2.1. Heights on weighted projective stacks

Given an $(n+1)$ -tuple of positive integers ${\textbf {w}}=(w_0,\dots ,w_n)$ , the weighted projective stack ${\mathcal P}({\textbf {w}})$ is the quotient stack

$$\begin{align*}{\mathcal P}({\textbf{w}})\stackrel{\mathrm{def}}{=}[({\mathbb A}^{n+1}-\{0\})/\mathbb{G}_m], \end{align*}$$

where the multiplicative group scheme, $\mathbb {G}_m$ , acts on the punctured affine space, ${\mathbb A}^{n+1}-\{0\}$ , as follows:

$$ \begin{align*} \mathbb{G}_m\times({\mathbb A}^{n+1}-\{0\}) &\to ({\mathbb A}^{n+1}-\{0\})\\ (\lambda,(x_0,\dots,x_n)) &\mapsto \lambda\ast_{\textbf{w}} (x_0,\dots,x_n)\stackrel{\mathrm{def}}{=}(\lambda^{w_0}x_0,\dots, \lambda^{w_n}x_n). \end{align*} $$

In the special case when ${\textbf {w}}=(1,\dots ,1)$ , this recovers the usual projective space ${\mathbb P}^n$ .

The point $[(a_0,\dots ,a_n)]\in {\mathcal P}(w_0,\dots ,w_n)$ has stabilizer $\mu _m$ , where $m=\gcd (w_i : a_i\neq 0)$ . When K is a field of characteristic zero, we have that $\mu _m$ is finite and reduced over K. It follows that, over fields of characteristic zero, ${\mathcal P}(w_0,\dots ,w_n)$ is a Deligne–Mumford stack.

For any field F, let ${\mathcal P}({\textbf {w}})(F)$ denote the set of isomorphism classes of the groupoid of F-points of ${\mathcal P}({\textbf {w}})$ . More concretely, ${\mathcal P}({\textbf {w}})(F)$ is in canonical bijection with the quotient

$$\begin{align*}(F^{n+1}-\{0\})/F^\times, \end{align*}$$

where $F^\times $ acts on $F^{n+1}-\{0\}$ by the weighted action

$$ \begin{align*} F^\times \times (F^{n+1}-\{0\}) &\to (F^{n+1}-\{0\})\\ (\lambda, (a_0,\dots,a_n)) &\mapsto (\lambda^{w_0}a_0,\dots,\lambda^{w_n}a_n). \end{align*} $$

Throughout this article, unless otherwise stated, we assume all weighted projective stacks ${\mathcal P}({\textbf {w}})$ are defined over a number field K.

For each finite place $v\in \text {Val}_0(K)$ , let ${\mathfrak p}_v$ be the corresponding prime ideal, and let $\pi _v$ be a uniformizer for the completion $K_v$ of K at v. For $x=(x_0,\dots ,x_n)\in K^{n+1}-\{0\}$ , set $|x_i|_{{\textbf {w}},v}\stackrel {\mathrm{def}}{=}|\pi _v|_v^{\lfloor v(x_i)/w_i\rfloor }$ , and set

$$\begin{align*}|x|_{{{\textbf{w}}},v}\stackrel{\mathrm{def}}{=}\begin{cases} \max_i\{|x_i|_{{\textbf{w}},v}\} & \text{ if } v\in \text{Val}_0(K)\\ \max_i\{|x_i|_v^{1/w_i}\} & \text{ if } v\in \text{Val}_\infty(K). \end{cases} \end{align*}$$

Definition 2.1.1 (Height).

The (exponential) height of a point $x=[x_0:\cdots :x_n]\in {\mathcal P}({\textbf {w}})(K)$ is defined as

$$\begin{align*}\text{Ht}_{\textbf{w}}(x)\stackrel{\mathrm{def}}{=}\prod_{v\in \text{Val}(K)} |(x_0,\dots,x_n)|_{{\textbf{w}},v}. \end{align*}$$

It is straightforward to check that this height function does not depend on the choice of representative of x.

Definition 2.1.2 (Scaling ideal).

Define the scaling ideal ${\mathfrak I}_{{\textbf {w}}}(x)$ of $x=(x_0,\dots ,x_n)\in K^{n+1}-\{0\}$ to be the fractional ideal

$$\begin{align*}{\mathfrak I}_{{\textbf{w}}}(x)\stackrel{\mathrm{def}}{=}\prod_{v\in \text{Val}_0(K)} {\mathfrak p}_v^{\min_i\{\lfloor v(x_i)/w_i\rfloor\}}. \end{align*}$$

The scaling ideal ${\mathfrak I}_{{\textbf {w}}}(x)$ can be characterized as the intersection of all fractional ideals ${\mathfrak a}$ of ${\mathcal O}_K$ such that $x\in {\mathfrak a}^{w_0}\times \cdots \times {\mathfrak a}^{w_n}\subseteq K^{n+1}$ . It has the property that

$$\begin{align*}{\mathfrak I}_{\textbf{w}}(x)^{-1}=\{a\in K : a^{w_i}x_i\in {\mathcal O}_K \text{ for all } i\}. \end{align*}$$

The height can be written in terms of the scaling ideal as follows:

$$\begin{align*}\text{Ht}_{\textbf{w}}([x_0:\dots:x_n])=\frac{1}{N_{K/{\mathbb Q}}({\mathfrak I}_{\textbf{w}}(x))}\prod_{v\in \text{Val}_\infty(K)} \max_i\{|x_i|_v^{1/w_i}\}. \end{align*}$$

It is straightforward to check that this agrees with the height from Definition 2.1.1.

The (compactified) moduli stack of elliptic curves, ${\mathcal X}_{\text {GL}_2({\mathbb Z})}$ , is isomorphic to the weighted projective stack ${\mathcal P}(4,6)$ . This isomorphism can be given explicitly as

$$ \begin{align*} {\mathcal X}_{\text{GL}_2({\mathbb Z})} &\xrightarrow{\sim} {\mathcal P}(4,6)\\ y^2=x^3+Ax+B & \mapsto [A:B]. \end{align*} $$

Under this isomorphism, the tautological bundle on ${\mathcal P}(4,6)$ corresponds to the Hodge bundle on ${\mathcal X}_{\text {GL}_2({\mathbb Z})}$ . The usual naive height of an elliptic curve E over K with a reduced integral short Weierstrass equation $y^2=x^3+Ax+B$ is defined as

$$\begin{align*}\text{Ht}(E)\stackrel{\mathrm{def}}{=}\prod_{v\in \text{Val}_\infty(K)} \max\{|A|_v^3,|B|_v^2\}. \end{align*}$$

One can show that this naive height is the same as the stacky height on ${\mathcal X}_{\text {GL}_2({\mathbb Z})}$ with respect to the twelfth power of the Hodge bundle [Reference Ellenberg, Satriano and Zureick-BrownESZB23, §3.3]. For any short Weierstrass equation $y^2=x^3+ax+b$ of E, the naive height is given by

$$\begin{align*}\text{Ht}(E)\stackrel{\mathrm{def}}{=} \text{Ht}_{(4,6)}([a:b])^{12}. \end{align*}$$

3.1. O-minimal geometry

We briefly cover the basics of o-minimal geometry which will be needed in this article. A general reference for this subsection is [Reference van den DriesvdD98, §1]. Let $m_L$ denote Lebesgue measure on ${\mathbb R}^n$ .

Definition 3.1.1 (Semi-algebraic set).

A (real) semi-algebraic subset of ${\mathbb R}^n$ is a finite union of sets of the form

$$\begin{align*}\{x\in {\mathbb R}^n: f_1(x)=\cdots=f_k(x)=0 \text{ and } g_1(x)>0,\dots,g_l(x)>0\}, \end{align*}$$

where $f_1,\dots ,f_k,g_1,\dots ,g_l\in {\mathbb R}[X_1,\dots ,X_n]$ .

Note that the intersection of definable sets is definable by property (i).

The class of semialgebraic sets is an example of an o-minimal structure (see, for example, [Reference van den DriesvdD98, §2]). The main structure we will use is ${\mathbb R}_{\exp }$ , which is defined to be the smallest structure in which the real exponential function, $\exp \colon {\mathbb R}\to {\mathbb R}$ , is definable. Observe that the function $\log (x)$ is definable in ${\mathbb R}_{\exp }$ since its graph,

$$\begin{align*}\Gamma(\log(x))=\{(x,\log(x)) : x\in {\mathbb R}_{>0}\}=\{(\exp(y), y) : y\in {\mathbb R}\}\subset {\mathbb R}^2, \end{align*}$$

is clearly definable in ${\mathbb R}_{\exp }$ .

From now on, we will call a subset of ${\mathbb R}^n$ definable if it is definable in some o-minimal structure.

3.2. Weighted geometry-of-numbers

The following proposition is a version of the Principle of Lipschitz, which gives estimates for the number of lattice points in a weighted homogeneous space.

Proposition 3.2.1. Let $R\subset {\mathbb R}^{n}$ be a bounded set definable in an o-minimal structure. Let $\Lambda \subset {\mathbb R}^n$ be a rank n lattice with successive minima $\lambda _1,\dots ,\lambda _n$ (with respect to the origin-centered unit ball in ${\mathbb R}^n$ ). Set

$$\begin{align*}R(B)=B\ast_{{\textbf{w}}}R=\{B\ast_{{\textbf{w}}}x : x\in R\}. \end{align*}$$

Then, for sufficiently largeFootnote ² B,

$$\begin{align*}\#(\Lambda\cap R(B))=\frac{m_L(R)}{\det \Lambda} B^{|{\textbf{w}}|} + O\,\left(\lambda_n\det(\Lambda)^{-1} B^{|{\textbf{w}}|-w_{\min}}\right), \end{align*}$$

where the implied constant depends only on R and ${\textbf {w}}$ .

