbb {Q}$ be a non-CM elliptic curve of adelic level $m_E$ . Let $n = n_1n_2$ where $n_1 = \gcd (n,m_E^\infty )$ and $(n_2,m_E) = 1$ . By (32), (36), and Lemma 4.5, we have

$$ \begin{align*} \begin{split} C^{\operatorname{\mathrm{prime}}}_{E,n,k} = \frac{\delta^{\operatorname{\mathrm{prime}}}_{E,n,k}(L)}{\prod_{\ell \mid L}(1-1/\ell)} & \cdot \frac{1}{\phi(n_2)} \prod_{\substack{\ell \nmid m_E \\ \ell \mid n \\ \ell \nmid k-1}} \left( 1 - \frac{\ell^2+\ell}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right) \prod_{\substack{\ell \nmid m_E \\ \ell \mid (n,k-1)}} \left( 1 - \frac{\ell}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right) \\ &\cdot \prod_{\ell \nmid nm_E}\left( 1 - \frac{\ell^2-\ell-1}{(\ell-1)^3(\ell+1)}\right). \end{split} \end{align*} $$

We now propose the average counterpart of $C^{\operatorname {\mathrm {prime}}}_{E,n,k}$ ,

(42)

The formula for $C_{n,k}^{\operatorname {\mathrm {prime}}}$ coincides with $C^{\operatorname {\mathrm {prime}}}_{E,n,k}$ if one takes $m_E = 1$ , similar to the case for $C_{n,k}^{\operatorname {\mathrm {cyc}}}$ in (23). Parallel to Proposition 4.1, we show that $C_{n,k}^{\operatorname {\mathrm {prime}}}$ behaves as expected when we sum over k.

Proposition 4.6 For any positive integer n, we have

$$ \begin{align*}\sum_{\substack{1 \leq k \leq n \\ (n,k) = 1}}C^{\operatorname{\mathrm{prime}}}_{n,k} = C^{\operatorname{\mathrm{prime}}},\end{align*} $$

where $C^{\operatorname {\mathrm {prime}}}$ and $C^{\operatorname {\mathrm {prime}}}_{n,k}$ are defined in (33) and (42), respectively.

Proof For notational convenience, we define

$$ \begin{align*}f_1(\ell) = 1 - \frac{\ell^2+\ell}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}, \hspace{0.5cm} f_2(\ell) = 1 - \frac{\ell}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}.\end{align*} $$

To show the desired equation, we need to verify that

(43)

First, we prove that (43) is true for $n = p^a$ , a prime power. Observe that

$$ \begin{align*} F(p^a) &= \frac{1}{\phi(p^a)}\left(p^{a-1}(f_1(p)(p-2) + f_2(p))\right)\\ &= \frac{p^{a-1}}{\phi(p^a)} \left[ (p-2)\left( 1 - \frac{p^2+p}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/p\mathbb{Z})|}\right) + \left( 1 - \frac{p}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/p\mathbb{Z})|}\right) \right] \\ &= 1 - \frac{p^2-p-1}{(p-1)^3(p+1)}. \end{align*} $$

Let us prove that F is multiplicative. Let n be coprime to $p^a$ , a prime power. We see that

$$ \begin{align*} F(p^an) &= \frac{1}{\phi(p^an)}\sum_{\substack{1 \leq k \leq p^a n \\ (k,pn) = 1}} \prod_{\substack{\ell \mid pn \\ k \not \equiv 1 (\ell)}}f_1(\ell) \prod_{\substack{\ell \mid pn \\ k \equiv 1 (\ell)}}f_2(\ell) \\ &= \frac{1}{\phi(p^a)} \frac{1}{\phi(n)}\left[\sum_{\substack{1 \leq k \leq p^an \\ (k,pn) = 1 \\ k \not \equiv 1 (p)}}f_1(p) \prod_{\substack{\ell \mid n \\ k \not \equiv 1(\ell)}}f_1(\ell) \prod_{\substack{\ell \mid n \\ k \equiv 1 (\ell)}} f_2(\ell) + \sum_{\substack{1 \leq k \leq p^an \\ (k,pn) = 1 \\ k \equiv 1 (p)}} f_2(p) \prod_{\substack{\ell \mid n \\ k \not \equiv 1 (\ell)}} f_1(\ell) \prod_{\substack{\ell \mid n \\ k \equiv 1(\ell)}}f_2(\ell)\right] \\ &= \frac{f_1(p) (p-2)p^{a-1}}{\phi(p^a)} \frac{1}{\phi(n)}\sum_{\substack{1 \leq k \leq n \\ (k,n) = 1}} \prod_{\substack{\ell \mid n \\ k \not \equiv 1 (\ell)}} f_1(\ell) \prod_{\substack{\ell \mid n \\ k \equiv 1 (\ell)}}f_2(\ell) \\ &\hspace{1in} + \frac{f_2(p) p^{a-1}}{\phi(p^a)} \frac{1}{\phi(n)} \sum_{\substack{1 \leq k \leq n \\ (k,n) = 1}} \prod_{\substack{\ell \mid n \\ k \not \equiv 1 (\ell)}} f_1(\ell) \prod_{\substack{\ell \mid n \\ k \equiv 1 (\ell)}}f_2(\ell) \\ &=\frac{1}{\phi(p^a)}\left(p^{a-1}(f_1(p)(p-2) + f_2(p))\right) \cdot F(n) = F(p^a)F(n). \end{align*} $$

This completes the proof.

4.3 Applying Zywina’s approach for the cyclicity problem

Zywina [Reference Zywina62] refined the Koblitz conjecture by improving the heuristic explanation for the constant $C^{\operatorname {\mathrm {prime}}}_E$ . In essence, he interprets the desired property of a prime of Koblitz reduction in terms of Galois representations, examines the ratio of elements with the desired property in each finite level $G_E(m)$ , and considers the limit of that ratio as m approaches infinity. In this subsection, we apply Zywina’s approach to determine the heuristic densities of primes of cyclic reduction for E and verify their concurrence with the densities proposed by Serre and Akbal–Güloğlu.

