Constants in Titchmarsh divisor problems for elliptic curves

TL;DR

This paper investigates the constants arising in the asymptotic analysis of elliptic curve divisor sums, providing bounds, explicit formulas for generic cases, and moments over families of elliptic curves, enhancing understanding of their individual and average behaviors.

Contribution

It offers bounds and explicit formulas for the constants in elliptic divisor sums, advancing the analysis of their asymptotic and individual properties.

Findings

Upper bounds for constants C(E) are established.

Explicit formulas for C(E) are derived for Serre curves.

Moments of C(E) are computed over families of elliptic curves.

Abstract

Inspired by the analogy between the group of units $\mathbb{F}_p^{\times}$ of the finite field with $p$ elements and the group of points $E(\mathbb{F}_p)$ of an elliptic curve $E/\mathbb{F}_p$, E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum $\sum_{p \leq x} \tau(p + a) \sim C x$. In this paper, we present a comprehensive study of the constants $C(E)$ emerging in the asymptotic study of these elliptic curve divisor sums. Specifically, by analyzing the division fields of an elliptic curve $E/\mathbb{Q}$, we prove upper bounds for the constants $C(E)$ and, in the generic case of a Serre curve, we prove explicit closed formulae for $C(E)$ amenable to concrete computations. Moreover, we compute the moments of the constants $C(E)$ over two-parameter families of elliptic curves $E/\mathbb{Q}$. Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics.

Equations

$$\sum _{p\le x}\tau (p+a)\sim \frac{\zeta (2)\zeta (3)}{\zeta (6)}\prod _{\ell \mid a}\left(1-\frac{\ell }{{\ell }^2-\ell +1}\right)x,$$

$$\overline{E}({\mathbb{F}}_p)\simeq \mathbb{Z}/d_{1,p}\mathbb{Z}\times \mathbb{Z}/d_{2,p}\mathbb{Z},$$

$$\sum _{p\le x}{d}_{1,p}$$

$$C_{d_1,\text{non-CM}}(E)$$

$$C_{d_1,\text{CM}}(E)$$

$$\sum _{p\le x}\tau (d_{1,p})\sim C_{\tau (d_1)}(E)\mathrm{li}(x),$$

$$C_{\tau (d_1)}(E):=\sum _{m\ge 1}\frac{1}{[\mathbb{Q}(E[m]):\mathbb{Q}]}.$$

$$\sum _{p\le x}{d}_{2,p}\sim \frac{1}{2}{C}_{d_2}(E)\mathrm{li}\left(x^2\right),$$

$$C_{d_2}(E):=\sum _{m\ge 1}\frac{(-1{)}^{\omega (m)}\varphi (\mathrm{rad}m)}{m[\mathbb{Q}(E[m]):\mathbb{Q}]},$$

$$C_{d_1}$$

$$C_{\tau (d_1)}$$

$$C_{d_2}$$

$$\frac{1}{|\mathcal{C}(A,B)|}\sum _{E\in \mathcal{C}(A,B)}\sum _{\genfrac{}{}{0ex}{}{p\le x}{p\nmid \Delta (E)}}\tau (d_{1,p}(E))=C_{\tau (d_1)}\mathrm{li}(x)+O\left(\frac{x}{(\log x{)}^c}\right),$$

$$\frac{1}{|\mathcal{C}(A,B)|}\sum _{E\in \mathcal{C}(A,B)}\sum _{\genfrac{}{}{0ex}{}{p\le x}{p\nmid \Delta (E)}}{d}_{2,p}(E)=\frac{1}{2}{C}_{d_2}\mathrm{li}(x^2)+O\left(\frac{x^2}{(\log x{)}^c}\right).$$

$$\frac{1}{|\mathcal{C}(A,B)|}\sum _{\genfrac{}{}{0ex}{}{E\in \mathcal{C}(A,B)}{E\text{without CM}}}\sum _{\genfrac{}{}{0ex}{}{p\le x}{p\nmid \Delta (E)}}{d}_{1,p}(E)=C_{d_1}\mathrm{li}(x)+o\left(\frac{x}{\log x}\right).$$

$$C_{d_1,\text{CM}}(E)\ll 1$$

$$C_{d_2}(E)\le C_{\tau (d_1)}(E)\ll 1.$$

$$C_{d_1,\text{non-CM}}(E){\ll }_{\epsilon }{\left(\max \left\{|a{|}^3,|b{|}^2\right\}\log {\left(\max \left\{|a{|}^3,|b{|}^2\right\}\right)}^{\gamma }\right)}^{\epsilon }\text{for any}\epsilon >0$$

$$C_{d_2}(E)\le C_{\tau (d_1)}(E)\ll \log \log \left\{\max \left\{|a{|}^3,|b{|}^2\right\}\log {\left(\max \left\{|a{|}^3,|b{|}^2\right\}\right)}^{\gamma }\right\}.$$

$$m_{E}=\left\{\begin{array}[]{cc}2\left|\Delta_{\text{sf}}(E)\right|&\;\text{if $\Delta_{\text{sf}}(E)\equiv 1(\operatorname{mod}4)$,}\\ 4\left|\Delta_{\text{sf}}(E)\right|&\;\text{otherwise},\end{array}\right.$$

$$C_{d_1,non-CM}(E)$$

$$C_{\tau (d_1)}(E)$$

$$C_{d_2}(E)$$

$$\underset{x\to \infty }{\lim }\frac{1}{|\mathcal{C}(A(x),B(x))|}\sum _{E\in \mathcal{C}(A(x),B(x))}C(E)=C.$$

$$\frac{1}{|\mathcal{C}(A,B)|}\sum _{E\in \mathcal{C}(A,B)}{\left|C(E)-C\right|}^n$$

$${\phi }_{E,m}:G_{\mathbb{Q}}⟶{\mathrm{GL}}_2(\mathbb{Z}/m\mathbb{Z}).$$

$$\mathbb{Q}(E[m])={\overline{\mathbb{Q}}}^{\mathrm{Ker}{\phi }_{E,m}}\text{and}[\mathbb{Q}(E[m]):\mathbb{Q}]=\left|\mathrm{Im}({\phi }_{E,m})\right|.$$

$${\phi }_E:G_{\mathbb{Q}}⟶{\mathrm{GL}}_2(\hat{\mathbb{Z}}),$$

$${\phi }_{E,m^{\infty }}:G_{\mathbb{Q}}⟶{\mathrm{GL}}_2({\mathbb{Z}}_m),$$

$$\mathbb{Q}(i),\mathbb{Q}\left(\sqrt{-2}\right),\mathbb{Q}\left(\sqrt{-3}\right),\mathbb{Q}\left(\sqrt{-7}\right),\mathbb{Q}\left(\sqrt{-11}\right),\mathbb{Q}\left(\sqrt{-19}\right),\mathbb{Q}\left(\sqrt{-43}\right),\mathbb{Q}\left(\sqrt{-67}\right),\mathbb{Q}\left(\sqrt{-163}\right),$$

$$-3,-4,-7,-8,-11,-12,-16,-19,-27,-28,-43,-67,-163.$$

$$0,2^6\cdot 3^3,-3^3\cdot 5^3,2^6\cdot 5^3,-2^{15},2^4\cdot 3^3\cdot 5^3,2^3\cdot 3^3\cdot 11^3,$$

Full text

ACC’s work on this material was partially supported by the Simons Collaboration Grant under Award No. 318454. IV’s work was partially supported by the NSF Graduate Research Fellowship Program and grant DMS-1601946.