Proof. This will follow from a general version of the Principle of Lipschitz due to Barroero and Widmer [Reference Barroero and WidmerBW14, Theorem 1.3]. Since $m_L(R(B))=B^{|{\textbf {w}}|} m_L(R)$ , [Reference Barroero and WidmerBW14, Theorem 1.3] gives the desired leading term. Let $V_j(R(B))$ denote the sum of the j-dimensional volumes of the orthogonal projections of $R(B)$ onto each j-dimensional coordinate subspace of ${\mathbb R}^n$ . The error term given in [Reference Barroero and WidmerBW14, Theorem 1.3] is

(1) $$ \begin{align} O\,\left(1+\sum_{j=1}^{n-1} \frac{V_j(R(B))}{\lambda_1\cdots \lambda_j}\right), \end{align} $$

where the implied constant depends only on R. In our case, one observes that $V_i(R(B))=O(V_{j}(R(B)))$ for all $i\leq j$ . Moreover, $V_{n-1}(R(B))=O(\sum _{i\leq n} B^{|{\textbf {w}}|-w_i})=O(B^{|{\textbf {w}}|-w_{\min }})$ , where the implied constant depends only on R and ${\textbf {w}}$ . By dimension considerations, we see that if $B^{|{\textbf {w}}|-w_{\min }}=O(V_i(R(B)))$ , then we must have $i\geq n-1$ . From these observations, it follows that

$$\begin{align*}O\,\left(1+\sum_{j=1}^{n-1} \frac{V_j(R(B))}{\lambda_1\cdots \lambda_j}\right)=O\,\left(\frac{V_{n-1}(R(B))}{\lambda_1\cdots\lambda_{n-1}}\right). \end{align*}$$

By Minkowski’s second theorem [Reference Minkowski and ApproximationenMin07] (for a more modern reference see, for example, [Reference CasselsCas59, p. 203]),

$$\begin{align*}\lambda_1\cdots\lambda_n\geq \frac{2^n}{n!\cdot m_L(\mathscr{B}_n)}\det(\Lambda), \end{align*}$$

where $\mathscr {B}_n$ is the unit ball in ${\mathbb R}^n$ . From this, we obtain the desired expression for the error term,

$$\begin{align*}O\,\left(\frac{V_{n-1}(R(B))}{\lambda_1\cdots\lambda_{n-1}}\right)=O\,\left(\frac{\lambda_n B^{|{\textbf{w}}|-w_{\min}}}{\det(\Lambda)}\right).\\[-44pt] \end{align*}$$

Definition 3.2.2 (Boxes).

A ( $\mathbf {{\mathbb Z}_p^n}$ )-box is a subset ${\mathcal B}_p\subset {\mathbb Z}_p^n$ for which there exist closed balls

$$\begin{align*}{\mathcal B}_{p,j}=\{x\in {\mathbb Z}_p : |x-a_j|_p\leq b_{p,j}\}\subset {\mathbb Z}_p, \end{align*}$$

where $a_j\in {\mathbb Z}_p$ and $b_{p,j}\in \{p^{-k}: k\in {\mathbb Z}_{\geq 0}\}$ , such that ${\mathcal B}_p$ equals the Cartesian product $\prod _{j=1}^n {\mathcal B}_{p,j}$ . Let S be a finite set of primes. An S-box is a subset ${\mathcal B}\subset \prod _{p\in S} {\mathbb Z}_p^n$ for which there exist ${\mathbb Z}_p^n$ -boxes ${\mathcal B}_p$ such that

$$\begin{align*}{\mathcal B}=\prod_{p\in S} {\mathcal B}_{p}=\prod_{p\in S}\prod_{j=1}^n {\mathcal B}_{p,j}. \end{align*}$$

Lemma 3.2.3 (Box Lemma).

Let $\Omega _\infty \subset {\mathbb R}^n$ be a bounded subset definable in an o-minimal structure. Let S be a finite set of primes and ${\mathcal B}=\prod _{p\in S} {\mathcal B}_p$ an S-box. Then, for each $B\in {\mathbb R}_{>0}$ , we have

$$ \begin{align*} &\#\left\{x\in {\mathbb Z}^n \cap \left(B\ast_{\textbf{w}} \Omega_\infty\right): x\in \prod_{p\in S} \mathcal{B}_p \right\}\\ &\hspace{1cm}=\left(m_L(\Omega_\infty)\prod_{p\in S} m_p(\mathcal{B}_p)\right)B^{|{\textbf{w}}|} + O\,\left(\max_j\left\{\prod_{p\in S} b_{p,j}^{-1}\right\}\left(\prod_{p \in S} m_p(\mathcal{B}_p)\right)B^{|{\textbf{w}}|-w_{\min}} \right), \end{align*} $$

where the implied constant depends only on $\Omega _\infty $ and ${\textbf {w}}$ .

Proof. We have that $\mathcal {B}_p=\prod _{j=1}^n \mathcal {B}_{p,j}$ with $\mathcal {B}_{p,j}=\{x\in {\mathbb Z}_p : |x-a_j|_p\leq b_{p,j}\}$ , where $a_j\in {\mathbb Z}_p$ and $b_{p,j}\in \{p^{-k} : k\in {\mathbb Z}_{\geq 0}\}$ . Observe that for each $j\in \{1,\dots ,n\}$ , the set ${\mathbb Z}\cap \mathcal {B}_{p,j}$ is a translate of the sublattice $\langle b_{p,j}^{-1}\rangle \subseteq {\mathbb Z}$ . By the Chinese Remainder Theorem, the set ${\mathbb Z}\cap \prod _{p\in S} \mathcal {B}_{p,j}$ is a translate of the sublattice $\left \langle \prod _{p\in S} b_{p,j}^{-1}\right \rangle \subseteq {\mathbb Z}$ . Taking a Cartesian product, we find that the set

$$\begin{align*}\prod_{j=1}^n\left( {\mathbb Z}\cap \prod_{p\in S} \mathcal{B}_{p,j}\right)={\mathbb Z}^n\cap \prod_{p\in S} \mathcal{B}_p \end{align*}$$

is a translate of a sub-lattice of ${\mathbb Z}^n$ whose successive minima are $\prod _{p\in S} b_{p,j}^{-1}$ . In particular, the n-th successive minimum of the sublattice is

$$\begin{align*}\max_j\left\{\prod_{p\in S} b_{p,j}^{-1}\right\}, \end{align*}$$

and the determinant of the sublattice is

$$\begin{align*}\prod_{j=1}^n\prod_{p\in S} b_{p,j}^{-1}=\prod_{p\in S}m_p(\mathcal{B}_p)^{-1}. \end{align*}$$

The lemma then follows from Proposition 3.2.1.

Let $\Lambda $ be a free ${\mathbb Z}$ -module of finite rank n. Set $\Lambda _\infty =\Lambda \otimes _{\mathbb Z} {\mathbb R}$ and $\Lambda _p=\Lambda \otimes _{\mathbb Z} {\mathbb Z}_p$ . Equip $\Lambda _\infty $ and $\Lambda _p$ with Haar measures $m_\infty $ and $m_p$ , respectively, normalized so that $m_p(\Lambda _p)=1$ for all but finitely many primes p. As $\Lambda _p\cong {\mathbb Z}_p^n$ , we define a ( $\mathbf {\Lambda _p)}$ -box to be a subset of $\Lambda _p$ isomorphic to a $({\mathbb Z}_p^n)$ -box.

Lemma 3.2.4. Let $\Lambda $ be a free ${\mathbb Z}$ -module of finite rank n, let $\Omega _\infty \subset \Lambda _\infty $ be a bounded definable subset, and, for each prime p in a finite subset S, let $\Omega _p=\prod _j \{x\in {\mathbb Z}_p : |x-a_{p,j}|_p\leq \omega _{p,j}\}\subset \Lambda _p$ be a $(\Lambda _p)$ -box. Let ${\textbf {w}}=(w_1,\dots ,w_n)$ be an n-tuple of positive integers. Then,

$$ \begin{align*} &\#\{x\in \Lambda\cap \left(B \ast_{\textbf{w}} \Omega_\infty\right): x\in \Omega_p \text{ for all primes } p\in S\}\\ &\hspace{1cm}= \left(\frac{m_\infty(\Omega_\infty)}{m_\infty(\Lambda_\infty/\Lambda)}\prod_{p\in S} \frac{m_p(\Omega_p)}{m_p(\Lambda_p)}\right)B^{|{\textbf{w}}|} + O\,\left(\max_j\left\{\prod_{p\in S} \omega_{p,j}^{-1}\right\}\left(\prod_{p\in S} \frac{m_p(\Omega_p)}{m_p(\Lambda_p)}\right)B^{|{\textbf{w}}|-w_{\min}}\right), \end{align*} $$

where the implied constant depends only on $\Lambda $ , $\Omega _\infty $ , and ${\textbf {w}}$ .

If K is a number field of degree d over ${\mathbb Q}$ with discriminant $\Delta _K$ , then its ring of integers ${\mathcal O}_K$ may naturally be viewed as a rank d lattice in $K_\infty \stackrel {\mathrm{def}}{=}{\mathcal O}_K\otimes _{{\mathbb Z}} {\mathbb R}=\prod _{v|\infty } K_v$ . This lattice has covolume $|\Delta _K|^{1/2}$ with respect to the usual Haar measure $m_\infty $ on $K_\infty $ (which differs from Lebesgue measure on $K_\infty \cong {\mathbb R}^{r_1+2r_2}$ by a factor of $2^{r_2}$ ) [Reference NeukirchNeu99, Chapter I Proposition 5.2]. More generally, any nonzero integral ideal ${\mathfrak a}\subseteq {\mathcal O}_K$ may be viewed as a lattice in $K_\infty $ with covolume $N_{K/{\mathbb Q}}({\mathfrak a})|\Delta _K|^{1/2}$ . For an n-tuple of positive integers ${\textbf {w}}=(w_1,\dots ,w_n)$ , define the lattice

$$\begin{align*}{\mathfrak a}^{\textbf{w}}\stackrel{\mathrm{def}}{=}{\mathfrak a}^{w_1}\times\cdots \times {\mathfrak a}^{w_n}\subset K_\infty^{n}, \end{align*}$$

where we view each ${\mathfrak a}^{w_i}$ as a subset of $K_\infty $ . This lattice has covolume $N_{K/{\mathbb Q}}({\mathfrak a})^{|{\textbf {w}}|}|\Delta _K|^{n/2}$ . For example, in the case that ${\mathfrak a}={\mathcal O}_K$ and $w_i=1$ for all i, one has the lattice ${\mathfrak a}^{\textbf {w}}={\mathcal O}_K^{n}$ of covolume $|\Delta _K|^{n/2}$ .