Let $E/\mathbb {Q}$ be a non-CM elliptic curve and fix a good prime $p \neq 2$ . We now give a criterion for p to be a prime of cyclic reduction for E.Footnote ⁵ Let $\operatorname {\mathrm {Frob}}_p$ denote a Frobenius element in $\operatorname {\mathrm {Gal}}(\overline {\mathbb {Q}}/\mathbb {Q})$ at p. By [Reference Cojocaru and Ram Murty20, Lemma 2.1], we have that

$$ \begin{align*} \widetilde{E}_p(\mathbb{F}_p) \text{ is cyclic } & \iff \forall \text{ primes } \ell \neq p, \, \widetilde{E}_p(\mathbb{F}_p)\\ &\qquad\quad \text{ does not contain a subgroup isomorphic to } \mathbb{Z}/\ell \mathbb{Z} \oplus \mathbb{Z}/\ell \mathbb{Z} \\ & \iff \forall \text{ primes } \ell \neq p, \, \rho_{E,\ell}(\operatorname{\mathrm{Frob}}_p) \not\equiv I \quad\pmod{\ell} \\ &\iff \forall m \in \mathbb{N} \text{ with } p \nmid m \text{ and } \forall \text{ prime } \ell \mid m, \, \rho_{E,\ell}(\operatorname{\mathrm{Frob}}_p) \not\equiv I \quad\pmod{\ell}. \end{align*} $$

Drawing a parallel to (26), we consider the set

and the ratio

Taking the limit of $\delta ^{\operatorname {\mathrm {cyc}}}_E(m)$ over all positive integers, ordered by divisibility, we expect to obtain the heuristic density of primes of cyclic reduction.

Proposition 4.7 Let $E/\mathbb {Q}$ be a non-CM elliptic curve of adelic level $m_E$ . Then $\delta ^{\operatorname {\mathrm {cyc}}}_{E}(\cdot )$ , as an arithmetic function, satisfies the following properties:

1. (1) for any positive integer m, $\delta ^{\operatorname {\mathrm {cyc}}}_E(m) = \delta ^{\operatorname {\mathrm {cyc}}}_E(\operatorname {\mathrm {rad}}(m))$ ;

2. (2) for any prime $\ell \nmid m_E$ and integer d coprime to $\ell $ , $\delta ^{\operatorname {\mathrm {cyc}}}_{E}(d\ell ) = \delta ^{\operatorname {\mathrm {cyc}}}_E(d) \cdot \delta ^{\operatorname {\mathrm {cyc}}}_E(\ell )$ .

Therefore, the heuristic density of primes of cyclic reduction can be expressed as follows,

$$ \begin{align*} \lim_{m \to \infty} \delta_E^{\operatorname{\mathrm{cyc}}}(m) = \delta_{E}^{\operatorname{\mathrm{cyc}}}(\operatorname{\mathrm{rad}}(m_E)) \cdot \prod_{\ell \nmid m_E} \delta_E^{\operatorname{\mathrm{cyc}}}(\ell). \end{align*} $$

Remark 4.8 One can easily check that for $\ell \nmid m_E$ ,

(44) $$ \begin{align} \delta_E^{\operatorname{\mathrm{cyc}}}(\ell) = 1 - \frac{1}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|} \sim 1 - \frac{1}{\ell^4}, \hspace{0.5cm} \text{ as } \ell \to \infty \end{align} $$

and hence the infinite product converges absolutely.

We now verify that the limit $\lim _{m \to \infty } \delta _E^{\operatorname {\mathrm {cyc}}}(m)$ appearing in Proposition 4.7 coincides with the cyclicity constant $C^{\operatorname {\mathrm {cyc}}}_E$ originally defined by Serre [Reference Serre51, pp. 465–468].

Proposition 4.9 Let $E/\mathbb {Q}$ be a non-CM elliptic curve. Then we have

$$ \begin{align*}C^{\operatorname{\mathrm{cyc}}}_E = \delta_E^{\operatorname{\mathrm{cyc}}}(\operatorname{\mathrm{rad}}(m_E)) \cdot \prod_{\ell \nmid m_E} \left( 1 - \frac{1}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right).\end{align*} $$

Proof Let $R = \operatorname {\mathrm {rad}}(m_E)$ . By (19) and (44), it suffices to check

$$ \begin{align*}\sum_{d\mid m_E} \frac{\mu(d)}{[\mathbb{Q}(E[d]) : \mathbb{Q}]} = \delta^{\operatorname{\mathrm{cyc}}}_E(R).\end{align*} $$

Let m be a positive integer and $d \mid m$ . We define

From the definition, one may observe that $G_E(R) \cap \Psi ^{\operatorname {\mathrm {cyc}}}(R) = S^{\prime }_E(R)$ . Thus, we have

$$ \begin{align*}\delta_E^{\operatorname{\mathrm{cyc}}}(R) = \frac{|S^{\prime}_E(R)|}{|G_E(R)|}.\end{align*} $$

Also, note that $S_E^{(d)}(d) = \{I\}$ . Let $\varpi \colon G_E(m) \to G_E(d)$ be the natural reduction map. Then,

$$ \begin{align*}\frac{|S^{(d)}_E(m)|}{|G_E(m)|} = \frac{\left|\varpi^{-1}(S^{(d)}_E(d))\right|}{\left|\varpi^{-1}(G_E(d))\right|} = \frac{\left|S_E^{(d)}(d)\right|}{\left|G_E(d)\right|} = \frac{1}{|G_E(d)|}.\end{align*} $$

Observe that $S^{\prime }_E(R) = G_E(R) - \bigcup _{\ell \mid R} S^{(\ell )}_E(R)$ . By the principle of inclusion–exclusion, we obtain

$$ \begin{align*}\delta_E^{\operatorname{\mathrm{cyc}}}(R) = \frac{|S^{\prime}_E(R)|}{|G_E(R)|} = \sum_{d \mid R} \frac{\mu(d)}{|G_E(d)|} = \sum_{d\mid m_E} \frac{\mu(d)}{[\mathbb{Q}(E[d]):\mathbb{Q}]}.\end{align*} $$

This completes the proof.