1. Introduction

The Titchmarsh divisor problem concerns the asymptotic behaviour of the sum $\displaystyle\sum_{p\leq x}\tau(p+a)$, as a function of $x$, where $p$ denotes a rational prime, $\tau(n):=\#\{d\geq 1:d\mid n\}$ denotes the divisor function, and $a$ denotes a fixed integer. The study of this sum has spanned over five decades and is intimately related to some of the most significant research in analytic number theory (see [BoFrIw], [Fo], [Ha], [Li], and [Ti]). By results about primes in arithmetic progressions which have become standard, we now know that, as $x\rightarrow\infty$,

[TABLE]

where $\zeta(\cdot)$ denotes the Riemann zeta function and $\ell$ denotes a rational prime.

Divisor problems similar to Titchmarsh’s may be formulated in other settings, such as the setting of elliptic curves, as we now describe. Let $E/\mathbb{Q}$ be an elliptic curve defined over the field of rational numbers. For a prime $p$ of good reduction for $E$, let $\overline{E}/\mathbb{F}_{p}$ be the reduction of $E$ modulo $p$. From the basic theory of elliptic curves (see [Si, Ch. III, §6]), it is known that the group $\overline{E}(\mathbb{F}_{p})$ may be expressed as a product of two cyclic groups,

[TABLE]

where $d_{1,p}=d_{1,p}(E),d_{2,p}=d_{2,p}(E)$ are uniquely determined positive integers satisfying $d_{1,p}\mid d_{2,p}$. Determining the asymptotic behaviour of sums over $p\leq x$ of arithmetic functions evaluated at the elementary divisors $d_{1,p}$ and $d_{2,p}$ may be viewed as Titchmarsh divisor problems for elliptic curves. Such problems unravel striking similarities, but also intriguing contrasts, to the original Titchmarsh divisor problem, as illustrated in [AkGh], [AkFe], [Co], [CoMu], [Fe], [FeMu], [FrKu], [FrPo], [Ki], [Ko], [Po], and [Wu].

The focus of our paper is on the constants emerging in the following three Titchmarsh divisor problems for elliptic curves. In all the expressions below, the letters $p$ and $\ell$ denote primes, with $p$ being a prime of good reduction for the given elliptic curve.

Conjecture 1**.**

(Kowalski [Ko, Section 3.2])

Let $E/\mathbb{Q}$ be an elliptic curve. Then, as $x\rightarrow\infty$,

[TABLE]

where

[TABLE]

with $\phi(m):=\#\{1\leq k\leq m:(k,m)=1\}$ denoting the Euler function of $m$, $\mathbb{Q}(E[m])$ denoting the $m$-division field of $E$, and $\operatorname{li}(x)\colonequals\displaystyle\int_{2}^{x}\frac{1}{\operatorname{log}{t}}\ dt$ denoting the standard logarithmic integral.

Conjecture 2**.**

(Akbary-Ghioca [AkGh, Section 1])

Let $E/\mathbb{Q}$ be an elliptic curve, with or without complex multiplication. Then, as $x\rightarrow\infty$,

[TABLE]

where

[TABLE]

Conjecture 3**.**

(Freiberg-Kurlberg [FrKu, Section 1])

Let $E/\mathbb{Q}$ be an elliptic curve, with or without complex multiplication. Then, as $x\rightarrow\infty$,

[TABLE]

where

[TABLE]

with $\omega(m):=\displaystyle\sum_{\ell\mid m}1$ denoting the number of distinct prime factors of $m$ and $\operatorname{rad}(m):=\displaystyle\prod_{\ell\mid m}\ell$ denoting the product of distinct prime factors of $m$.

The constants $C_{d_{1},\text{non-CM}}(E)$, $C_{d_{1},\text{CM}}(E)$, $C_{\tau(d_{1})}(E)$, and $C_{d_{2}}(E)$ appearing in these conjectures are deeply related to the arithmetic of the elliptic curve $E/\mathbb{Q}$ and are heuristically derived via the Chebotarev density theorem by considering the action of a Frobenius element at $p$ on $E[m]$.

Conjecture 1 was investigated by Freiberg and Pollack in [FrPo] in the case that $E$ has complex multiplication; precisely, they proved that $\displaystyle\sum_{p\leq x}d_{1,p}\asymp_{E}x$. A similar result is not yet known if $E$ does not have complex multiplication, even under the Generalized Riemann Hypothesis. Conjecture 2 was investigated by Akbary and Ghioca in [AkGh]; precisely, they proved (6) under the Generalized Riemann Hypothesis if $E$ is without complex multiplication and unconditionally if $E$ is with complex multiplication. Conjecture 3 was investigated by Freiberg and Kurlberg in [FrKu]; precisely, they proved (8) under the Generalized Riemann Hypothesis if $E$ is without complex multiplication and unconditionally if $E$ is with complex multiplication. The proofs of the main results in [AkGh] and [FrKu] rely upon the methods of [CoMu] and have been refined in several subsequent works, including [AkFe], [Fe], [FeMu], [Ki], and [Wu].

Using ideas originating in [FoMu], Conjectures 1 - 3 may also be investigated on average over elliptic curves in families. For this, we consider parameters $A,B>2$, the family ${\mathcal{C}}(A,B)$ of $\mathbb{Q}$-isomorphism classes of elliptic curves $E=E_{a,b}$ defined by $Y^{2}=X^{3}+aX+b$ with $a,b\in\mathbb{Z}$, $|a|\leq A,|b|\leq B$, $\Delta(E)=\Delta(a,b):=-16(4a^{3}+27b^{2})\neq 0$, and average the Titchmarsh divisor sums of Conjectures 1 - 3 over $E\in{\mathcal{C}}(A,B)$.

In this context, it is natural to consider the following idealized versions of the constants $C_{d_{1},\text{non-CM}}(E)$, $C_{\tau(d_{1})}(E)$, and $C_{d_{2}}(E)$,

[TABLE]

These idealized constants turn out to be the average constants, as explained below.

In [AkFe, Cor 1.6], Akbary and Felix proved that for any $c>1$ and $x>e$, there exists $c_{1}>0$ such that, for any $A=A(x)$, $B=B(x)$ with $A,B>\operatorname{exp}(c_{1}(\operatorname{log}{x})^{1/2})$ and $AB>x(\operatorname{log}{x})^{4+2c}$, we have

[TABLE]

[TABLE]

These results confirm Conjectures 2 - 3 on average.

Conjecture 1 is also expected to hold on average; that is, for suitably large $A=A(x)$, $B=B(x)$, we expect

[TABLE]

While this average is open, Akbary and Felix proved related results supporting it (see [AkFe, Remark 1.7]).

In this paper we investigate the constants $C_{d_{1},\text{non-CM}}(E)$, $C_{d_{1},\text{CM}}(E)$, $C_{\tau(d_{1})}(E)$, and $C_{d_{2}}(E)$, on their own and in relation to the idealized constants $C_{d_{1}}$, $C_{\tau(d_{1})}$, and $C_{d_{2}}$. Specifically, using properties of the division fields of $E/\mathbb{Q}$, which we derive from the celebrated open image theorems for elliptic curves with complex multiplication (due to Weil), respectively without complex multiplication (due to Serre), we prove:

Theorem 4**.**

Let $E/\mathbb{Q}$ be an elliptic curve.