For any rational prime p, we have ${\mathcal O}_K\otimes _{{\mathbb Z}} {\mathbb Z}_p=\prod _{{\mathfrak p}|p} {\mathcal O}_{K,{\mathfrak p}}$ . Equip each ${\mathcal O}_{K,{\mathfrak p}}$ with the Haar measure $m_{\mathfrak p}$ , normalized so that $m_{\mathfrak p}({\mathcal O}_{K,{\mathfrak p}})=1$ . This measure induces a measure on ${\mathcal O}_{K,{\mathfrak p}}^n$ , which we will also denote by $m_{\mathfrak p}$ . Similarly, the measure $m_\infty $ on $K_\infty $ induces a measure on $K_\infty ^n$ , which we will denote by $m_\infty $ .

For $v\in \text {Val}_0(K)$ a finite place, let $q_v$ denote the size of the residue field ${\mathcal O}_{K}/{\mathfrak p}_v$ . Let ${\mathfrak a}_{\mathfrak p}$ denote the image of ${\mathfrak a}$ in ${\mathcal O}_{K,{\mathfrak p}}$ .

We now apply Lemma 3.2.4 to the rank $nd$ lattice ${\mathfrak a}^{\textbf {w}}$ , and with the $(nd)$ -tuple of weights $\tilde {{\textbf {w}}}=(w_1,\dots ,w_1,w_2,\dots ,w_2,\dots ,w_n,\dots ,w_n)$ , obtained by repeating each of the weights in the n-tuple ${\textbf {w}}$ exactly d times. Observing that $|\tilde {{\textbf {w}}}|=d|{\textbf {w}}|$ and $\tilde {w}_{\min }=w_{\min }$ , we obtain the following proposition:

Proposition 3.2.7. Let $\Omega _\infty \subset K_\infty ^{n}$ be a bounded definable subset. Let S be a finite set of prime ideals of ${\mathcal O}_K$ . For each ${\mathfrak p}\in S$ , let $\Omega _{\mathfrak p}=\prod _j \{x\in {\mathcal O}_{K,{\mathfrak p}} : |x-a_{{\mathfrak p},j}|_{\mathfrak p}\leq \omega _{{\mathfrak p},j}\}\subset {\mathcal O}_{K,{\mathfrak p}}^n$ be an irreducible local condition. Then,

$$ \begin{align*} \#\{x\in {\mathfrak a}^{\textbf{w}}\cap \left(B \ast_{\textbf{w}} \Omega_\infty\right) : x\in \Omega_{\mathfrak p} \text{ for all primes } {\mathfrak p}\in S\}=\kappa B^{d|{\textbf{w}}|}+ O\,\left(\epsilon B^{d|{\textbf{w}}|-w_{\min}} \right), \end{align*} $$

where

$$ \begin{align*} \kappa&=\frac{m_\infty(\Omega_\infty)}{N_{K/{\mathbb Q}}({\mathfrak a})^{|{\textbf{w}}|}|\Delta_K|^{n/2}} \left(\prod_{{\mathfrak p}\in S} \frac{m_{\mathfrak p}(\Omega_{\mathfrak p}\cap {\mathfrak a}_{\mathfrak p}^{\textbf{w}})}{m_{\mathfrak p}({\mathfrak a}_{\mathfrak p}^{\textbf{w}})}\right),\\ \epsilon&=\max_j\left\{\prod_{{\mathfrak p}\in S} \omega_{{\mathfrak p},j}^{-1}\right\}\left(\prod_{{\mathfrak p}\in S} \frac{m_{\mathfrak p}(\Omega_{\mathfrak p}\cap {\mathfrak a}_{\mathfrak p}^{\textbf{w}})}{m_{\mathfrak p}({\mathfrak a}_{\mathfrak p}^{\textbf{w}})}\right), \end{align*} $$

and where the implied constant depends only on K, $\Omega _\infty $ and ${\textbf {w}}$ .

5.1. Counting elliptic curves with a prescribed local condition

An application of Theorem 4.0.5 gives the following result:

Corollary 5.1.1. Let ${\mathcal E}_K(B)$ denote the set of isomorphism classes of elliptic curves over K with height less than B. Then,

$$\begin{align*}\#{\mathcal E}_K(B)=\kappa B^{5/6}+O(B^{5/6-1/3d}), \end{align*}$$

where

$$\begin{align*}\kappa=\frac{h_K (2^{r_1+r_2}\pi^{r_2})^{2} R_K 10^{r_1+r_2-1}\gcd(2,\varpi_K)}{\varpi_K |\Delta_K| \zeta_K(10)}. \end{align*}$$

We now use Theorem 4.0.5 to count elliptic curves with a prescribed local condition.

Theorem 1.1.2.

Let K be a degree d number field, and let ${\mathfrak p}\subset {\mathcal O}_K$ be a prime ideal of norm q such that $2\nmid q$ and $3\nmid q$ . Let ${\mathscr L}$ be one of the local conditions listed in Table 1. Then the number of elliptic curves over K with naive height less than B and which satisfy the local condition ${\mathscr L}$ at ${\mathfrak p}$ is

$$\begin{align*}\kappa\kappa_{\mathscr L} B^{5/6}+O\,\left(\epsilon_{\mathscr L} B^{\frac{5}{6}-\frac{1}{3d}}\right), \end{align*}$$

where

$$\begin{align*}\kappa=\frac{h_K \left(2^{r_1+r_2}\pi^{r_2}\right)^{2}R_K10^{r_1+r_2-1}\gcd(2,\varpi_K)}{\varpi_{K} |\Delta_K|\zeta_K(10)}, \end{align*}$$

and where $\kappa _{\mathscr L}$ and $\epsilon _{\mathscr L}$ are as in Table 1.

Proof. Good reduction case. Note that the number of elliptic curves over ${\mathbb F}_q$ with good reduction is

$$\begin{align*}\#\{(a,b)\in {\mathbb F}_q\times{\mathbb F}_q : 4a^3+27b^2\not\equiv 0 \quad\pmod{q}\}=q^2-q. \end{align*}$$

For each pair $(a,b)$ in the above set, fix a lift in ${\mathcal O}_{K,v}^2$ , which, by an abuse of notation, we will also denote $(a,b)$ . Then, consider the irreducible local condition

$$\begin{align*}\Omega_{{\mathfrak p},0}^{\mathrm{aff}}(a,b)\stackrel{\mathrm{def}}{=}\left\{(x_0,x_1)\in {\mathcal O}_{K,v}^{2} : |x_0-a|_v\leq \frac{1}{q},\ |x_1-b|_v\leq \frac{1}{q}\right\}. \end{align*}$$

We find the ${\mathfrak p}$ -adic measure

$$\begin{align*}m_{\mathfrak p}(\Omega_{{\mathfrak p},0}^{\mathrm{aff}}(a,b))=\frac{1}{q^2}. \end{align*}$$

For $t\in {\mathbb Z}_{\geq 0}$ , the sets

$$\begin{align*}\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(a,b)=\pi_v^t\ast_{(4,6)}\Omega_{{\mathfrak p},0}^{\mathrm{aff}}(a,b)=\left\{(x_0,x_1)\in {\mathcal O}_{K,v}^{2} : |x_0-q^{4t}a|_v\leq \frac{1}{q^{4t+1}},\ |x_1-q^{6t}b|_v\leq \frac{1}{q^{6t+1}}\right\} \end{align*}$$

each have ${\mathfrak p}$ -adic measure

$$\begin{align*}m_{\mathfrak p}(\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(a,b))=\frac{1}{q^{10t+2}}. \end{align*}$$

Applying Theorem 4.0.5 to each $\Omega _{{\mathfrak p},0}^{\mathrm{aff}}(a,b)$ , and summing over all the $q^2-q$ possible $(a,b)$ pairs, it follows that the number of elliptic curves of bounded height over K with good reduction at ${\mathfrak p}$ is

(5) $$ \begin{align} \kappa\kappa_{\mathscr L} B^{5/6}+O\,\left(\epsilon B^{\frac{5}{6}-\frac{1}{3d}}\right), \end{align} $$

where

$$\begin{align*}\kappa_{\mathscr L}=\sum_{(a,b)}\sum_{t=0}^\infty m_{{\mathfrak p}}(\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(a,b))=(q^2-q)\sum_{t=0}^\infty \frac{1}{q^{10t+2}}=(q^2-q)\frac{1}{q^2}\frac{q^{10}}{q^{10}-1}= \frac{q-1}{q}\frac{q^{10}}{q^{10}-1}, \end{align*}$$

and

$$\begin{align*}\epsilon=\sum_{(a,b)}\sum_{t=0}^\infty\max\{q^{4t+1},q^{6t+1}\}m_{{\mathfrak p}}(\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(a,b)) =(q^2-q)\sum_{t=0}^\infty \frac{q^{6t+1}}{q^{10t+2}} =\frac{q^5-q^4}{q^4-1}. \end{align*}$$

As the error term can be rewritten as $O(q B^{\frac {5}{6}-\frac {1}{3d}})$ , we may replace the coefficient $\epsilon $ in the asymptotic (5) with $\epsilon _{\mathscr L}=q$ .