Now, we construct a heuristic density of primes of cyclic reduction that lie in an arithmetic progression. Consider

We define

Drawing parallels from Zywina/s approach, we consider the limit

(45) $$ \begin{align} \lim_{m \to \infty} \delta_{E,n,k}^{\operatorname{\mathrm{cyc}}}(m), \end{align} $$

where the limit is taken over all positive integers, ordered by divisibility. We’ll prove in Proposition 4.12 that (45) coincides with $C_{E,n,k}^{\operatorname {\mathrm {cyc}}}$ as defined in [Reference Akbal and Güloğlu1]. To do so, we’ll first give some properties of $\delta _{E,n,k}^{\operatorname {\mathrm {cyc}}}(\cdot )$ .

Proposition 4.10 Let $E/\mathbb {Q}$ be a non-CM elliptic curve of adelic level $m_E$ . Fix a positive integer n. Set L as in (7). Then, $\delta _{E,n,k}^{\operatorname {\mathrm {cyc}}}(\cdot )$ , as an arithmetic function, satisfies the following properties:

1. (1) Let $L \mid L^{\prime } \mid L^\infty $ . Then, $\delta _{E,n,k}^{\operatorname {\mathrm {cyc}}}(L) = \delta _{E,n,k}^{\operatorname {\mathrm {cyc}}}(L^{\prime })$ ;

2. (2) Let $\ell ^\alpha $ be a prime power and d be a positive integer with $(\ell , Ld) = 1$ . Then, $\delta _{E,n,k}^{\operatorname {\mathrm {cyc}}}(d\ell ^a) = \delta _{E,n,k}^{\operatorname {\mathrm {cyc}}}(d) \cdot \delta _{E,n,k}^{\operatorname {\mathrm {cyc}}}(\ell ^\alpha )$ .

3. (3) Let $\ell ^\alpha \parallel n$ and $(\ell , L) = 1$ . Then, for any $\beta> \alpha $ , $\delta ^{\operatorname {\mathrm {cyc}}}_{E,n,k}(\ell ^\beta ) = \delta ^{\operatorname {\mathrm {cyc}}}_{E,n,k}(\ell ^\alpha )$ . Further, if $\ell \nmid nL$ , we have $\delta _{E,n,k}^{\operatorname {\mathrm {cyc}}}(\ell ^\beta ) = \delta ^{\operatorname {\mathrm {cyc}}}_E(\ell )$ .

Therefore, (45) can be expressed as follows,

(46)

and the product converges absolutely.

The next lemma allows us to describe (46) explicitly.

Lemma 4.11 Let $E/\mathbb {Q}$ be a non-CM elliptic curve of adelic level $m_E$ . Suppose $\ell ^a \parallel n$ and $\ell \nmid m_E$ . For any k coprime to n, we have

$$ \begin{align*}\delta^{\operatorname{\mathrm{cyc}}}_{E,n,k}(\ell^a) = \begin{cases} \frac{1}{\phi(\ell^a)} & \text{ if } \ell \mid n \text{ and } \ell \nmid (k-1) \\ \frac{1}{\phi(\ell^a)}\left ( 1 - \frac{\phi(\ell)}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right) & \text{ if } \ell \mid (n,k-1). \end{cases}\end{align*} $$

Let $E/\mathbb {Q}$ be a non-CM elliptic curve of adelic level $m_E$ . Let $n = n_1n_2$ where $n_1 = \gcd (n,m_E^\infty )$ and $(n_2,m_E) = 1$ . By (44), (46), and Lemma 4.11, we obtain

(47) $$ \begin{align} \lim_{m\to \infty} \delta^{\operatorname{\mathrm{cyc}}}_{E,n,k}(m) = \frac{\delta_{E,n,k}^{\operatorname{\mathrm{cyc}}}(L)}{\phi(n_2)} \prod_{\substack{\ell \nmid m_E \\ \ell \mid (n,k-1)}}\left( 1- \frac{\phi(\ell)}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right) \prod_{\ell \nmid nm_E} \left( 1 - \frac{1}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right). \end{align} $$

We now prove that (47) equals the cyclicity constant proposed by Akbal and Gülğlu.

Proposition 4.12 Let $E/\mathbb {Q}$ be a non-CM elliptic curve of adelic level $m_E$ and n be a positive integer. Let $n = n_1n_2$ where $n_1 = \gcd (n,m_E^\infty )$ and $(n_2,m_E) = 1$ . Then we have

$$ \begin{align*} C^{\operatorname{\mathrm{cyc}}}_{E,n,k} = \frac{\delta_{E,n,k}^{\operatorname{\mathrm{cyc}}}(L)}{\phi(n_2)} \prod_{\substack{\ell \nmid m_E \\ \ell \mid (n,k-1)}}\left( 1- \frac{\phi(\ell)}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right) \prod_{\ell \nmid nm_E} \left( 1 - \frac{1}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right). \end{align*} $$

Proof Define

Let $R = \operatorname {\mathrm {rad}}(m_E)$ . By [Reference Jones and Lee33, p. 13], (22) can be expressed as follows,

(48) $$ \begin{align} \frac{|S^{\prime}_{E,n,k}(R)|}{|\operatorname{\mathrm{Gal}}(\mathbb{Q}(E[R]) \mathbb{Q}(\zeta_n)/\mathbb{Q})|} \prod_{\substack{\ell \nmid m_E \\ \ell (n,k-1)}} \left(1-\frac{\phi(\ell)}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right) \prod_{\ell \nmid nm_E} \left(1-\frac{1}{|\operatorname{\mathrm{GL}}_2(\mathbb{Z}/\ell\mathbb{Z})|}\right). \end{align} $$