1. (i)

Assume that $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\not\simeq\mathbb{Z}$. Then

[TABLE]

and

[TABLE] 2. (ii)

Assume that $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\simeq\mathbb{Z}$. Let $Y^{2}=X^{3}+aX+b$ be a Weierstrass model for $E$ with $a,b\in\mathbb{Z}$. There exists a positive absolute constant $\gamma$ such that

[TABLE]

and

[TABLE]

The $\ll$-constants are absolute, while the $\ll_{\varepsilon}$-constant depends on $\varepsilon$.

Furthermore, by narrowing down our focus to generic elliptic curves, we prove explicit formulae for our Titchmarsh divisor constants:

Theorem 5**.**

Let $E/\mathbb{Q}$ be a Serre curve, that is, an elliptic curve over $\mathbb{Q}$ whose adelic Galois representation has maximal image. Let

[TABLE]

where $\Delta_{\text{sf}}(E)$ denotes the squarefree part of any Weierstrass discriminant of $E$. Then

[TABLE]

Finally, we use the above two results to prove that the average of the individual constants gives rise to the idealized constant:

Theorem 6**.**

For any $A(x),B(x)>2$, tending to infinity with $x$ such that the ratios of their logarithms remain bounded, we have

[TABLE]

Here, the pair $(C(E),C)$ is, respectively, $(C_{d_{1}}(E),C_{d_{1}})$, $(C_{\tau(d_{1})}(E),C_{\tau(d_{1}}))$, and $(C_{d_{2}}(E),C_{d_{2}})$, with $C_{d_{1}}(E)$ denoting $C_{d_{1},\text{non-CM}}(E)$ if $E$ is without complex multiplication, and $C_{d_{1},\text{CM}}(E)$ if $E$ is with complex multiplication.

Remark 7**.**

**

1. (i)

The constant $\gamma$ occurring in Theorem 4 is known as the Masser-Wüstholz constant and originates in [MaWu]; it has been studied computationally in [Ka]. 2. (ii)

The constant $m_{E}$ occurring in Theorem 5 was first introduced in [Jo10] in relation to Serre’s open image theorem from [Se72]. In Section 2, we will revisit its original definition (see Theorem 14) and we will confirm that it satisfies equation (16) above (see Proposition 20). 3. (iii)

In Section 5, we will actually prove a stronger result than (20) by bounding, from above,

[TABLE]

for any integer $n\geq 1$ and for any $A,B>2$; see equations (47), (48), (49), and (50). 4. (iv)

It is a difficult problem to calculate the constants $C_{d_{1},\text{non-CM}}(E)$, $C_{d_{1},\text{CM}}(E)$, $C_{\tau(d_{1})}(E)$, and $C_{d_{2}}(E)$ for an arbitrary elliptic curve $E/\mathbb{Q}$. The underlying reason is the difficulty in attaining a complete understanding, explicit in terms of $E$, of the non-trivial intersections between the division fields of $E$. This may be rephrased as a question about understanding the constant $m_{E}$, alluded to in Theorem 6 and introduced in the upcoming Theorem 12 and Theorem 14 of Section 2. In Theorem 4 we succeed in bounding the constants $C_{d_{1},\text{non-CM}}(E)$, $C_{d_{1},\text{CM}}(E)$, $C_{\tau(d_{1})}(E)$, and $C_{d_{2}}(E)$ by either working with all twists of $E/\mathbb{Q}$ in the complex multiplication case, or by using an upper bound for $m_{E}$ in the other case. 5. (v)

Theorem 5 shows that the constants $C_{d_{1},\text{non-CM}}(E)$, $C_{\tau(d_{1})}(E)$, and $C_{d_{2}}(E)$ of Conjectures 1 - 3 are never equal to the idealized constants $C_{d_{1}}$, $C_{\tau(d_{1})}$, and $C_{d_{2}}$. This is in contrast to other questions about reductions of elliptic curves, such as Koblitz’s Conjecture about the primality of $|\overline{E}(\mathbb{F}_{p})|$. Moreover, the explicit formulae of Theorem 5 can be used to study Conjectures 1 - 3 numerically, as done in [CoFiInYi]. Combining these formulae with Theorem 6, we may also naturally expect a positive bias in the remainder term associated to an (appropriately formulated) average version of these conjectures; see Section 6. 6. (vi)

Theorem 6 contributes to the research on averages of constants arising in the study of reductions of elliptic curves over $\mathbb{Q}$, as pursued in [AkDaHaTh], [BaCoDa], [BaSh], [CoIwJo], and [Jo09]. It also complements research on averages of constants arising in the study of all elliptic curves over the field $\mathbb{F}_{p}$, as pursued in [DaKoSm], [Ge], [Ho], [KaPe], and [Vl]. The connection between the former “global” viewpoint and the latter “local” viewpoint involves the question of to what extent the reductions of a fixed elliptic curve $E/\mathbb{Q}$ behave like random elliptic curves over $\mathbb{F}_{p}$. In our average approach we follow the methods of [Jo09] and realize the global-to-local connection via Theorem 5 and Jones’ result that most elliptic curves over $\mathbb{Q}$ are Serre curves [Jo10]. 7. (vii)

A natural question is whether a conjecture similar to Conjecture 2 may be formulated regarding the behaviour of $\tau(d_{2,p})$. Since we have not found such an investigation in the literature, we relegate it to future research.

Notation. Throughout the paper, we follow the following standard notation.

$\bullet$ The letters $p$ and $\ell$ denote rational primes. The letters $d,k,m,n$ denote rational integers. The letters $\phi$, $\tau$, $\omega$ denote the Euler function, the divisor function, and the prime factor counting function. For an integer $m$, $|m|$ denotes its absolute value, $\operatorname{rad}(m)$ its radical, and $v_{\ell}(m)$ its valuation at a prime $\ell$. For nonzero integers $m$, $n$, we write $m\mid n^{\infty}$ for the assumption that every prime divisor of $m$ also divides $n$. The symbol $\mathbb{N}$ stands for all natural numbers, including [math].

$\bullet$ For two functions $f,g:D\longrightarrow\mathbb{R}$, with $D\subseteq\mathbb{C}$ and $g$ positive, we write $f(x)=\operatorname{O}(g(x))$ or $f(x)\ll g(x)$ if there is a positive constant $c_{1}$ such that $|f(x)|\leq c_{1}g(x)$ for all $x\in D$. If $c_{1}$ depends on another specified constant $c_{2}$, we may write $f(x)=\operatorname{O}_{c_{2}}(g(x))$ or $f(x)\ll_{c_{2}}g(x)$. If $c:=\displaystyle\lim_{x\rightarrow\infty}\frac{f(x)}{g(x)}$ exists, we write $f(x)\sim c\ g(x)$.

$\bullet$ For a field $K$, we write $\overline{K}$ for a fixed algebraic closure and $G_{K}$ for the absolute Galois group $\operatorname{Gal}\left(\overline{K}/K\right)$.