Kodaira type III* case. By Tate’s algorithm [Reference TateTat75], an elliptic curve over $K_{\mathfrak p}$ given in short Weierstrass form, $E:y^2=x^3+ax+b$ , has type III* reduction if and only if $v_{\mathfrak p}(a)=3$ and $v_{\mathfrak p}(b)\geq 5$ . Observe that

$$\begin{align*}\#\{(a,b)\in {\mathcal O}_K/{\mathfrak p}^5\times {\mathcal O}_K/{\mathfrak p}^5 : v_{\mathfrak p}(a)=3,\ v_{\mathfrak p}(b)\geq 5\}=q(q-1). \end{align*}$$

For each pair $(a,b)$ in the above set, fix a lift in ${\mathcal O}_{K,v}^2$ , which, by an abuse of notation, we will also denote $(a,b)$ . Then, consider the irreducible local condition

$$\begin{align*}\Omega_{{\mathfrak p},0}^{\mathrm{aff}}(a,b)\stackrel{\mathrm{def}}{=} \left\{(x_0,x_1)\in {\mathcal O}_{K,v}^{2} : |x_0-a|_v\leq \frac{1}{q^5},\ |x_1-b|_v\leq \frac{1}{q^5}\right\}. \end{align*}$$

We find that the ${\mathfrak p}$ -adic measure of this set is

$$\begin{align*}m_{\mathfrak p}(\Omega_{{\mathfrak p},0}^{\mathrm{aff}}(a,b))=\frac{1}{q^{10}}. \end{align*}$$

The sets

$$\begin{align*}\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(a,b)= \pi_v^t\ast_{(4,6)}\Omega_{{\mathfrak p},0}^{\mathrm{aff}}(a,b)=\left\{(x_0,x_1)\in {\mathcal O}_{K,v}^{2} : |x_0-q^{4t}a|_v\leq \frac{1}{q^{4t+5}},\ |x_1-q^{6t}b|_v\leq \frac{1}{q^{6t+5}}\right\} \end{align*}$$

each have ${\mathfrak p}$ -adic measure

$$\begin{align*}m_{\mathfrak p}(\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(a,b))=\frac{1}{q^{10t+10}}. \end{align*}$$

Applying Theorem 4.0.5 to each $\Omega _{{\mathfrak p},0}^{\mathrm{aff}}(a,b)$ , and summing over all pairs $(a,b)$ , yields the following asymptotic for the number of elliptic curves of bounded height over K with reduction of Kodaira type III* at ${\mathfrak p}$ :

(6) $$ \begin{align} \kappa\kappa_{\mathscr L} B^{5/6}+O\,\left(\epsilon B^{\frac{5}{6}-\frac{1}{3d}}\right), \end{align} $$

where

$$\begin{align*}\kappa_{\mathscr L} =\sum_{(a,b)}\sum_{t=0}^\infty m_{{\mathfrak p}}(\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(a,b))=q(q-1)\sum_{t=0}^\infty \frac{1}{q^{10t+10}}=(q^2-q)\frac{1}{q^{10}}\frac{q^{10}}{q^{10}-1}=\frac{q-1}{q^9}\frac{q^{10}}{q^{10}-1}, \end{align*}$$

and

$$\begin{align*}\epsilon=\sum_{(a,b)}\sum_{t=0}^\infty\max\{q^{4t+5},q^{6t+5}\}m_{{\mathfrak p}}(\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(a,b)) =(q^2-q)\sum_{t=0}^\infty \frac{q^{6t+5}}{q^{10t+10}} =\frac{q-1}{q^{4}-1}. \end{align*}$$

As the error term can be rewritten as $O(q^{-3}B^{\frac {5}{6}-\frac {1}{3d}})$ , we may replace the coefficient $\epsilon $ in the asymptotic (6) with $\epsilon _{\mathscr L}=1/q^{3}$ .

The other cases can be proven similarly to those given above.

We now turn to estimating the number of elliptic curves with a prescribed trace of Frobenius. For $b,c\in {\mathbb F}_q$ , let $E_{b,c}$ denote the short Weierstrass model $y^2=x^3+bx+c$ , with disciminant $\Delta (b,c)=-16(4b^3+27c^2)$ and trace of Frobenius $a_q(E_{b,c})$ . We study the counting function

(7) $$ \begin{align} H(a,q)\stackrel{\mathrm{def}}{=}\#\{(b,c)\in {\mathbb F}_q\times {\mathbb F}_q : \Delta(b,c)\neq 0,\ a_q(E_{b,c})=a\}. \end{align} $$

This function is closely related to Hurwitz-Kronecker class numbers, which we now recall. Let K be an imaginary quadratic field with ring of integers ${\mathcal O}_K$ , and let ${\mathcal O}$ be an order in K with class number $h({\mathcal O})$ . The Hurwitz-Kronecker class number of ${\mathcal O}$ , denoted $H(\text {disc}({\mathcal O}))$ , is defined as

(8) $$ \begin{align} H(\text{disc}({\mathcal O}))\stackrel{\mathrm{def}}{=}\sum_{{\mathcal O}\subset {\mathcal O}' \subset {\mathcal O}_K} \frac{h({\mathcal O}')}{\# {\mathcal O}^{\prime\times}}. \end{align} $$

This definition of the Hurwitz-Kronecker class number agrees with the definition given in [Reference LenstraLen87] but is half as large as it is sometimes defined (such as in [Reference CoxCox89]). Let $q=p^n$ be a power of a prime $p>3$ . Then, a beautiful result, following largely from work of Deuring [Reference DeuringDeu41], expresses the number of elliptic curves over a finite field ${\mathbb F}_q$ with a prescribed trace of Frobenius in terms of Hurwitz-Kronecker class numbers:

(9) $$ \begin{align} \begin{aligned} H(a,q)=\begin{cases} (q-1) H(a^2-4q) & \text{ if } |a|<2\sqrt{q} \text{ and } p\nmid a\\ (q-1)H(-4p) & \text{ if } n \text{ is odd and } a=0\\ \frac{q-1}{12}\left(p+6-4\genfrac{(}{)}{}{}{-3}{p}-3\genfrac{(}{)}{}{}{-4}{p}\right) & \text{ if } n \text{ is even and } a^2=4q\\ (q-1)\left(1-\genfrac{(}{)}{}{}{-3}{p}\right) & \text{ if } n \text{ is even and } a^2=q\\ (q-1)\left(1-\genfrac{(}{)}{}{}{-4}{p}\right) & \text{ if } n \text{ is even and } a=0\\ 0 & \text{ otherwise.} \end{cases} \end{aligned} \end{align} $$

This formula can be derived from [Reference SchoofSch87, Theorem 4.6], which counts the number of isomorphism classes of elliptic curves over ${\mathbb F}_q$ with a prescribed trace of Frobenius. From this, one can deduce formula (9) by noting that the isomorphism class of each $E_{b,c}$ contains exactly $q-1$ elliptic curves. To see that each isomorphism class contains exactly $q-1$ elliptic curves, note that $E_{b,c}\cong E_{d,e}$ if and only if $(d,e)=(\lambda ^4b, \lambda ^6c)$ for some $\lambda \in {\mathbb F}_q^\times $ , and since $p>3$ , the pairs $(\lambda ^4b, \lambda ^6c)$ will be distinct for distinct $\lambda \in {\mathbb F}_q^\times $ . (See [Reference CoxCox89, Theorem 14.18] for a proof of formula (9) in the case that $q=p$ is a prime.)

Theorem 1.1.3.

Let K be a degree d number field, and let ${\mathfrak p}$ be a prime ideal of ${\mathcal O}_K$ of degree n above a rational prime $p>3$ . Set $q=p^n$ . Let $a\in {\mathbb Z}$ be an integer satisfying $|a|\leq 2\sqrt {q}$ . Then, the number of elliptic curves over K with naive height less than B, good reduction at ${\mathfrak p}$ , and which have trace of Frobenius a at ${\mathfrak p}$ is

$$\begin{align*}\kappa\kappa_{n,a} B^{5/6}+O\,\left(\epsilon_{n,a} B^{\frac{5}{6}-\frac{1}{3d}}\right), \end{align*}$$

where

$$\begin{align*}\kappa=\frac{h_K \left(2^{r_1+r_2}\pi^{r_2}\right)^{2}R_K10^{r_1+r_2-1}\gcd(2,\varpi_K)}{\varpi_{K} |\Delta_K|\zeta_K(10)}, \end{align*}$$

and where $\kappa _{n,a}$ and $\epsilon _{n,a}$ are as in Table 2.