Thus, it suffices to verify that

$$ \begin{align*}\frac{|S^{\prime}_{E,n,k}(R)|}{|\operatorname{\mathrm{Gal}}(\mathbb{Q}(E[R])\mathbb{Q}(\zeta_n))/\mathbb{Q}|} = \frac{\delta_{E,n,k}^{\operatorname{\mathrm{cyc}}}(L)}{\phi(n_2)}.\end{align*} $$

By the Weil pairing, we have $\mathbb {Q}(\zeta _{n_2}) \subseteq \mathbb {Q}(E[n_2])$ . Thus, we see that $\mathbb {Q}(E[R])\mathbb {Q}(\zeta _{n_1})$ and $\mathbb {Q}(\zeta _{n_2})$ must be linearly disjoint by Lemma 2.2, and hence

$$ \begin{align*}\operatorname{\mathrm{Gal}}(\mathbb{Q}(E[R])\mathbb{Q}(\zeta_n)/\mathbb{Q}) \simeq \operatorname{\mathrm{Gal}}(\mathbb{Q}(E[R]\mathbb{Q}(\zeta_{n_1}))/\mathbb{Q}) \times \operatorname{\mathrm{Gal}}(\mathbb{Q}(\zeta_{n_2})/\mathbb{Q}).\end{align*} $$

Under the isomorphism, the set $S^{\prime }_{E,n,k}(R)$ can be identified as $S^{\prime }_{E,n_1,k}(R) \times \left \{\sigma _k\right \}$ , and hence $|S^{\prime }_{E,n,k}(R)| = |S^{\prime }_{E,n_1,k}(R)|$ . Thus, we have

$$ \begin{align*}\frac{|S^{\prime}_{E,n,k}(R)|}{|\operatorname{\mathrm{Gal}}(\mathbb{Q}(E[R])\mathbb{Q}(\zeta_n)/\mathbb{Q})|} = \frac{1}{\phi(n_2)} \cdot \frac{|S^{\prime}_{E,n_1,k}(R)|}{|\operatorname{\mathrm{Gal}}(\mathbb{Q}(E[R])\mathbb{Q}(\zeta_{n_1})/\mathbb{Q})|}.\end{align*} $$

Remark that $\mathbb {Q}(E[R]) \subseteq \mathbb {Q}(E[L])$ and $\mathbb {Q}(\zeta _{n_1}) \subseteq \mathbb {Q}(E[L])$ by the definition of L. Thus, the usual restriction $\varpi \colon G_E(L) \to \operatorname {\mathrm {Gal}}(\mathbb {Q}(E[R])\mathbb {Q}(\zeta _{n_1})/\mathbb {Q})$ gives a surjective group homomorphism.

Viewing $G_E(L)$ as a subgroup of $\operatorname {\mathrm {GL}}_2(\mathbb {Z}/L\mathbb {Z})$ , we may observe that

$$ \begin{align*} \varpi^{-1}(S^{\prime}_{E,n_1,k}(R)) &= \left\{\widetilde{\sigma} \in G_E(L) : \widetilde{\sigma}|_{\mathbb{Q}(\zeta_{n_1})} = \sigma_{k}, \widetilde{\sigma}|_{\mathbb{Q}(E[\ell])} \not \equiv 1 \quad\pmod {\ell} \text{ for all } \ell \mid L \right\} \\ &= G_E(L) \cap \left\{M \in \operatorname{\mathrm{GL}}_2(\mathbb{Z}/L\mathbb{Z}) : \operatorname{det} M \equiv k \quad\pmod {n_1}, M \not \equiv I \quad\pmod {\ell} \text{ for all } \ell \mid L \right\} \\ &= G_E(L) \cap \Psi^{\operatorname{\mathrm{cyc}}}_{n,k}(L). \end{align*} $$

Therefore,

$$ \begin{align*}\frac{|S^{\prime}_{E,n_1,k}(R)|}{|\operatorname{\mathrm{Gal}}(\mathbb{Q}(E[R])\mathbb{Q}(\zeta_{n_1})/\mathbb{Q})|} = \frac{\left|\varpi^{-1}\left(S^{\prime}_{E,n_1,k}(R)\right)\right|}{\left|\varpi^{-1}(\operatorname{\mathrm{Gal}}(\mathbb{Q}(E[R])\mathbb{Q}(\zeta_{n_1})/\mathbb{Q}))\right|} = \frac{\left| G_E(L)\cap \Psi^{\operatorname{\mathrm{cyc}}}_{n,k}(L)\right|}{\left|G_E(L)\right|} = \delta^{\operatorname{\mathrm{cyc}}}_{E,n,k}(L).\end{align*} $$

This completes the proof.

6.1 Bounding the Koblitz constant for non-CM, non-Serre curves

In this subsection, we will determine an upper bound for $C^{\operatorname {\mathrm {prime}}}_{E,n,k}$ in the case of non-CM, non-Serre curves.

Let $E/\mathbb {Q}$ be a non-CM, non-Serre curve, defined by the model (4), of adelic level $m_E$ . Let L be defined as in (7). Then we write $L = L_1L_2$ such that $L_2$ is the product of prime powers $\ell ^{\alpha } \parallel L$ with $\ell \not \in \{2,3,5\}$ and $G_E(\ell ) \simeq \operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell \mathbb {Z})$ . By [Reference Cojocaru17, Appendix, Theorem 1], $G_E(L_2) \simeq \operatorname {\mathrm {GL}}_2(\mathbb {Z}/L_2\mathbb {Z})$ . Let $\varpi \colon \operatorname {\mathrm {GL}}_2(\mathbb {Z}/L\mathbb {Z}) \to \operatorname {\mathrm {GL}}_2(\mathbb {Z}/L_2\mathbb {Z})$ be the natural reduction map. Note that

$$ \begin{align*}\varpi\left(G_E(L) \cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(L)\right) \subseteq G_E(L_2) \cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(L_2).\end{align*} $$

Since $\varpi $ is a surjective group homomorphism, we have

(54) $$ \begin{align} \begin{aligned} \delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(L) &= \frac{\left|G_E(L) \cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(L)\right|}{\left|G_E(L)\right|} \\ &\leq \frac{\left| \varpi^{-1}\left(G_E(L_2) \cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(L_2)\right)\right|}{|\varpi^{-1}(G_E(L_2))|} = \frac{\left| G_E(L_2) \cap \Psi^{\operatorname{\mathrm{prime}}}_{n,k}(L_2)\right|}{|G_E(L_2)|} = \delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(L_2). \end{aligned} \end{align} $$

Since $\rho _{E,L_2}$ is surjective, we apply the same argument as in the proof of Lemma 4.5 and obtain

$$ \begin{align*}\frac{\delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(L_2)}{\prod_{\ell \mid L_2}(1-1/\ell)}\leq 1.\end{align*} $$

Before proceeding to bound the constant $C_{E, n, k}^{\operatorname {\mathrm {prime}}}$ , we first state a standard analytic result.