2. Generalities about elliptic curves

In this section, we review the main properties of elliptic curves needed in the proofs of our main results. While many of these properties are standard (Theorem 8 - Corollary 15), the ones towards the end of the section (Lemma 16 - Theorem 22) are less known, but crucial to our approach.

2.1. General notation

For an elliptic curve $E/\mathbb{Q}$, we use the following notation. We denote by $j=j(E)$ the $j$-invariant of $E$. We denote by $\operatorname{End}_{\overline{\mathbb{Q}}}(E)$ and $\operatorname{Aut}_{\overline{\mathbb{Q}}}(E)$ the endomorphism ring and the automorphism ring of $E$ over $\overline{\mathbb{Q}}$. We denote by $E(\mathbb{Q})$ and $E(\overline{\mathbb{Q}})$ the groups of $\mathbb{Q}$-rational and $\overline{\mathbb{Q}}$-rational points of $E$, and by $E(\mathbb{Q})_{\text{tors}}$ and $E_{\text{tors}}:=E(\overline{\mathbb{Q}})_{\text{tors}}$ their respective torsion subgroups.

For an integer $m\geq 1$, we denote by $E[m]\colonequals E(\overline{\mathbb{Q}})[m]$ the group of $m$-division points of $E(\overline{\mathbb{Q}})$. This has the structure of a free $\mathbb{Z}/m\mathbb{Z}$-module of rank $2$, with a $\mathbb{Z}/m\mathbb{Z}$-linear action of the absolute Galois group $G_{\mathbb{Q}}$. Thus, fixing an isomorphism $(\mathbb{Z}/m\mathbb{Z})^{2}\to E[m]$, which amounts to choosing a $\mathbb{Z}/m\mathbb{Z}$-basis for $E[m]$, we obtain a Galois representation

[TABLE]

Note that the $m$-division field $\mathbb{Q}(E[m])$ of $E$, defined by adjoining to $\mathbb{Q}$ the $x$ and $y$ coordinates of the points in $E[m]$, satisfies

[TABLE]

Choosing compatible bases for all $E[m]$, we form the inverse limit over $m$ ordered by divisibility and obtain a continuous Galois representation

[TABLE]

where $\hat{\mathbb{Z}}:=\displaystyle\varprojlim_{m}\mathbb{Z}/m\mathbb{Z}$. By the Chinese Remainder Theorem, we have $\hat{\mathbb{Z}}\simeq\displaystyle\prod_{\ell}\mathbb{Z}_{\ell}$, and, post-composing with projections, we obtain the Galois representations

[TABLE]

where $\mathbb{Z}_{m}\colonequals\displaystyle\prod_{\ell\mid m}\mathbb{Z}_{\ell}$.

2.2. Morphisms

Theorem 8**.**

([Cox, Thm 7.30]) Let $E/\mathbb{Q}$ be an elliptic curve. The following statements hold.

1. (i)

Either $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\simeq\mathbb{Z}$, in which case we say that $E$ is without complex multiplication, or $\operatorname{End}_{\overline{\mathbb{Q}}}(E)$ is an order $O$ in an imaginary quadratic field $K$, in which case we say that $E$ is with complex multiplication by $K$. 2. (ii)

If $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\simeq O\not\simeq\mathbb{Z}$ and $K$ is the field of fractions of $O$, then $O$ and the ring of integers of $K$ have class number 1. In particular, by the Baker-Heegner-Stark Theorem, $K$ is one of nine possible fields,

[TABLE]

while $O$ is one of thirteen possible orders, of discriminant

[TABLE]

Corollary 9**.**

([Cox, p. 261])

Let $E/\mathbb{Q}$ be an elliptic curve. If $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\not\simeq\mathbb{Z}$, then $j(E)$ is one of thirteen possible integers,

[TABLE]

[TABLE]

listed in the same order as the above list of possible discriminants.

Note that two elliptic curves of the same $j$-invariant, while isomorphic over $\overline{\mathbb{Q}}$, may fail to be isomorphic over $\mathbb{Q}$; we call such curves twists of each other.

Proposition 10**.**

([Si, Chapter 3, Cor 10.2])

Let $E/\mathbb{Q}$ be an elliptic curve. Then the set of all twists of $E/\mathbb{Q}$ is classified by the Galois cohomology group

[TABLE]

Corollary 11**.**

Let $E/\mathbb{Q}$, $E^{\prime}/\mathbb{Q}$ be elliptic curves with $j(E)=j(E^{\prime})$. Then there exists an extension $L/\mathbb{Q}$ with $[L:\mathbb{Q}]\leq|\operatorname{Aut}_{\overline{\mathbb{Q}}}(E)|\leq 6$ such that $E\simeq_{L}E^{\prime}$.

2.3. Open image theorems

Theorem 12**.**

([We55], [We55bis])

Let $E/\mathbb{Q}$ be an elliptic curve such that $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\simeq O\not\simeq\mathbb{Z}$ is an order in the imaginary quadratic field $K$. Consider the projective limit $\widehat{O}:=\displaystyle\varprojlim_{m}O/mO$. Then:

1. (i)

$\widehat{O}$* is a free $\hat{\mathbb{Z}}$-module of rank $2$; consequently, after choosing a $\hat{\mathbb{Z}}$-basis of $\widehat{O}$, we obtain an embedding*

[TABLE] 2. (ii)

$K(E_{\text{tors}})=\mathbb{Q}(E_{\text{tors}})$* is a free $\widehat{O}$-module of rank 1, acted on $\widehat{O}$-linearly by $G_{K}$;* 3. (iii)

$\varphi_{E}(G_{K})$* is contained in $\iota(\widehat{O}^{\times})$, while $\varphi_{E}(G_{\mathbb{Q}})$ is contained in the normalizer in $\operatorname{GL}_{2}(\hat{\mathbb{Z}})$ of $\iota(\widehat{O}^{\times})$;* 4. (iv)

upon identifying $\widehat{O}^{\times}$ with $\iota(\widehat{O}^{\times})$, its image in $\operatorname{GL}_{2}(\hat{\mathbb{Z}})$, the representation

[TABLE]

has open image in $\widehat{O}^{\times}$, that is,

[TABLE]

In particular, there exists a smallest integer $m_{E}\geq 1$ such that

[TABLE]

where $\operatorname{pr}:\widehat{O}^{\times}\longrightarrow(O/m_{E}\;O)^{\times}$ is the natural projection.

Corollary 13**.**

Let $E/\mathbb{Q}$ be an elliptic curve such that $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\simeq O\not\simeq\mathbb{Z}$. With notation as above, for any integer $m\geq 1$, written uniquely as $m=m_{1}m_{2}$ for some integers $m_{1}$, $m_{2}$ with $m_{1}\mid m_{E}^{\infty}$ and $\operatorname{gcd}(m_{2},m_{E})=1$, we have

[TABLE]

where $d:=\operatorname{gcd}(m_{1},m_{E})$ and $\operatorname{pr}_{m_{1},d}:(O/m_{1}\;O)^{\times}\longrightarrow(O/d\;O)^{\times}$ is the natural projection.