Proof. For each of the $H(a,q)$ pairs $(b,c)$ in the set

$$\begin{align*}\{(b,c)\in {\mathbb F}_q\times {\mathbb F}_q : \Delta(b,c)\neq 0,\ a_q(E_{b,c})=a\}, \end{align*}$$

fix a lift in ${\mathcal O}_{K,v}^2$ , which, by an abuse of notation, we will also denote $(b,c)$ . Then consider the irreducible local condition

$$\begin{align*}\Omega_{{\mathfrak p},0}^{\mathrm{aff}}(b,c)\stackrel{\mathrm{def}}{=}\left\{(x_0,x_1)\in {\mathcal O}_{K,v}^{2} : |x_0-b|_v\leq \frac{1}{q},\ |x_1-c|_v\leq \frac{1}{q}\right\}. \end{align*}$$

We find that the ${\mathfrak p}$ -adic measure of each of these local conditions is

$$\begin{align*}m_{\mathfrak p}(\Omega_{{\mathfrak p},0}^{\mathrm{aff}}(b,c))=\frac{1}{q^2}. \end{align*}$$

More generally, for $t\in {\mathbb Z}_{\geq 0}$ , the sets

$$\begin{align*}\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(b,c)=\pi_v^t\ast_{(4,6)}\Omega_{{\mathfrak p},0}^{\mathrm{aff}}(b,c)=\left\{(x_0,x_1)\in {\mathcal O}_{K,v}^{2} : |x_0-q^{4t}b|_v\leq \frac{1}{q^{4t+1}},\ |x_1-q^{6t}c|_v\leq \frac{1}{q^{6t+1}}\right\} \end{align*}$$

each have ${\mathfrak p}$ -adic measure

$$\begin{align*}m_{\mathfrak p}(\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(b,c))=\frac{1}{q^{10t+2}}. \end{align*}$$

Applying Theorem 4.0.5 to each $\Omega _{{\mathfrak p},0}^{\mathrm{aff}}(b,c)$ , and summing over all the $H(a,q)$ possible $(b,c)$ pairs, it follows that the number of elliptic curves of bounded height over K with good reduction at ${\mathfrak p}$ and trace of Frobenius a at ${\mathfrak p}$ is

(10) $$ \begin{align} \kappa\kappa' B^{5/6}+O\,\left(\epsilon B^{\frac{5}{6}-\frac{1}{3d}}\right), \end{align} $$

where

$$\begin{align*}\kappa'=\sum_{(b,c)}\sum_{t=0}^\infty m_{{\mathfrak p}}(\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(b,c))=H(a,q)\sum_{t=0}^\infty \frac{1}{q^{10t+2}}=H(a,q)\frac{1}{q^2}\frac{q^{10}}{q^{10}-1}, \end{align*}$$

and

$$\begin{align*}\epsilon=\sum_{(b,c)}\sum_{t=0}^\infty\max\{q^{4t+1},q^{6t+1}\}m_{{\mathfrak p}}(\Omega_{{\mathfrak p},t}^{\mathrm{aff}}(b,c)) =H(a,q)\sum_{t=0}^\infty \frac{q^{6t+1}}{q^{10t+2}} =H(a,q)\frac{q^3}{q^4-1}. \end{align*}$$

From this, the desired asymptotics can now be derived using (9).

For example, in the case that a is an integer satisfying $|a|< 2\sqrt {q}$ and $p\nmid a$ , we have that $H(a,q)=(q-1)H(a^2-4q)$ . Thus,

$$\begin{align*}\kappa'=\frac{H(a,q)}{q^2}\frac{q^{10}}{q^{10}-1}=\frac{(q-1)H(a^2-4q)}{q^2}\frac{q^{10}}{q^{10}-1}=\kappa_{n,a} \end{align*}$$

and

$$\begin{align*}\epsilon=(q-1)H(a^2-4q)\frac{q^3}{q^4-1}. \end{align*}$$

As the error term can be rewritten as $O\left (H(a^2-4q)B^{\frac {5}{6}-\frac {1}{3d}}\right )$ , we may replace the coefficient $\epsilon $ in the asymptotic (10) with $\epsilon _{n,a}=H(a^2-4q)$ .

6.1. L-functions of elliptic curves

In this subsection, we recall basic facts about L-functions of elliptic curves. Our exposition mostly follows [Reference MillerMil02, Appendix A].

Let E be an elliptic curve of discriminant $\Delta _E$ over a number field K. For each finite place $v\in \text {Val}_0(K)$ , let $E_v$ denote the reduction of E at v, let $q_v$ be the order of the residue field $k_v$ of K at v, let $N_v=\# E_v({\mathbb F}_{q_v})$ , and let $a_v=q_v+1-N_v$ . If E has bad reduction at v, set

$$\begin{align*}b_v\stackrel{\mathrm{def}}{=} \begin{cases} 1 & \text{ if }E\text{ has split multiplicative reduction at }v,\\ -1 & \text{ if }E\text{ has nonsplit multiplicative reduction at }v,\\ 0 & \text{ if }E\text{ has additive reduction at }v. \end{cases} \end{align*}$$

For each $v\in \text {Val}_0(K)$ , define a polynomial $L_v(E/K,T)$ by

$$\begin{align*}L_v(E/K,T)\stackrel{\mathrm{def}}{=}\begin{cases} 1-a_v T+ q_v T^2 & \text{ if }E\text{ has good reduction at }v,\\ 1-b_vT & \text{ if }E\text{ has bad reduction at }v. \end{cases} \end{align*}$$

The Hasse–Weil $\mathbf {L}$ -function $L(E/K,s)$ of E over K is defined as the Euler product

$$\begin{align*}L(E/K,s)\stackrel{\mathrm{def}}{=} \prod_{v\in \text{Val}_0(K)} L_v(E/K, q_v^{-s})^{-1}. \end{align*}$$

The logarithmic derivative of $L(E/K,s)$ is

$$\begin{align*}\frac{L'}{L}(E/K,s)=-\sum_{\substack{v\in \text{Val}_0(K)\\ v(\Delta_E)>0}} \frac{d}{ds}\left(\log(1-b_v q_v^{-s})\right)-\sum_{\substack{v\in \text{Val}_0(K)\\ v(\Delta_E)=0}} \frac{d}{ds}\left(\log(1-a_v q_v^{-s}+q_v^{1-2s})\right). \end{align*}$$

The first sum can be rewritten as

$$\begin{align*}-\sum_{\substack{v\in \text{Val}_0(K)\\ v(\Delta_E)>0}} \frac{d}{ds}\left(\log(1-b_v q_v^{-s})\right)=\sum_{\substack{v\in \text{Val}_0(K)\\ v(\Delta_E)>0}} \frac{d}{ds}\left(\sum_{k=1}^\infty \frac{b_v^k}{k q_v^{ks}}\right)=-\sum_{\substack{v\in \text{Val}_0(K)\\ v(\Delta_E)>0}} \sum_{k=1}^\infty \frac{b_v^k \log(q_v)}{q_v^{ks}}. \end{align*}$$

For the second sum, factor $1-a_v T+q_v T^2$ as $(1-\alpha _v T)(1-\beta _v T)$ , where $\alpha _v+\beta _v=a_v$ and $\alpha _v\beta _v=q_v$ . Then,

$$\begin{align*}-\sum_{\substack{v\in \text{Val}_0(K)\\ v(\Delta_E)=0}} \frac{d}{ds}\left(\log(1-a_v q_v^{-s}+q_v^{1-2s})\right)=-\sum_{\substack{v\in \text{Val}_0(K)\\ v(\Delta_E)=0}} \sum_{k=1}^\infty \frac{(\alpha_v^k+\beta_v^k) \log(q_v)}{q_v^{ks}}. \end{align*}$$

We thus obtain the following expression for the logarithmic derivative of $L(E/K,s)$ :

$$ \begin{align*} \frac{L'}{L}(E/K,s)=-\sum_{\substack{v\in \text{Val}_0(K)\\ v(\Delta_E)>0}} \sum_{k=1}^\infty \frac{b_v^k \log(q_v)}{q_v^{ks}}-\sum_{\substack{v\in \text{Val}_0(K)\\ v(\Delta_E)=0}} \sum_{k=1}^\infty \frac{(\alpha_v^k+\beta_v^k) \log(q_v)}{q_v^{ks}}. \end{align*} $$

Define the von Mangoldt function on integral ideals ${\mathfrak a}\subseteq {\mathcal O}_K$ as

$$\begin{align*}\Lambda_K({\mathfrak a})\stackrel{\mathrm{def}}{=} \begin{cases} \log(q_v) & \text{ if } {\mathfrak a}={\mathfrak p}_v^k \text{ for some } k,\\ 0 & \text{ otherwise.} \end{cases} \end{align*}$$

For prime powers, set

$$\begin{align*}\widehat{a_E}({\mathfrak p}_v^k)=\begin{cases} \alpha_v^k+\beta_v^k & \text{ if }E\text{ has good reduction at }{\mathfrak p}_v,\\ b_v^k & \text{ if }E\text{ has bad reduction at }{\mathfrak p}_v, \end{cases} \end{align*}$$

and for ${\mathfrak a}$ not a power of a prime, set $\widehat {a_E}({\mathfrak a})=0$ . Then our expression for the logarithmic derivative of $L(E/K,s)$ becomes

(11) $$ \begin{align} \frac{L'}{L}(E/K,s)=-\sum_{{\mathfrak a}\subseteq {\mathcal O}_K} \frac{\widehat{a_E}({\mathfrak a})\Lambda_K({\mathfrak a})}{N_{K/{\mathbb Q}}({\mathfrak a})^s}, \end{align} $$

where the sum is over the nonzero integral ideals of ${\mathcal O}_K$ .

Let $\Gamma (s)\stackrel {\mathrm{def}}{=}\int _0^\infty t^{s-1}e^{-t} dt$ be the usual gamma function. Let $\Gamma _K(s)$ be the gamma factor

$$\begin{align*}\Gamma_K(s)\stackrel{\mathrm{def}}{=}\left((2\pi)^{-s} \Gamma(s)\right)^{[K:{\mathbb Q}]}. \end{align*}$$

Let $\mathfrak {f}_{E/K}$ be the conductor of E over K, and define a constant

$$\begin{align*}A_{E/K}\stackrel{\mathrm{def}}{=} N_{K/{\mathbb Q}}(\mathfrak{f}_{E/K}) \Delta_K^2. \end{align*}$$

Define the completed Hasse–Weil $\mathbf {L}$ -function, $\Lambda (E/K, s)$ , as

$$\begin{align*}\Lambda(E/K,s)\stackrel{\mathrm{def}}{=} A_{E/K}^{s/2}\Gamma_K(s) L(E/K,s). \end{align*}$$

We now assume that our elliptic curve E is modular over K, so that the L-function $L(E/K,s)$ is automorphic. This implies that the Hasse–Weil conjecture holds true for E, and thus,

$$\begin{align*}\Lambda(E/K,s)=\epsilon(E) \Lambda(E/K, 2-s), \end{align*}$$

where $\epsilon (E)\in \{\pm 1\}$ is the root number. We see that the logarithmic derivative of $\Lambda (E/K, s)$ is

(12) $$ \begin{align} \frac{\Lambda'}{\Lambda}(E/K, s)=\frac{\log(A_{E/K})}{2}+\frac{\Gamma_K'}{\Gamma_K}(s)+\frac{L'}{L}(E/K,s)=-\frac{\Lambda'}{\Lambda}(E/K,2-s). \end{align} $$

We now state the explicit formula for L-functions of elliptic curves over number fields:

Proposition 6.1.1 (Explicit Formula).