Lemma 6.1 For any positive integer M, we have

$$ \begin{align*}\prod_{\ell \mid M} \left(1-\frac{1}{\ell}\right)^{-1} \ll \max\{1, \log \log M\}.\end{align*} $$

From Lemma 4.5, Lemma 6.1, (32), (36), and (54), we obtain

(55) $$ \begin{align} \begin{aligned} C^{\operatorname{\mathrm{prime}}}_{E,n,k} &= \frac{\delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(L)}{\prod_{\ell \mid L}(1-1/\ell)} \prod_{\substack{\ell^\alpha \parallel n \\ \ell \nmid m_E}}\frac{\delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(\ell^\alpha)}{1-1/\ell} \prod_{\ell \nmid nm_E}\frac{\delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(\ell)}{1-1/\ell} \\ &\leq \frac{1}{\prod_{\ell \mid L_1}(1-1/\ell)} \cdot \frac{\delta_{E,n,k}^{\operatorname{\mathrm{prime}}}(L_2)}{\prod_{\ell \mid L_2}(1-1/\ell)} \leq \prod_{\ell \mid L_1} \frac{1}{1-1/\ell} \ll \max\{1, \log \log \operatorname{\mathrm{rad}}(L_1)\}. \end{aligned} \end{align} $$

Our next task is to bound $\operatorname {\mathrm {rad}} (L_1)$ in terms of a and b appearing in the short Weierstrass model (4) of E. Write $j_E\in \mathbb {Q}$ to denote the j-invariant of E and for the Weil height of $j_E$ . If $\ell \mid L_1$ , then either $\ell \leq 5$ or $\rho _{E,\ell }$ is not surjective. By the main theorem of [Reference Masser and Wüstholz41], there exist absolute constant $\kappa $ and $\lambda $ for which $\rho _{E,\ell }$ is surjective for all $\ell> \kappa (\max \{1,h\})^\lambda $ . Since $\operatorname {\mathrm {rad}}(L_1)$ is squarefree, we have

(56) $$ \begin{align} & \operatorname{\mathrm{rad}}(L_1) \leq 30\prod_{\ell \leq \kappa (\max\{1,h\})^\lambda} \ell \nonumber \\ & \implies \log \operatorname{\mathrm{rad}} (L_1) \ll \sum_{\ell \leq \kappa(\max\{1,h\})^\lambda} \log \ell \ll (\max\{1,h\})^\lambda \log \max\{1, h\}. \end{align} $$

Since E is given by the model (4), we have that

(57) $$ \begin{align} h=h(j_{E}) \ll \log \max \{|a|^3,|b|^2\}. \end{align} $$

Combining (55), (56), and (57), we obtain the following result.

Proposition 6.2 Let $E/\mathbb {Q}$ be a non-CM, non-Serre curve given by (4). Then we have

$$ \begin{align*}C^{\operatorname{\mathrm{prime}}}_{E,n,k} \ll \log \log \max\{|a|^3,|b|^2\}.\end{align*} $$

6.2 Bounding the Koblitz constant for CM curves

In this subsection, we focus on CM elliptic curves $E/\mathbb {Q}$ . The goal is to show that the constant $C^{\operatorname {\mathrm {prime}}}_{E, n, k}$ is bounded independent of the choice of the CM curve (Proposition 6.7). We keep the notation from Section 2.3.

Let $E/\mathbb {Q}$ be an elliptic curve with CM by an order $\mathcal {O}$ in an imaginary quadratic field $K = \mathbb {Q}(\sqrt {-D})$ . Let p be a prime of Koblitz reduction for $E/\mathbb {Q}$ . Since $[K:\mathbb {Q}] = 2$ , the prime p either splits completely, stays inert, or ramifies over $K/\mathbb {Q}$ .

If p does not split over $K/\mathbb {Q}$ , then by Deuring’s criterion [Reference Deuring23], p is a supersingular prime for E and we have $a_p(E) = 0$ . Therefore,

$$ \begin{align*}|\widetilde{E}_p(\mathbb{F}_p)| =p + 1,\end{align*} $$

which is an even number if $p> 2$ . Thus, an odd supersingular prime cannot be a prime of Koblitz reduction for E.

Now suppose p splits completely in K and let $\mathbf {p}$ be a prime lying above p. We consider two cases depending on the value of D modulo $4$ . Following the notation of [Reference Wan and Xi58, Chapter 2.2], when $D \equiv 1,2 \ \pmod 4$ , let $M,N\in \mathbb {Z}$ be such that $\mathbf {p}$ is generated by $M+N\sqrt {-D}$ for some $M,N\in \mathbb {Z}$ . In this case, the Frobenius trace satisfies $a_p(E) = 2M$ , so $|\widetilde {E}_p(\mathbb {F}_p)| = p+1-a_p(E)$ is always even for odd primes p. Therefore, $\pi _E^{\operatorname {\mathrm {prime}}}(x)$ is uniformly bounded.