Theorem 14**.**

([Se72])

Let $E/\mathbb{Q}$ be an elliptic curve such that $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\simeq\mathbb{Z}$. Then $\varphi_{E}$ has open image in $\operatorname{GL}_{2}\left(\hat{\mathbb{Z}}\right)$, that is,

[TABLE]

In particular, there exists a smallest integer $m_{E}\geq 1$ such that

[TABLE]

where $\operatorname{pr}:\operatorname{GL}_{2}\left(\hat{\mathbb{Z}}\right)\longrightarrow\operatorname{GL}_{2}(\mathbb{Z}/m_{E}\mathbb{Z})$ is the natural projection.

Corollary 15**.**

Let $E/\mathbb{Q}$ be an elliptic curve such that $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\simeq\mathbb{Z}$. With notation as above, for any integer $m\geq 1$, written uniquely as $m=m_{1}m_{2}$ for some integers $m_{1}$, $m_{2}$ with $m_{1}|m_{E}^{\infty}$ and $\operatorname{gcd}(m_{2},m_{E})=1$, we have

[TABLE]

where $d:=\operatorname{gcd}(m_{1},m_{E})$ and $\operatorname{pr}_{m_{1},d}:\operatorname{GL}_{2}(\mathbb{Z}/m_{1}\mathbb{Z})\longrightarrow\operatorname{GL}_{2}(\mathbb{Z}/d\mathbb{Z})$ is the natural projection.

A useful consequence to the above two open image theorems is:

Lemma 16**.**

Let $E/\mathbb{Q}$ be an elliptic curve and let $m_{E}$ be as in Theorem 12, respectively Theorem 14. Let $\ell\mid m_{E}$ and $d\mid m_{E}$ with $\ell\nmid d$ (recall that $\ell$ denotes a rational prime).

1. (i)

If $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\not\simeq\mathbb{Z}$, then

[TABLE]

where $K\simeq\operatorname{End}_{\overline{\mathbb{Q}}}(E)\otimes\mathbb{Q}$. 2. (ii)

If $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\simeq\mathbb{Z}$, then

[TABLE]

Proof.

We apply Corollary 13, respectively Corollary 15, with $m:=\ell^{v_{\ell}(m_{E})+\delta}d$, in which case $m_{1}=m$ and $m_{2}=1$, giving $\operatorname{gcd}(m_{1},m_{E})=\ell^{v_{\ell}(m_{E})}d$. We also observe that, since $v_{\ell}(m_{E})>0$,

[TABLE]

Lemma 16 follows.

∎

In applications, it is desirable to explicitly bound $m_{E}$ in terms of $E$; such a bound is illustrated in the result below:

Proposition 17**.**

Let $E/\mathbb{Q}$ be an elliptic curve such that $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\simeq\mathbb{Z}$. Let $m_{E}$ be as in Theorem 14, let $Y^{2}=X^{3}+aX+b$ be a Weierstrass model for $E$ with $a,b\in\mathbb{Z}$, and denote by $\Delta(a,b):=-16(4a^{3}+27b^{2})$ the discriminant of this model. Then there exists a positive absolute constant $\gamma$ such that

[TABLE]

where the $\ll$-constant is absolute.

Proof.

The bound

[TABLE]

follows from the main result in [Jo17], while the bound

[TABLE]

follows from [Zy, Theorem 1.1] (see also [MaWu]). The bound $\operatorname{rad}(|\Delta(a,b)|)\ll\max\{|a|^{3},|b|^{2}\}$ is straightforward. ∎

2.4. Serre curves

Lemma 18**.**

([Se72, Section 5.5])

Let $E/\mathbb{Q}$ be an elliptic curve. Then $\left|\operatorname{GL}_{2}\left(\hat{\mathbb{Z}}\right):\varphi_{E}(G_{\mathbb{Q}})\right|\geq 2.$

In particular, no elliptic curve $E/\mathbb{Q}$ satisfies $\left|\operatorname{GL}_{2}(\mathbb{Z}/m\mathbb{Z}):\varphi_{E,m}(G_{\mathbb{Q}})\right|=1$ for all integers $m\geq 1$. Rather, the best we can hope for is captured in the following definition:

Definition 19**.**

An elliptic curve $E/\mathbb{Q}$ is called a Serre curve if

[TABLE]

Equivalently, an elliptic curve is a Serre curve if and only if

[TABLE]

Given $E/\mathbb{Q}$ and denoting by $\Delta_{\text{sf}}(E)$ the squarefree part of the discriminant $\Delta(E)$ of any Weierstrass model for $E$, the bound in Lemma 18 arises from the containments

[TABLE]

where

[TABLE]

The existence of an integer $d_{E}$ satisfying (24) is guaranteed by the Kronecker-Weber Theorem, since $\mathbb{Q}\left(\sqrt{\Delta(E)}\right)$ is abelian over $\mathbb{Q}$; this value of $d_{E}$ minimizes $|d_{E}|$ subject to (24).) It follows that

[TABLE]

where the two maps

[TABLE]

are defined as follows: $\epsilon$ is the projection modulo $2$ followed by the signature character on the permutation group $S_{3}$ (which is also the unique non-trivial multiplicative character on $\operatorname{GL}_{2}(\mathbb{Z}/2\mathbb{Z})$); $\chi_{E}$ is the determinant map, followed by the reduction modulo $|d_{E}|$, and then followed by the Kronecker symbol $\left(\frac{d_{E}}{\cdot}\right)$.

Proposition 20**.**

Let $E/\mathbb{Q}$ be a Serre curve and let $\Delta(E)$ be the discriminant of any Weierstrass model for $E$. Then:

1. (i)

$\operatorname{End}_{\overline{\mathbb{Q}}}(E)\simeq\mathbb{Z}$; 2. (ii)

$E(\mathbb{Q})_{\text{tors}}$* is trivial;* 3. (iii)

the integer $m_{E}$ introduced in Theorem 14 satisfies

[TABLE]

where $\Delta_{\text{sf}}(E)$ denotes the squarefree part of $\Delta(E)$; 4. (iv)

for any integer $m\geq 1$,

[TABLE]

Proof.

(i) We proceed by contradiction. Suppose that $\operatorname{End}_{\overline{\mathbb{Q}}}(E)\simeq O\not\simeq\mathbb{Z}$ and let $K$ be the field of fractions of $O$. Then there exists an element $a\in O\backslash\mathbb{Z}$ and there exists a rational prime $\ell$ such that the characteristic polynomial of the action of $a$ on $E[\ell]$ has two distinct roots modulo $\ell$. Indeed, letting $\Delta_{a}$ denote the discriminant of this characteristic polynomial, one may take any $\ell$ satisfying $\left(\frac{\Delta_{a}}{\ell}\right)=1$. Then $G_{K}$ preserves the two eigenspaces of $a$, which implies that, written relative to an eigenbasis, we have

[TABLE]

We thus have $\left|\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z}):\varphi_{E,\ell}(G_{\mathbb{Q}})\right|\geq\ell(\ell+1)>2$, contradicting (23).