Let $\phi \colon {\mathbb C}\to {\mathbb C}$ be a partial function that is even and analytic in the strip $|\operatorname {Im}(s)|\leq \frac {1}{2}+\epsilon $ for some $\epsilon>0$ , and assume that on this strip $|\phi (s)|\ll (1+|s|)^{-(1+\delta )}$ for some $\delta>0$ as $|\text {Re}(s)|\to \infty $ . For $x\in {\mathbb R}$ , assume that $\phi (x)\in {\mathbb R}$ , and set

$$\begin{align*}\widehat{\phi}(t)\stackrel{\mathrm{def}}{=}\int_{\mathbb R} \phi(x) e^{-2\pi i tx} dx, \end{align*}$$

the Fourier transform. Let E be a modular elliptic curve over a number field K. Suppose that the L-function $L(E/K,s)$ associated to E satisfies the Generalized Riemann Hypothesis, so that all nontrivial zeros $\rho _j$ of $L(E/K,s)$ are of the form

$$\begin{align*}\rho_j=1+i\gamma_j \text{ for some } \gamma_j\in {\mathbb R}. \end{align*}$$

Let $r(E)\stackrel {\mathrm{def}}{=}\text {ord}_{s=1}(L(E/K,s))$ be the analytic rank of E. Then, we have

$$ \begin{align*} &r(E)\phi(0)+\sum_{\rho_j\neq 1} \phi(\gamma_j)\\ &\hspace{5mm} =\frac{\widehat{\phi}(0)}{2\pi}\log(A_{E/K})+\frac{1}{\pi}\int_{{\mathbb R}}\frac{\Gamma_K'}{\Gamma_K}(1+iy) \phi(y) dy -\frac{1}{\pi} \sum_{{\mathfrak a}\subseteq {\mathcal O}_K} \frac{ \widehat{a_E}({\mathfrak a}) \Lambda_K({\mathfrak a})}{N_{K/{\mathbb Q}}({\mathfrak a})}\cdot\widehat{\phi}\left(\frac{\log(N_{K/{\mathbb Q}}({\mathfrak a}))}{2\pi}\right), \end{align*} $$

where the sum on the left-hand side is over the zeros $\rho _j\neq 1$ with relevant multiplicity.

Proposition 6.1.1 is a slight generalization of [Reference Iwaniec and KowalskiIK04, Theorem 5.12] (see also [Reference MestreMes86] and [Reference MillerMil02, Appendix A] for the specific case of L-functions of elliptic curves). In [Reference Iwaniec and KowalskiIK04], the condition on the test function $\phi $ is that it is a Schwartz function. However, as observed in [Reference Goldston and GonekGG07, Lemma 1], the proof of [Reference Iwaniec and KowalskiIK04, Theorem 5.12] can be adapted to work for the set of functions $\phi $ satisfying the conditions in Proposition 6.1.1. We will use the following modified version of the explicit formula:

Corollary 6.1.2 (Modified Explicit Formula).

Maintaining the notation from Proposition 6.1.1 and letting X be a parameter, we have that

(13) $$ \begin{align} \begin{split} &r(E)\phi(0)+\sum_{\gamma_j\neq 0} \phi\left(\gamma_j\frac{\log(X)}{2\pi}\right) \\ &\hspace{1mm}= \frac{\widehat{\phi}(0) \log(A_{E/K})}{\log(X)} -\frac{2}{\log(X)} \sum_{{\mathfrak a}\subseteq {\mathcal O}_K} \frac{ \widehat{a_E}({\mathfrak a}) \Lambda_K({\mathfrak a})}{N_{K/{\mathbb Q}}({\mathfrak a})}\cdot\widehat{\phi}\left(\frac{\log(N_{K/{\mathbb Q}}({\mathfrak a}))}{\log(X)}\right)+O\,\left(\frac{1}{\log(X)}\right). \end{split} \end{align} $$

Proof. From the explicit formula (Proposition 6.1.1), it follows that

(14) $$ \begin{align} r(E)\phi(0)+\sum_{\gamma_j\neq 0} \phi\left(\gamma_j\frac{\log(X)}{2\pi}\right) &= \frac{\widehat{\phi}(0) \log(A_{E/K})}{\log(X)}+\frac{1}{\pi}\int_{{\mathbb R}}\frac{\Gamma_K'}{\Gamma_K}(1+iy) \phi\left(y\frac{\log(X)}{2\pi}\right) dy \nonumber\\& \quad -\frac{2}{\log(X)} \sum_{{\mathfrak a}\subseteq {\mathcal O}_K} \frac{ \widehat{a_E}({\mathfrak a}) \Lambda_K({\mathfrak a})}{N_{K/{\mathbb Q}}({\mathfrak a})}\cdot\widehat{\phi}\left(\frac{\log(N_{K/{\mathbb Q}}({\mathfrak a}))}{\log(X)}\right). \end{align} $$

A classical estimate for the logarithmic derivative of the gamma function is

$$\begin{align*}\left| \frac{\Gamma'}{\Gamma}(1+iy)\right|=O(\log(|y|+2)). \end{align*}$$

From this, and the fact that $\phi $ is absolutely integrable (since $|\phi (s)|\ll (1+|s|)^{-(1+\delta )}$ ), it follows that

$$\begin{align*}\int_{{\mathbb R}}\frac{\Gamma_K'}{\Gamma_K}(1+iy) \phi\left(y\frac{\log(X)}{2\pi}\right) dy=O\,\left(\frac{1}{\log(X)}\right). \end{align*}$$

Substituting this into (14) gives the modified explicit formula.

We now use the modified explicit formula to obtain an expression that bounds the analytic rank. Let E be an elliptic curve with minimal discriminant ${\mathfrak D}_{E/K}$ and of height less than or equal to X. For each such E, we have that $N_{K/{\mathbb Q}}({\mathfrak D}_{E/K})\ll X$ , and therefore,

$$\begin{align*}\log\left(N_{K/{\mathbb Q}}({\mathfrak D}_{E/K})\right)\leq \log(X)+O(1). \end{align*}$$

But since the conductor ${\mathfrak f}_{E/K}$ divides the minimal discriminant ${\mathfrak D}_{E/K}$ , one has $N_{K/{\mathbb Q}}({\mathfrak f}_{E/K})\leq N_{K/{\mathbb Q}}({\mathfrak D}_{E/K})$ , and thus,

$$\begin{align*}\log(N_{K/{\mathbb Q}}({\mathfrak f}_{E/K}))/\log(X)\leq 1+O(1/\log X). \end{align*}$$

It follows that $\log (A_{E/K})/\log (X)\leq 1+O(1/\log X)$ . By Corollary 6.1.2, this observation gives the inequality

$$ \begin{align*} &r(E)\phi(0)+\sum_{\gamma_j\neq 0} \phi\left(\gamma_j\frac{\log(X)}{2\pi}\right)\\ &\hspace{1cm}\leq \widehat{\phi}(0)-\frac{2}{\log(X)} \sum_{{\mathfrak a}\subseteq {\mathcal O}_K} \frac{ \widehat{a_E}({\mathfrak a}) \Lambda_K({\mathfrak a})}{N_{K/{\mathbb Q}}({\mathfrak a})}\cdot\widehat{\phi}\left(\frac{\log(N_{K/{\mathbb Q}}({\mathfrak a}))}{\log(X)}\right) +O\,\left(\frac{1}{\log X}\right). \end{align*} $$

We further simplify by rewriting the sum on the right-hand side as

$$\begin{align*}\sum_{{\mathfrak a}\subseteq {\mathcal O}_K} \frac{ \widehat{a_E}({\mathfrak a}) \Lambda_K({\mathfrak a})}{N_{K/{\mathbb Q}}({\mathfrak a})}\cdot\widehat{\phi}\left(\frac{\log(N_{K/{\mathbb Q}}({\mathfrak a}))}{\log(X)}\right) =\sum_{v\in \text{Val}_0(K)} \sum_{k\geq 1} \frac{ \widehat{a_E}({\mathfrak p}_v^k) \log(q_v)}{q_v^k}\cdot\widehat{\phi}\left(\frac{k \log(q_v)}{\log(X)}\right). \end{align*}$$

By the Weil Conjectures for Elliptic curves, $|\alpha _v|=|\beta _v|=\sqrt {q_v}$ (see, for example, [Reference SilvermanSil09, §V Theorem 2.3.1(a)]). From this, and the fact that $|b_v|\leq 1$ , we have $|\widehat {a_E}({\mathfrak p}_v^k)|\leq 2q_v^{k/2}$ . Using this, we bound the $k\geq 3$ part of the sum:

$$ \begin{align*} \sum_{v\in \text{Val}_0(K)} \sum_{k\geq 3} \frac{ \widehat{a_E}({\mathfrak p}_v^k) \log(q_v)}{q_v^k}\cdot\widehat{\phi}\left(\frac{k \log(q_v)}{\log(X)}\right) & \leq \sum_{v\in \text{Val}_0(K)} \sum_{k\geq 3} \frac{ 2 \log(q_v)}{q_v^{k/2}} ||\widehat{\phi}||_\infty\\ &\leq \sum_{v\in \text{Val}_0(K)} \frac{ 2 \log(q_v)}{q_v^{3/2} (1-q_v^{-1/2})} ||\widehat{\phi}||_\infty\\ &=O(1), \end{align*} $$

where $||f||_\infty $ is the usual $L^\infty $ -norm defined as

$$\begin{align*}||f||_\infty\stackrel{\mathrm{def}}{=}\inf\left\{a\geq 0 : m_L(\{x : |f(x)|>a\})=0\right\}, \end{align*}$$

where $m_L$ is the Lebesgue measure on ${\mathbb R}$ . Also note that for any finite set of places $S\subset \text {Val}_0(K)$ , we clearly have

$$\begin{align*}\sum_{v\in S} \sum_{k=1}^2 \frac{ \widehat{a_E}({\mathfrak p}_v^k) \log(q_v)}{q_v^k}\cdot\widehat{\phi}\left(\frac{k \log(q_v)}{\log(X)}\right)=O(1). \end{align*}$$