On the other hand, if $D \equiv 3 \ \pmod 4$ , then we can let $M,N \in \mathbb {Z}$ be such that $M + N(1+\sqrt {-D})/2$ generates $\mathbf {p}$ . Let us define a binary quadratic form

$$ \begin{align*}f_D(x,y) = x^2 + xy + \left(\frac{1+D}{4}\right)y^2 \in \mathbb{Z}[x,y].\end{align*} $$

Then, one can check

$$ \begin{align*}p = N_{K/\mathbb{Q}}(\mathbf{p}) = f_D(M,N) \quad \text{ and } \quad |\widetilde{E}_p(\mathbb{F}_p)| = f_D(M-1,N).\end{align*} $$

Thus, we see that this is related to studying integer pairs $(M,N) \in \mathbb {Z}^2$ for which both $f_D(M,N)$ and $f_D(M-1,N)$ are primes. This setup is a special case of the multivariate Bateman–Horn conjecture [Reference Bateman and Horn7], which generalizes the Hardy–Littlewood conjecture to the setting of several variables [Reference Hardy and Littlewood29].

This idea can be used to note additional CM curves for which $\pi _E^{\operatorname {\mathrm {prime}}}(x)$ is bounded. Suppose $D \equiv 7 \ \pmod 8$ . (In fact, $K = \mathbb {Q}(\sqrt {-7})$ is the only CM field satisfying the property.) A direct calculation shows that there are no integer pairs $(M,N)$ for which both $f_D(M,N)$ and $f_D(M-1,N)$ are odd, and thus prime. Consequently, for this curve $\pi _E^{\operatorname {\mathrm {prime}}}(x)$ is uniformly bounded. An alternative way to see this is to observe that every elliptic curve E with CM field $\mathbb {Q}(\sqrt {-7})$ has torsion subgroup $\mathbb {Z}/2\mathbb {Z}$ .

We now turn to our original formulation of the prime-counting function. Note that $\mathbb {F}_p \simeq \mathbb {F}_{\mathbf {p}}$ and the $\widetilde {E}_p$ is isomorphic to $\widetilde {E}_{\mathbf {p}}$ as an elliptic curve over the base field. In particular,

$$\begin{align*}|\widetilde{E}_p(\mathbb{F}_p)| = |\widetilde{E}_{\mathbf{p}}(\mathbb{F}_{\mathbf{p}})|.\end{align*}$$

Thus, we obtain

The Koblitz conjecture in arithmetic progressions for CM elliptic curves can be formulated as follows.

Conjecture 6.3 Let $E/\mathbb {Q}$ be an elliptic curve with CM by an order $\mathcal {O}$ in an imaginary quadratic field K. Let $m_E$ be as in Lemma 2.6, n be a positive integer, and k be an integer coprime to n. Then there exists a constant $C^{\operatorname {\mathrm {prime}}}_{E/K,n,k}$ defined in (62) such that

(58) $$ \begin{align} \pi^{\operatorname{\mathrm{prime}}}_{E}(x;n,k) \sim \frac{C^{\operatorname{\mathrm{prime}}}_{E/K,n,k}}{2} \cdot \frac{x}{\log^2 x} \hspace{0.5cm} \text{ as } x\to \infty. \end{align} $$

If the constant vanishes, we interpret (58) as stating that there are only finitely many primes $p \equiv k \ \pmod n$ of Koblitz reduction for E.

Comparing with Conjecture 1.6, we have

(59) $$ \begin{align} C^{\operatorname{\mathrm{prime}}}_{E,n,k}=\frac{C^{\operatorname{\mathrm{prime}}}_{E/K,n,k}}{2}, \end{align} $$

where $C^{\operatorname {\mathrm {prime}}}_{E/K,n,k}$ is defined in (60).

We now introduce some notation used to determine the constant $C^{\operatorname {\mathrm {prime}}}_{E/K,n,k}$ . For a positive integer m, let us fix a $\mathbb {Z}/m\mathbb {Z}$ -basis of $\mathcal {O}/m\mathcal {O}$ . This allows us to view $\operatorname {\mathrm {GL}}_1(\mathcal {O}/m\mathcal {O})=(\mathcal {O}/m\mathcal {O})^{\times }$ a subgroup of $\operatorname {\mathrm {GL}}_2(\mathbb {Z}/m\mathbb {Z})$ . Let $\operatorname {det} \colon (\mathcal {O}/m\mathcal {O})^\times \to (\mathbb {Z}/m\mathbb {Z})^\times $ be the determinant map, defined in the natural way. Fixing a standard orthogonal basis of $\mathcal {O}/m\mathcal {O}$ , N is identified with the determinant map. Thus, drawing a parallel from (34), we are led to define

$$ \begin{align*}\Psi^{\operatorname{\mathrm{prime}}}_{K,n,k}(m) = \left\{g \in (\mathcal{O}/m\mathcal{O})^\times : \operatorname{det}(g-1) \in (\mathbb{Z}/m \mathbb{Z})^{\times}, \operatorname{det} g \equiv k \quad\pmod {\gcd(m,n)}\right\}.\end{align*} $$

Observe that $\rho _{E,m}(\operatorname {\mathrm {Frob}}_{\mathbf {p}}) \in G_E(m) \cap \Psi ^{\operatorname {\mathrm {prime}}}_{K,n,k}(m)$ if and only if $|\widetilde {E}_{\mathbf {p}}(\mathbb {F}_{\mathbf {p}})|$ is invertible in $\mathbb {Z}/m\mathbb {Z}$ and $\operatorname {det}(\rho _{E,m}(\operatorname {\mathrm {Frob}}_{\mathbf {p}})) \equiv k \ \pmod {\gcd (m,n)}$ . Hence, we are led to define

Drawing a parallel from (35), we set

(60)

where the limit is taken over all positive integers ordered by divisibility.