(ii) We proceed by contradiction. Suppose that $E(\mathbb{Q})_{\text{tors}}$ is non-trivial. Then there exists a non-trivial point $P\in E(\mathbb{Q})[\ell]$ for some rational prime $\ell$. Extending the set $\{P\}$ to an ordered $\mathbb{Z}/\ell\mathbb{Z}$-basis $\{P,Q\}$ of $E[\ell]$ and writing linear transformations relative to this basis, we obtain

[TABLE]

We thus have that $\left|\operatorname{GL}_{2}(\mathbb{Z}/\ell\mathbb{Z}):\varphi_{E,\ell}(G_{\mathbb{Q}})\right|\geq(\ell+1)(\ell-1)>2$, contradicting (23).

(iii) Since the subgroup of $\operatorname{GL}_{2}(\hat{\mathbb{Z}})$ where $\chi_{E}$ and $\epsilon$ agree is already of index $2$, and since $E$ is a Serre curve, we must have equality in (25). The subgroup defined therein is determined by its image at level

[TABLE]

verifying (16).

(iv) Let $d\mid m_{E}$ and denote by $\operatorname{pr}_{m_{E},d}:\operatorname{GL}_{2}(\mathbb{Z}/m_{E}\mathbb{Z})\longrightarrow\operatorname{GL}_{2}(\mathbb{Z}/d\mathbb{Z})$ the canonical projection. Since $\left|\operatorname{GL}_{2}(\mathbb{Z}/m_{E}\mathbb{Z}):\varphi_{E,m_{E}}(G_{\mathbb{Q}})\right|=2$, it follows from the minimality of $m_{E}$ that

[TABLE]

By Corollary 15, this proves (27). ∎

2.5. Two-parameter families of elliptic curves

Lemma 21**.**

Let $A,B>2$ and consider the family ${\mathcal{C}}(A,B)$ of $\mathbb{Q}$-isomorphism classes of elliptic curves $E=E_{a,b}$ defined by $Y^{2}=X^{3}+aX+b$ with $a,b\in\mathbb{Z}$ and $|a|\leq A,|b|\leq B$. Then

[TABLE]

More precisely,

[TABLE]

[TABLE]

and, for each of the $j$-invariants of Corollary 9 with $j\neq 0,1728$,

[TABLE]

for any $\varepsilon>0$. The $\ll$-constant is absolute, while the $\ll_{\varepsilon}$-constant depends on $\varepsilon$.

Proof.

We recall that associated to an elliptic curve $E_{a,b}/\mathbb{Q}$, and in particular to a Weierstrass equation $Y^{2}=X^{3}+aX+b$, we have the $j$-invariant $j(a,b):=1728\frac{4a^{3}}{4a^{3}+27b^{2}},$ which encodes the $\overline{\mathbb{Q}}$-isomorphism class of $E_{a,b}$: two elliptic curves $E_{a,b}/\mathbb{Q}$, $E_{a^{\prime},b^{\prime}}/\mathbb{Q}$ are $\overline{\mathbb{Q}}$-isomorphic if and only if $j(a,b)=j(a^{\prime},b^{\prime})$; furthermore, $E_{a,b}/\mathbb{Q}$, $E_{a^{\prime},b^{\prime}}/\mathbb{Q}$ are $\mathbb{Q}$-isomorphic if and only if

[TABLE]

In view of the above and of Corollary 9, it suffices to estimate the cardinality of

[TABLE]

for each of the 13 occurring $j$-invariants. We will consider the cases $j=0$, $j=1728$, and $j\neq 0,1728$ separately.

Note that $j(a,b)=0$ is equivalent to $a=0$. Thus

[TABLE]

(see [HaWr, p.355 ] for a standard approach towards such asymptotics).

Similarly, note that $j(a,b)=1728$ is equivalent to $b=0$. Thus

[TABLE]

Now let us fix $j\neq 0,1728$. Setting

[TABLE]

[TABLE]

and

[TABLE]

and noting that

[TABLE]

we obtain

[TABLE]

To estimate $N_{j}(A)$, let $a_{1}$, $a_{2}$ be two distinct integers counted in the set giving rise to $N_{j}(A)$. There exist exactly two rational numbers $u=u(a_{1},a_{2})\in\mathbb{Q}^{\times}$ such that

[TABLE]

in particular such that $u^{2}a_{2}^{3}$ is a cube in $\mathbb{Q}^{\times}$. Consequently, there exists a unique $v=v(u)\in\mathbb{Q}^{\times}$ such that $u=v^{3}$, thus such that

[TABLE]

Recalling that $|a_{1}|\leq A$ and $a_{2}\in\mathbb{Z}\backslash\{0\}$, we deduce that

[TABLE]

Let us write $v=\frac{n}{d}$ for uniquely determined coprime $n,d\in\mathbb{Z}\backslash\{0\}$. On one hand, the above bound on $|v|$ gives

[TABLE]

On the other hand, (33) gives $d^{6}a_{1}^{3}=n^{6}a_{2}^{3}$, hence

[TABLE]

Consequently, the number of $v\in\mathbb{Q}^{\times}$ satisfying (32) is

[TABLE]

for any $\varepsilon>0$ (see [HaWr, Thm. 315 p. 343] for the standard upper bound $\tau(a_{2})\ll_{\varepsilon}a_{2}^{\varepsilon}$), which implies that

[TABLE]

Similarly,

[TABLE]

Thus, by (31),

[TABLE]

for any $\varepsilon>0$. Combining this with (29) and (30) completes the proof of the lemma. ∎

Theorem 22**.**

([Jo10, Thm. 4])

Let $A,B>2$ and consider the family ${\mathcal{C}}(A,B)$ of $\mathbb{Q}$-isomorphism classes of elliptic curves $E=E_{a,b}$ defined by $Y^{2}=X^{3}+aX+b$ with $a,b\in\mathbb{Z}$ and $|a|\leq A,|b|\leq B$. Then there exists a positive absolute constant $\gamma^{\prime}$ such that

[TABLE]

where the $\ll$-constant is absolute.

3. Constants for non-Serre curves: proof of Theorem 4

In this section we prove upper bounds for the constants appearing in Conjectures 1 - 3, as stated in Theorem 4. The key ingredients in the proof are Corollaries 13 and 15, Lemma 16, Proposition 17, and the following lemma about arithmetic functions:

Lemma 23**.**

Let $f,g:\mathbb{N}\backslash\{0\}\longrightarrow\mathbb{C}^{\times}$ be arithmetic functions satisfying:

1. (i)

$g$* is multiplicative;* 2. (ii)

$\displaystyle\sum_{m\geq 1}|g(m)|$* converges.*

Assume that $\exists\;M\in\mathbb{N}\backslash\{0\}$ and $\exists\;\kappa\in\mathbb{C}$ with $\operatorname{Re}\kappa>0$ such that:

1. (iii)

$\forall\;m_{1}\mid M^{\infty}$* and $\forall m_{2}$ with $\operatorname{gcd}(m_{2},M)=1$, we have $f(m_{1}m_{2})=f(m_{1})g(m_{2})$;* 2. (iv)

$\forall\;d\mid M^{\infty}$, $\forall\;\ell\mid M$ with $\ell\nmid d$, and $\forall\;\delta\in\mathbb{N}$, we have $f(\ell^{v_{\ell}(M)+\delta}d)=\ell^{-\delta\kappa}f(\ell^{v_{\ell}(M)}d)$.

Then

[TABLE]

Proof.