Therefore, setting

$$\begin{align*}U_{k}(E,\phi, X)\stackrel{\mathrm{def}}{=} \sum_{\substack{v\in \text{Val}_0(K) \\ 2\nmid q_v,\ 3\nmid q_v}} \frac{\widehat{a_E}({\mathfrak p}_v^k)\log(q_v)}{q_v^k} \cdot \widehat{\phi}\left( \frac{k\log(q_v)}{\log(X)}\right), \end{align*}$$

we have

$$\begin{align*}r(E)\phi(0)+\sum_{\gamma_j\neq 0} \phi\left(\gamma_j\frac{\log(X)}{2\pi}\right) \leq \widehat{\phi}(0)-\frac{2}{\log(X)} ( U_1(E,\phi ,X)+U_2(E,\phi,X)) +O\,\left(\frac{1}{\log X}\right). \end{align*}$$

We now average over isomorphism classes of elliptic curves. For each elliptic curve E, let $\rho _{E,j}=1+i\gamma _{E,j}$ be the nontrivial zeros of the L-function $L(E/K,s)$ . Let ${\mathcal E}_K(B)$ denote the set of isomorphism classes of elliptic curves over K with height bounded by B. Setting

$$ \begin{align*} S_k(\phi,B) &\stackrel{\mathrm{def}}{=} \frac{2}{\log(B) \#{\mathcal E}_{K}(B)} \sum_{E\in {\mathcal E}_{K}(B)} U_k(E,\phi,B)\\ &=\frac{2}{\log(B) \#{\mathcal E}_{K}(B)} \sum_{\substack{v\in \text{Val}_0(K) \\ 2\nmid q_v,\ 3\nmid q_v}} \frac{\log(q_v)}{q_v^k} \cdot \widehat{\phi}\left( \frac{k\log(q_v)}{\log(B)}\right)\sum_{E\in{\mathcal E}_{K}(B)} \widehat{a_E}({\mathfrak p}_v^k), \end{align*} $$

we have that

(15) $$ \begin{align} \begin{split} &\phi(0) \frac{1}{\#\mathcal{E}_{K}(B)} \sum_{E\in {\mathcal E}_{K}(B)} r(E)+\frac{1}{\# {\mathcal E}_{K}(B)} \sum_{E\in{\mathcal E}_{K}(B)}\sum_{\gamma_{E,j}\neq 0}\phi\left(\gamma_{E,j}\frac{\log(B)}{2\pi}\right)\\& \quad \leq \widehat{\phi}(0)-S_1(\phi,B)-S_2(\phi,B)+O\,\left(\frac{1}{\log(B)}\right). \end{split} \end{align} $$

For our applications, we shall use the test function

$$\begin{align*}\phi(y)=\begin{cases} \left(\frac{\sin(\pi \nu y)}{2\pi y}\right)^2 & \text{ if } y\neq 0,\\ \nu^2/4 & \text{ if }y=0, \end{cases} \end{align*}$$

whose Fourier transform is

$$\begin{align*}\widehat{\phi}(t)=\begin{cases} \frac{1}{2}\left(\frac{\nu}{2}-\frac{|t|}{2}\right) &\text{ for } |t|\leq \nu\\ 0 &\text{ for } |t|>\nu. \end{cases} \end{align*}$$

Observe that this test function satisfies the conditions of Proposition 6.1.1 (despite not being a Schwartz function).

Our strategy will be to show that

$$\begin{align*}-S_1(\phi,B)-S_2(\phi,B)=\frac{\phi(0)}{2}+o(1). \end{align*}$$

This will imply the following upper bound for the average analytic rank of elliptic curves over the number field K:

$$\begin{align*}\frac{\widehat{\phi}(0)}{\phi(0)}+\frac{1}{2} = \frac{1}{\nu}+\frac{1}{2}. \end{align*}$$

6.2. Class number estimates

In this subsection, we give some estimates for the counting functions $H(a,q)$ , which were defined in Subsection 5.1 equation (7). These results will be used in the next subsection.

Proposition 6.2.1. For any $a\in {\mathbb Z}$ , we have

$$\begin{align*}H(a,q)\ll_{\epsilon} q^{3/2+\epsilon}, \end{align*}$$

as a function of q for any $\epsilon>0$ .

Proof. By formula (9) for $H(a,q)$ , it suffices to show $H(a^2-4q)\ll _{\epsilon } q^{1/2+\epsilon }$ and $H(-4p)\ll _{\epsilon } q^{1/2+\epsilon }$ , and for this, it will suffice to show that for all possible n, we have $H(n)\ll _{\epsilon } |n|^{1/2+\epsilon }$ . As n must be the discriminant of an order in an imaginary quadratic field, it must be negative, non-square and congruent to $0$ or $1$ modulo $4$ .

Let $\text {sqf}(n)$ denote the squarefree part of n (e.g., $\text {sqf}(12)=3$ ), and let $\text {sq}(n)$ be such that $n=\text {sqf}(n)\cdot \text {sq}(n)^2$ (e.g., $\text {sq}(12)=2$ ). For non-square negative integers D that are congruent to $0$ or $1$ modulo $4$ , let ${\mathcal O}_{D}$ denote the imaginary quadratic order of discriminant D. We can rewrite equation (8) as

$$\begin{align*}H(n)=\sum_{\substack{d|\text{sq}(n)\\ n/d^2\equiv 0,1\quad\pmod{4}}} \frac{h({\mathcal O}_{n/d^2})}{\#{\mathcal O}_{n/d^2}^{\times}}. \end{align*}$$

Define a quadratic character

$$ \begin{align*} \chi_D:{\mathbb Z}&\to {\mathbb C}^{\times}\\ m & \mapsto \genfrac{(}{)}{}{}{D}{m}. \end{align*} $$

Observe that

(16) $$ \begin{align} \lim_{s\to 1}(s-1)\zeta_{{\mathcal O}_D}(s) = \sum_{m=1}^\infty \frac{\chi_D(m)}{m} =\sum_{m=1}^\infty \chi_D(m) \int_m^{\infty}\frac{1}{x^2}dx. \end{align} $$

For any positive integer $r\in {\mathbb Z}_{\geq 1}$ , the Pólya–Vinogradov inequality gives

$$\begin{align*}\left|\sum_{m=1}^r \chi_D(m)\right|\leq 2\sqrt{|D|} \log(|D|). \end{align*}$$

In particular, the functions $f_r(x)=\sum _{m=1}^r \chi _D(m)/x^2$ are dominated by $2\sqrt {|D|}\log (|D|)/x^{2}$ on ${\mathbb R}_{\geq 1}$ . Therefore, by the dominated convergence theorem, we may switch the sum and integral in (16), so that

$$ \begin{align*} \sum_{m=1}^\infty \chi_D(m) \int_m^{\infty}\frac{1}{x^2}dx=\int_1^{\infty} \left(\sum_{m=1}^x \chi_D(m)\right) \frac{1}{x^2} dx. \end{align*} $$

Splitting this integral and again applying the Pólya–Vinogradov inequality, we find that for any $\epsilon>0$ ,

$$ \begin{align*} \lim_{s\to 1}(s-1)\zeta_{{\mathcal O}_D}(s) &= \int_1^{\sqrt{|D|}} \left(\sum_{m=1}^x \chi_D(m)\right) \frac{1}{x^2} dx + \int_{\sqrt{|D|}}^{\infty} \left(\sum_{m=1}^x \chi_D(m)\right) \frac{1}{x^2} dx \\ &\ll \int_1^{\sqrt{|D|}} \frac{x}{x^2} dx + \int_{\sqrt{|D|}}^{\infty} \frac{\sqrt{|D|}\log(|D|)}{x^2} dx \\ &\ll \log(|D|) + \log(|D|) \\ &\ll |D|^{\epsilon}. \end{align*} $$

From this and the analytic class number formula [Reference Jordan and PoonenJP20, Theorem 1.1], we have the following upper bound for class numbers of orders in imaginary quadratic fields:

$$\begin{align*}h({\mathcal O}_D)\ll |D|^{1/2+\epsilon}. \end{align*}$$

Let $\tau (n)$ denote the number of divisors of n, and recall that $\tau (n)\ll _{\epsilon } |n|^{\epsilon }$ for any $\epsilon>0$ . Combining the above observations, we have

$$\begin{align*}H(n)\leq \sum_{\substack{d|\text{sq}(n)\\ n/d^2\equiv 0,1\quad\pmod{4}}} h({\mathcal O}_{n/d^2})\ll \tau(\text{sq}(n)) |n|^{1/2+\epsilon}\ll_{\epsilon} |n|^{1/2+\epsilon}.\\[-58pt] \end{align*}$$

The next proposition gives estimates for certain sums involving $H(a,q)$ .