Lemma 6.4 Let $E/\mathbb {Q}$ be an elliptic curve with CM by an order $\mathcal {O}$ of conductor f in an imaginary quadratic field K. Let $m_E$ be as in Lemma 2.6. and be given as in (17). For each rational prime $\ell \nmid fm_E$ and $\ell \nmid n$ , we have

$$\begin{align*}\frac{\delta_{E/K,n,k}^{\operatorname{\mathrm{prime}}}(\ell)}{1-1/\ell}= \displaystyle 1 - \chi(\ell) \frac{\ell^2-\ell-1}{(\ell-\chi(\ell))(\ell-1)^2}. \end{align*}$$

For each prime $\ell \nmid f m_E$ and $\ell ^{\alpha } \parallel n$ , we have

$$ \begin{align*} \frac{\delta_{E/K,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha})}{1-1/\ell} = \begin{cases} \displaystyle \frac{1}{\phi(\ell^{\alpha})}\left( 1 - \chi(\ell) \frac{1}{(\ell-\chi(\ell))(\ell-1)} \right) & \text{ if } \ell^{\alpha} \parallel n \text{ and } k \equiv 1 \quad\pmod \ell, \\ \displaystyle \frac{1}{\phi(\ell^{\alpha})} \left(1 - \chi(\ell) \frac{\ell+1}{(\ell-\chi(\ell))(\ell-1)}\right) & \text{ if } \ell^{\alpha} \parallel n \text{ and } k \not \equiv 1 \quad\pmod \ell. \end{cases} \end{align*} $$

Proof First, we consider the case where $\ell \nmid nfm_E$ . By Lemma 2.6, we have $G_E(\ell )\simeq (\mathcal {O}_K/\ell \mathcal {O}_K)^{\times }$ and the condition $\operatorname {det} g\equiv k \ \pmod {\gcd (\ell , n)}$ trivially holds. Hence

$$\begin{align*}G_E(\ell)\cap \Psi_{K,n,k}^{\operatorname{\mathrm{prime}}}(\ell)=\{g\in (\mathcal{O}_K/\ell\mathcal{O}_K)^\times: \operatorname{det}(g-1)\not\equiv 0 \quad\pmod \ell \}. \end{align*}$$

Therefore, by Corollary 3.6, we get

$$\begin{align*}|\Psi_{K,n,k}^{\operatorname{\mathrm{prime}}}(\ell)|=(\ell-2)^2 \; \text{ or } \; |\Psi_{K,n,k}^{\operatorname{\mathrm{prime}}}(\ell)|=\ell^2-2 \end{align*}$$

depending on whether $\ell $ splits or is inert in K.

Now we assume $\ell ^{\alpha }\parallel n$ . Similarly, we have $G_E(\ell ^{\alpha })\simeq (\mathcal {O}_K/\ell ^{\alpha }\mathcal {O}_K)^{\times }$ and hence

$$\begin{align*}G_E(\ell^{\alpha})\cap \Psi_{K,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha})=\Psi_{K,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha}). \end{align*}$$

Then the condition $\operatorname {det} g\equiv k \ \pmod {\gcd (\ell ^{\alpha }, n)}$ becomes $\operatorname {det} g\equiv k \ \pmod {\ell ^{\alpha }}$ . So we get

$$\begin{align*}\Psi_{K,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha})=\left\{ g \in (\mathcal{O}_K/\ell^\alpha\mathcal{O}_K)^\times : \operatorname{det}(g-1) \not \equiv 0 \quad\pmod {\ell}, \operatorname{det} g \equiv k \quad\pmod {\ell^\alpha} \right\}. \end{align*}$$

If $k\equiv 1\ \pmod \ell $ , then by Corollary 3.6,

$$\begin{align*}|\Psi_{K,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha})|= \ell^{\alpha-1}(\ell-2) \; \text{ or } \; |\Psi_{K,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha})|= \ell^{\alpha} \end{align*}$$

depending on whether $\ell $ splits or is inert in K. If $k\not \equiv 1\ \pmod \ell $ , then

$$\begin{align*}|\Psi_{K,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha})|= \ell^{\alpha-1}(\ell-3) \; \text{ or } \; |\Psi_{K,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha})|= \ell^{\alpha-1}(\ell+1), \end{align*}$$

depending on whether $\ell $ splits or is inert in K.

For a CM elliptic curve $E/\mathbb {Q}$ with CM by an order $\mathcal {O}$ of conductor f, we set

(61)

To save notation, we will write $\ell ^{\alpha }$ instead of $\ell ^{\alpha _\ell }$ .

Proposition 6.5 Let $E/\mathbb {Q}$ have a CM by an order $\mathcal {O}$ of conductor f in an imaginary quadratic field K. Let be as given in (17). Let $m_E$ be as in Lemma 2.6. Let L be defined as in (61). Fix a positive integer n. Then, $\delta ^{\operatorname {\mathrm {prime}}}_{E/K,n,k}(\cdot )$ , as an arithmetic function, satisfies the following properties:

1. (1) Let $L \mid L^{\prime } \mid L^\infty $ . Then, $\delta ^{\operatorname {\mathrm {prime}}}_{E/K,n,k}(L) = \delta _{E/K,n,k}^{\operatorname {\mathrm {prime}}}(L^{\prime })$ ;

2. (2) Let $\ell ^\alpha $ be a prime power and d be a positive integer with $(\ell , Ld) = 1$ . Then, $\delta ^{\operatorname {\mathrm {prime}}}_{E/K,n,k}(d\ell ^\alpha ) = \delta ^{\operatorname {\mathrm {prime}}}_{E/K,n,k}(d) \cdot \delta ^{\operatorname {\mathrm {prime}}}_{E/K,n,k}(\ell ^\alpha )$ .

3. (3) Let $\ell ^\alpha \parallel n$ and $(\ell , L) = 1$ . Then, for any $\beta> \alpha $ , $\delta _{E/K,n,k}^{\operatorname {\mathrm {prime}}}(\ell ^\beta ) = \delta _{E/K,n,k}^{\operatorname {\mathrm {prime}}}(\ell ^\alpha )$ . Further, if $\ell \nmid nL$ , we have $\delta _{E/K,n,k}^{\operatorname {\mathrm {prime}}}(\ell ^\beta ) = \delta _{E/K,n,k}^{\operatorname {\mathrm {prime}}}(\ell )$ .