By (iii) we have the almost-product formula

[TABLE]

where, for each rational prime $\ell$, $g_{\ell}:=\displaystyle\sum_{r\geq 0}|g(\ell^{r})|=1+\sum_{r\geq 1}|g(\ell^{r})|$. Since

[TABLE]

it remains to bound $\displaystyle\sum_{m\mid M^{\infty}}|f(m)|$.

Let $M=\ell_{1}^{\alpha_{1}}\ldots\ell_{n}^{\alpha_{n}}$ be the prime factorization of $M$. For each subset $I\subseteq\{1,\ldots,n\}$, possibly empty, define

[TABLE]

and for each $m\mid M^{\infty}$, whose unique prime factorization we write as $m=\ell_{1}^{\beta_{1}}\ldots\ell_{n}^{\beta_{n}}$ with $\beta_{1}\geq 0,\ldots\beta_{n}\geq 0$, define

[TABLE]

Partitioning the integers $m\mid M^{\infty}$ according to the subsets $I_{m}\subseteq\{1,\ldots,n\}$ and using property (iv) of $f$, we obtain

[TABLE]

This completes the proof. ∎

Proof of part (i) of Theorem 4.

We begin by using Lemma 23 to verify that, for the arbitrary elliptic curve $E/\mathbb{Q}$ with complex multiplication by the order $O$ in the imaginary quadratic field $K$, we have

[TABLE]

By Corollary 13, for any integer $m\geq 1$, written uniquely as $m=m_{1}m_{2}$ with $m_{1}\mid m_{E}^{\infty}$ and $(m_{2},m_{E})=1$, we have

[TABLE]

where $\Phi_{O}$ denotes the Euler function on $O$, that is, $\Phi_{O}(n):=\left|(O/nO)^{\times}\right|$ for any positive integer $n$. Furthermore, by part (i) of Lemma 16, for any $\ell$, $d$ with $\ell\mid m_{E}$, $d\mid m_{E}$, $\ell\nmid d$, and for any $\delta\in\mathbb{N}$, we also have

[TABLE]

Defining

[TABLE]

we want to apply Lemma 23 with $M=m_{E}$ and $\kappa=2$.

Condition (i) follows from the Chinese Remainder Theorem. Indeed, if $m$ and $m^{\prime}$ are relatively prime positive integers, then the principal ideals $mO$ and $m^{\prime}O$ are co-maximal in the ring $O$, and so, by the Chinese Remainder Theorem, we have that

[TABLE]

from which it follows immediately that $g$ is multiplicative. Condition (ii) follows from the convergence of the series $\displaystyle\sum_{m\geq 1}\frac{1}{\phi(m)^{2}}$ since

[TABLE]

To justify the above inequality, note that

[TABLE]

and

[TABLE]

hence

[TABLE]

Conditions (iii) and (iv) follow from (37) - (38).

By Lemma 23, we obtain

[TABLE]

Having established (36), our goal is now to bound the sum $\displaystyle\sum_{m\geq 1}\frac{1}{[K(E[m]):K]}$ absolutely. To this end, let $j_{1},\ldots,j_{13}$ be the $j$-invariants of all elliptic curves over $\mathbb{Q}$ with complex multiplication, as per Corollary 9. For each $j_{i}$, choose an elliptic curve $E_{i}/\mathbb{Q}$ with $j(E_{i})=j_{i}$ and denote by $K_{i}$ its complex multiplication field. (Recalling Theorem 8, note that some of the fields $K_{1},\ldots,K_{13}$ occur with multiplicity since only nine of them are distinct.) Also note that, since (36) holds for any elliptic curve over $\mathbb{Q}$ with complex multiplication, it holds for each $E_{i}/\mathbb{Q}$. Thus we may define

[TABLE]

which is an absolute constant.

Now fix one of the fields $K_{i}$ and let $E_{i}^{\prime}/\mathbb{Q}$ be an elliptic curve with $j(E_{i}^{\prime})=j_{i}$. By Corollary 11, there exists a number field $L_{i}$ such that $[L_{i}:\mathbb{Q}]\leq 6$ and $E_{i}^{\prime}\simeq_{L_{i}}E_{i}$. Moreover, we claim that, for any integer $m\geq 1$,

[TABLE]

Indeed, we have

[TABLE]

and

[TABLE]

We deduce from (41) that for any elliptic curve $E^{\prime}/\mathbb{Q}$ with complex multiplication, say by $K^{\prime}$, we have

[TABLE]

In particular, this holds for the elliptic curve $E/\mathbb{Q}$, and so

[TABLE]

To relate $\displaystyle\sum_{m\geq 1}\frac{1}{[K(E[m]):K]}$ to the constants $C_{d_{2}}(E)$ and $C_{\tau(d_{1})}(E)$, note that

[TABLE]

Thus

[TABLE]

Using (42), we deduce that the constants $C_{d_{2}}(E)\leq C_{\tau(d_{1})}(E)$ are absolutely bounded by $6B_{\tau(d_{1})}$.

To bound

[TABLE]

we proceed much as above, as follows. Fix an arbitrary complex number $s=\sigma+it$ with $\sigma>0$ and define

[TABLE]

We want to apply Lemma 23 with $M=m_{E}$ and $\kappa_{s}=1+s$.

Condition (i) follows from the observation that a product of multiplicative functions is multiplicative. Condition (ii) follows from the calculation

[TABLE]

where $\chi_{O}$ is the character defined by

[TABLE]

and

[TABLE]

Conditions (iii) and (iv) follow, as before, from (37) - (38).

By Lemma 23, we obtain

[TABLE]

Furthermore, using

[TABLE]

we find that

[TABLE]

Note that this inequality holds for any elliptic curve $E/\mathbb{Q}$ with complex multiplication. In particular, if we let $j_{1},\ldots,j_{13}$ be the $j$-invariants of elliptic curves over $\mathbb{Q}$ with complex multiplication, as per Corollary 9, and if, as earlier, for each $1\leq i\leq 13$ we choose an elliptic curve $E_{i}/\mathbb{Q}$ with $j(E_{i})=j_{i}$ and $\operatorname{End}_{\overline{\mathbb{Q}}}(E_{i})\simeq O_{i}\not\simeq\mathbb{Z}$, denoting by $K_{i}$ the complex multiplication field of $E_{i}$ and applying (3), we obtain that

[TABLE]

Recalling (41) and arguing as in the first part of the proof, we obtain that

[TABLE]

which, as before, gives

[TABLE]

This completes the proof of part (i) of Theorem 4. $\square$

Proof of part (ii) of Theorem 4.

By Corollary 15, for any integer $m\geq 1$, written uniquely as $m=m_{1}m_{2}$ with $m_{1}\mid m_{E}^{\infty}$ and $(m_{2},m_{E})=1$, we have

[TABLE]

Furthermore, for any $\ell$, $d$, with $\ell\mid m_{E}$, $\ell\nmid d$, and for any $\delta\in\mathbb{N}$, by part (ii) of Lemma 16 we also have

[TABLE]

Defining $f(m):=\frac{1}{[\mathbb{Q}(E[m]):\mathbb{Q}]}$ and $g(m):=\frac{1}{|\operatorname{GL}_{2}(\mathbb{Z}/m\mathbb{Z})|}$, we apply Lemma 23 with $M=m_{E}$ and $\kappa=4$. Note that condition (i) follows immediately from the Chinese Remainder Theorem, condition (ii) follows immediately from

[TABLE]

and conditions (iii) and (iv) follow from (43) - (44). We obtain

[TABLE]

To analyze the last sum, recall that for any integer $m\geq 1$, denoting by $\zeta_{m}$ a primitive $m$-th root of unity, we have $\mathbb{Q}(\zeta_{m})\subseteq\mathbb{Q}(E[m])$; in particular, we have $\phi(m)\mid[\mathbb{Q}(E[m]):\mathbb{Q}]$. Then

[TABLE]

where, in the last line, we used the elementary inequality $1+t\leq\operatorname{exp}t$. To understand the last sum, we proceed in a standard way. We let $\ell_{i}$ denote the $i$-th prime and we recall that it satisfies the bound $\ell_{i}\leq i(\operatorname{log}i+\operatorname{log}\operatorname{log}i)$ (see [RoSc, Theorem 3, p. 69]). Then

[TABLE]

where, in the second line, we used Mertens’ Theorem $\displaystyle\sum_{\ell\leq n}\frac{1}{\ell}=\operatorname{log}\operatorname{log}n+\operatorname{O}(1)$, and, in the last line, we used the bound $\omega(n)\leq\operatorname{log}n$.

The desired bounds for $C_{d_{2}}(E)$, $C_{\tau(d_{1})}(E)$ are obtained by plugging the above estimate in (45) and by invoking Proposition 17.

To estimate $C_{d_{1}}(E):=C_{d_{1},\text{non-CM}}(E)$, defining $f(m):=\frac{\phi(m)}{[\mathbb{Q}(E[m]):\mathbb{Q}]}$ and $g(m):=\frac{\phi(m)}{|\operatorname{GL}_{2}(\mathbb{Z}/m\mathbb{Z})|}$, we apply Lemma 23 with $M=m_{E}$ and $\kappa=3$. Note that condition (i) follows immediately from the Chinese Remainder Theorem, condition (ii) follows immediately from

[TABLE]

and conditions (iii) and (iv) follow from (43) - (44). We obtain

[TABLE]

Recalling that $\phi(m)\mid[\mathbb{Q}(E[m]):\mathbb{Q}]$, we deduce that

[TABLE]

for any $\varepsilon>0$. The desired bound for $C_{d_{1}}(E)$ is obtained by plugging the above estimates in (46) and by invoking Proposition 17. $\square$

4. Constants for Serre curves: proof of Theorem 5

In this section we prove closed formulae relating the constants of Conjectures 1 - 3 to the idealized constants (10) - (12), as stated in Theorem 5. The key ingredients in the proof are Corollary 15 and the following lemma:

Lemma 24**.**

Let $f,g:\mathbb{N}\backslash\{0\}\longrightarrow\mathbb{C}^{\times}$ be arithmetic functions satisfying the following:

1. (i)

$g$* is multiplicative;* 2. (ii)

$\displaystyle\sum_{m\geq 1}|g(m)|$* converges.*

Assume that $\exists\;M\in\mathbb{N}\backslash\{0\}$, $\exists\;\alpha\in(0,\infty)$ and $\exists\;\kappa\in\mathbb{N}\backslash\{0,1\}$ such that:

1. (iii)

$\forall m\in\mathbb{N}\backslash\{0\}$, we have

[TABLE] 2. (iv)

$\forall\;m\mid M^{\infty}$, we have $g(mM)=m^{-\kappa}g(M).$

Then

[TABLE]

where, for any rational prime $\ell$, $g_{\ell}:=\displaystyle\sum_{r\geq 0}g(\ell^{r})$.

Proof.

This result can be found in [Ko, Lemma 3.12]; we give its proof for completeness:

[TABLE]

∎

Proof of Theorem 5.

The proposition follows from part (iv) of Proposition 16, Lemma 24 with $M=m_{E}$ and $f$, $g$, $\alpha$, $\kappa$ as described below, and from summing geometric series. Note that, by definition, $\displaystyle\prod_{\ell}g_{\ell}$ is the idealized constant $C_{d_{1}}$, $C_{\tau(d_{1})}$, respectively $C_{d_{2}}$.

[TABLE]

$\square$

5. Averaging the constants over a family: proof of Theorem 6

In this section we prove Theorem 6 using the results of Sections 2-4 and following the approach initiated in [Jo09]. For this, let $A,B>2$ and $n\in\mathbb{N}\backslash\{0\}$ be fixed and consider the moment

[TABLE]

where the pair $(C(E),C)$ is, respectively, $(C_{d_{1}}(E),C_{d_{1}})$, $(C_{\tau(d_{1})}(E),C_{\tau(d_{1}}))$, and $(C_{d_{2}}(E),C_{d_{2}})$, and where $C_{d_{1}}(E)$ is $C_{d_{1},\text{non-CM}}(E)$ if $E$ is without complex multiplication, and $C_{d_{1},\text{CM}}(E)$ if $E$ is with complex multiplication. Our strategy is to partition ${\mathcal{C}}(A,B)$ into the subset of elliptic curves with complex multiplication (CM curves), the subset of elliptic curves without complex multiplication and which are not Serre curves (non-CM $\&$ non-Serre curves), and the subset of Serre curves; we then estimate each emerging subsum via different techniques.

To handle the contribution from elliptic curves with complex multiplication, using part (i) of Theorem 4 and Lemma 21, we obtain

[TABLE]

To handle the contribution from elliptic curves without complex multiplication, which are not Serre curves, using part (ii) of Theorem 4, as well as Theorem 22, we obtain

[TABLE]

for $(C(E),C)$ equal to $(C_{\tau(d_{1})}(E),C_{\tau(d_{1}}))$ or $(C_{d_{2}}(E),C_{d_{2}})$, and

[TABLE]

for $(C(E),C)$ equal to $(C_{d_{1},\text{non-CM}}(E),C_{d_{1}})$.

To handle the contribution from Serre curves, using Theorem 5 and part (iii) of Proposition 20, we obtain

[TABLE]

Next, following the approach of [Jo09, Section 4.2], we choose a parameter $z=z(A,B)$ and partition the Serre curves $E$ in ${\mathcal{C}}(A,B)$ according to whether $|\Delta_{\text{sf}}(E)|\leq z$ or not. For the first subsum we use [Jo09, Lemma 22, p. 705], while for the second subsum we use trivial estimates:

[TABLE]

Upon choosing

[TABLE]

we deduce that

[TABLE]

Putting together (47), (48), (49), and (50), it suffices to show that if $A=A(x)$ and $B=B(x)$ are functions tending to infinity with $x$ and such that

[TABLE]

then there exists an $\varepsilon>0$ such that

[TABLE]

Assumption (51) implies that

[TABLE]

and so

[TABLE]

For any $\varepsilon<\varepsilon_{0}$, we therefore have

[TABLE]

Exponentiating finishes the argument.

Acknowledgments. This research started during the Arizona Winter School 2016: Analytic Methods in Arithmetic Geometry, organized at the University of Arizona, Tucson, USA, during March 12-16, 2016. We thank the conference organizers Alina Bucur, David Zureick-Brown, Bryden Cais, Mirela Ciperiani, and Romyar Sharifi for all their time and support, and we thank the National Science Foundation for sponsoring our participation in this conference.