Therefore, (60) can be expressed as

(62) $$ \begin{align} C^{\operatorname{\mathrm{prime}}}_{E/K,n,k} = \frac{\delta_{E/K,n,k}^{\operatorname{\mathrm{prime}}}(L)}{\prod_{\ell \mid L} (1-1/\ell)} \cdot \prod_{\substack{\ell \nmid fm_E \\ \ell^{\alpha} \parallel n}}\frac{\delta_{E/K,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha})}{1-1/\ell} \cdot \prod_{\substack{ \ell \nmid nfm_E}} \left(1 - \chi(\ell)\frac{\ell^2-\ell-1}{(\ell-\chi(\ell))(\ell-1)^2} \right). \end{align} $$

Proof One can prove (1)–(3) following the same strategy as in the proof of Proposition 4.4. One only needs to replace $m_E$ by $fm_E$ and $\operatorname {\mathrm {GL}}_2(\mathbb {Z}/\ell ^{\alpha }\mathbb {Z})$ by $(\mathcal {O}/\ell ^{\alpha }\mathcal {O})^\times $ . Therefore, from these results, we get

$$ \begin{align*} C^{\operatorname{\mathrm{prime}}}_{E/K,n,k} = & \frac{\delta_{E/K,n,k}^{\operatorname{\mathrm{prime}}}(L)}{\prod_{\ell \mid L} (1-1/\ell)} \cdot \prod_{\substack{\ell \nmid fm_E \\ \ell^{\alpha} \parallel n}}\frac{\delta_{E/K,n,k}^{\operatorname{\mathrm{prime}}}(\ell^{\alpha})}{1-1/\ell} \cdot \prod_{\substack{\ell \nmid nfm_E}} \frac{\delta_{E/K,n,k}^{\operatorname{\mathrm{prime}}}(\ell)}{1-1/\ell}. \end{align*} $$

Now, we see that (62) follows from Lemma 6.4.

Remark 6.6 Given that $\ell \nmid nfm_E$ , we observe that

$$ \begin{align*} \frac{\delta^{\operatorname{\mathrm{prime}}}_{E/K,n,k}(\ell)}{1-1/\ell} &= 1 - \chi_K(\ell) \frac{\ell^2-\ell-1}{(\ell-\chi_K(\ell))(\ell-1)^2} \\ &= \left( 1 - \frac{\chi_K(\ell)}{\ell} + \mathbf{O}\left(\frac{1}{\ell^2}\right)\right) \\ &= \left(1 - \frac{\chi_K(\ell)}{\ell}\right)\left(1+\mathbf{O}\left(\frac{1}{\ell^2}\right)\right). \end{align*} $$

Thus, we have

$$ \begin{align*} \prod_{\substack{\ell \nmid fm_En}} \left( 1 - \chi_K(\ell) \frac{\ell^2-\ell-1}{(\ell-\chi_K(\ell))(\ell-1)^2}\right) = \prod_{\substack{ \ell \nmid fm_En}} \left(1-\frac{\chi_K(\ell)}{\ell}\right)\left(1+\mathbf{O}\left(\frac{1}{\ell^2}\right)\right). \end{align*} $$

Note that this is a product of an Euler factorization of $L(s,\chi _K)^{-1}$ at $s = 1$ (with some correction factor) and an absolutely convergent product. Since $L(1,\chi _K) \neq 0$ for a non-trivial character $\chi _K$ , the infinite product in (62) is conditionally convergent.

By (58), (59), Lemma 6.4, and Proposition 6.5, we can explicitly formulate the conjectural Koblitz constant for CM elliptic curves. Let $n = n_1n_2$ where $n_1 \mid (fm_E)^\infty $ and $(n_2,fm_E) = 1$ . (In particular, $n_2$ is the product of $\ell ^{\alpha }$ for which $\ell ^{\alpha } \parallel n$ with $\ell \nmid L$ .) We have

(63) $$ \begin{align} \begin{aligned} C_{E,n,k}^{\operatorname{\mathrm{prime}}} = \frac{1}{2} \cdot \frac{1}{\phi(n_2)} \cdot \frac{\delta_{E/K, n,k}^{\operatorname{\mathrm{prime}}}(L)}{\prod_{\ell \mid L}(1-1/\ell)} & \cdot \prod_{\substack{\ell^{\alpha} \parallel n \\ \ell \nmid L \\ \ell\mid k-1}} \left( 1 - \chi_K(\ell) \frac{\ell}{(\ell-\chi_K(\ell))(\ell-1)}\right) \\ &\cdot \prod_{\substack{\ell^{\alpha} \parallel n \\ \ell \nmid L \\ \ell\nmid k-1}} \left(1-\chi_K(\ell) \frac{\ell+1}{(\ell-\chi_K(\ell))(\ell-1)}\right) \\ &\cdot \prod_{\ell \nmid nfm_E}\left(1-\chi_K(\ell)\frac{\ell^2-\ell-1}{(\ell-\chi_K(\ell))(\ell-1)^2}\right). \end{aligned} \end{align} $$

Proposition 6.7 For any CM elliptic curve $E/\mathbb {Q}$ , we have

$$ \begin{align*}C^{\operatorname{\mathrm{prime}}}_{E,n,k} \ll_n 1.\end{align*} $$

Proof Note that the finite product terms in (63) are all bounded by $1$ . By definition, we have

$$ \begin{align*}\delta_{E/K,n,k}^{\operatorname{\mathrm{prime}}}(L) \leq 1,\end{align*} $$

and hence,

$$ \begin{align*}\frac{\delta_{E/K,n,k}^{\operatorname{\mathrm{prime}}}(L)}{\prod_{\ell \mid L}(1-1/\ell)} \ll \max\{1, \log \log \operatorname{\mathrm{rad}}(fm_E)\} \ll 1,\end{align*} $$

by Proposition 2.8 and Lemma 6.1. Finally, the infinite product, up to a correction factor depending on n, is universally bounded, since there are only finitely many possibilities for K.