Proposition 6.2.2. Let q be a power of a prime $p>3$ . For any $\epsilon>0$ , we have the following estimates:

$$ \begin{align*} &\sum_{|a|\leq 2\sqrt{q}} H(a,q)= q^2-q,\\ &\sum_{|a|\leq 2\sqrt{q}} aH(a,q)=0,\\ &\sum_{|a|\leq 2\sqrt{q}} a^2 H(a,q)=q^3+O\,\left(q^{\frac{5}{2}+\epsilon}\right). \end{align*} $$

Proof. The first sum: We have that

$$\begin{align*}\sum_{|a|\leq 2\sqrt{q}} H(a,q)=\#\{(b,c)\in {\mathbb F}_q\times{\mathbb F}_q : \Delta(b,c)\neq 0\}=q^2-q. \end{align*}$$

The second sum: Pairing positive and negative terms, we have that

$$\begin{align*}\sum_{|a|\leq 2\sqrt{q}} a H(a,q)=\sum_{0<a \leq 2\sqrt{q}} \left(a H(a,q)-a H(a,q)\right)= 0, \end{align*}$$

where we have used that $H(a,q)=H(-a,q)$ .

The third sum: This sum is the most subtle. As observed in Subsection 5.1, each isomorphism class of elliptic curves over ${\mathbb F}_q$ contains exactly $q-1$ elliptic curves, which can be thought of as the size of the orbit of an elliptic curve E with respect to the weighted ${\mathbb G}_m({\mathbb F}_q)$ action, with weights $4$ and $6$ . Let ${\mathcal E}_{{\mathbb F}_q}$ denote the set of isomorphism classes of elliptic curves over ${\mathbb F}_q$ . By the orbit stabilizer theorem,

$$\begin{align*}H(a,q)=\sum_{\substack{E\in {\mathcal E}_{{\mathbb F}_q}\\ a_q(E)=a}}\frac{q-1}{\#\text{Aut}_{{\mathbb F}_q}(E)}. \end{align*}$$

Therefore, we can rewrite the third sum as

$$\begin{align*}\sum_{|a|\leq 2\sqrt{q}} a^2 H(a,q) =(q-1)\sum_{|a|\leq 2\sqrt{q}}\sum_{{\substack{E\in {\mathcal E}_{{\mathbb F}_q}\\ a_q(E)=a}}} \frac{a^2}{\#\text{Aut}_{{\mathbb F}_q}(E)} =(q-1)\sum_{E\in {\mathcal E}_{{\mathbb F}_q}} \frac{a_q(E)^2}{\#\text{Aut}_{{\mathbb F}_q}(E)}. \end{align*}$$

The final sum can be estimated using work of Birch [Reference BirchBir68] and Ihara [Reference IharaIha67], as we now explain. Write $q=p^r$ . Define an indicator function

$$\begin{align*}\delta_{2{\mathbb Z}}(r)=\begin{cases} 1 & \text{ if }r\text{ is even}\\ 0 & \text{ if }r\text{ is odd}. \end{cases} \end{align*}$$

Let $T_k(m)$ be the m-th Hecke operator on the space of weight k cusp forms of level 1, and let $\text {Tr}(T_k(m))$ denote the trace of $T_k(m)$ .

The following result is due to Birch (in the $r=1$ case) [Reference BirchBir68] and Ihara (for $r>1$ ) [Reference IharaIha67] (see [Reference Kaplan and PetrowKP17, Theorem 2]):

Theorem 6.2.3 (Birch-Ihara).

With notation as above, we have that

$$ \begin{align*} \frac{1}{q}\sum_{E\in {\mathcal E}_{{\mathbb F}_q}} \frac{a_q(E)^2}{\#\text{Aut}_{{\mathbb F}_q}(E)} \end{align*} $$

equals

$$ \begin{align*} &\frac{1}{q}\left(p^3\text{Tr}(T_4(p^{r-2}))-\text{Tr}(T_4(q)) +\frac{1}{2}\left( p^3 \sum_{i=0}^{r-2} \min\{p^i,p^{r-2-i}\}^3-\sum_{i=0}^{r} \min\{p^i, p^{r-i}\}^3\right)\right)\\& \quad + p \text{Tr}(T_2(p^{r-2})) - \text{Tr}(T_2(q)) + \frac{1}{2}\left(p \sum_{i=0}^{r-2} \min\{p^i,p^{r-2-i}\} - \sum_{i=0}^r \min\{p^i,p^{r-i}\} \right) \\& \quad + \sum_{i=0}^r p^i-p\sum_{i=0}^{r-2}p^i. \end{align*} $$

We now show that for any $\epsilon>0$ ,

$$ \begin{align*} \sum_{E\in {\mathcal E}_{{\mathbb F}_q}} \frac{a_q(E)^2}{\#\text{Aut}_{{\mathbb F}_q}(E)} = q^2 + O(q^{3/2+\epsilon}). \end{align*} $$

Using the Ramanujan–Peterson–Deligne bound for Hecke eigenvalues [Reference DeligneDel74], one can show that for any prime power $p^s$ and any $\epsilon>0$ ,

$$\begin{align*}\text{Tr}(T_k(p^s))\ll_k p^{s(k-1)/2}s\ll_k p^{s((k-1)/2+\epsilon)} \end{align*}$$

(see, for example, the introduction of [Reference PetrowPet18]). In particular,

$$ \begin{align*} p^3\text{Tr}(T_4(p^{r-2}))-\text{Tr}(T_4(q)) \ll p^3 p^{(r-2)(3/2+\epsilon)} + q^{3/2+\epsilon} \ll q^{3/2+\epsilon} \end{align*} $$

and

$$ \begin{align*} p \text{Tr}(T_2(p^{r-2})) - \text{Tr}(T_2(q))\ll p p^{(r-2)(1/2+\epsilon)}+ q^{1/2+\epsilon} \ll q^{1/2+\epsilon}. \end{align*} $$

We now estimate the sums in Theorem 6.2.3. As a function of q, the sum $\sum _{i=0}^{r} \min \{p^i, p^{r-i}\}^3$ has on the order of $\log (q)$ terms, and each term is bounded above by $\left (p^{r/2}\right )^3=q^{3/2}$ . Therefore,

$$ \begin{align*} \sum_{i=0}^{r} \min\{p^i, p^{r-i}\}^3\ll q^{3/2}\log(q)\ll q^{3/2+\epsilon}. \end{align*} $$

Similar reasoning shows that

$$ \begin{align*} p^3 \sum_{i=0}^{r-2} \min\{p^i,p^{r-2-i}\}^3\ll p^3\log(q)p^{3(r-2)/2}\ll q^{3/2+\epsilon}. \end{align*} $$

We also compute

$$ \begin{align*} p \sum_{i=0}^{r-2} \min\{p^i,p^{r-2-i}\} - \sum_{i=0}^r \min\{p^i,p^{r-i}\} =-\sum_{i\in\{0,r\}} \min\{p^i,p^{r-i}\} =-2, \end{align*} $$

and

$$ \begin{align*} \sum_{i=0}^r p^i-p\sum_{i=0}^{r-2}p^i=\sum_{i\in \{0,r\}} p^i = 1 + q. \end{align*} $$

Combining the above estimates, we find that

$$ \begin{align*} \frac{1}{q}\sum_{E\in {\mathcal E}_{{\mathbb F}_q}} \frac{a_q(E)^2}{\#\text{Aut}_{{\mathbb F}_q}(E)} &= {\mathcal O}\left(\frac{q^{3/2+\epsilon}}{q}\right) + O(q^{1/2+\epsilon})-\frac{2}{2}+(1+q)= q + O(q^{1/2+\epsilon}). \end{align*} $$

From this, we obtain

$$\begin{align*}\sum_{|a|\leq 2\sqrt{q}} a^2 H(a,q)=(q-1)\left(q^2 + O(q^{3/2+\epsilon})\right)=q^3+O(q^{5/2+\epsilon}).\\[-51pt] \end{align*}$$

6.3. Estimating $S_1(\phi ,B)$ and $S_2(\phi ,B)$

Lemma 6.3.1. Let ${\mathfrak p}\subset {\mathcal O}_K$ be a prime ideal of norm $N_{K/{\mathbb Q}}({\mathfrak p})=q$ such that $2\nmid q$ and $3\nmid q$ . Then, for any $\epsilon>0$ , we have the following upper bound:

$$\begin{align*}\sum_{E\in {\mathcal E}_K(B)} \widehat{a_E}({\mathfrak p})\ll_{\epsilon} \frac{1}{q} B^{5/6}+q^{3/2+\epsilon} B^{5/6-1/3d}. \end{align*}$$

Proof. Let ${\mathcal E}_K^{\text {mult}}(B)$ denote the set of elliptic curves over K of naive height bounded by B and multiplicative reduction at ${\mathfrak p}$ . Then, we have

$$\begin{align*}\sum_{E\in {\mathcal E}_K(B)}\widehat{a_E}({\mathfrak p})=\sum_{|a|\leq 2\sqrt{q}} \sum_{\substack{E\in{\mathcal E}_K(B)\\ a_{\mathfrak p}(E)=a}} \widehat{a_E}({\mathfrak p}) + \sum_{E\in {\mathcal E}_K^{\text{mult}}(B)} \widehat{a_E}({\mathfrak p}). \end{align*}$$

By Theorem 1.1.2, we have that

(17) $$ \begin{align} \left|\sum_{E\in {\mathcal E}_K^{\text{mult}}(B)} \widehat{a_E}({\mathfrak p})\right|\ll \#{\mathcal E}_K^{\text{mult}}(B) \ll \frac{1}{q}B^{5/6}+B^{5/6-1/3d}. \end{align} $$

By Proposition 6.2.2, Proposition 6.2.1 and the proof of Theorem 1.1.3, we have that

$$ \begin{align*} \sum_{|a|\leq 2\sqrt{q}} \sum_{\substack{E\in{\mathcal E}_K(B)\\ a_{\mathfrak p}(E)=a}} \widehat{a_E}({\mathfrak p}) &=\sum_{|a|\leq 2\sqrt{q}} a\left(\kappa\frac{H(a,q)}{q^2}\frac{q^{10}}{q^{10}-1}B^{5/6}+O\,\left(q^{-1}H(a,q)B^{5/6-1/3d}\right)\right)\\ &=0+\sum_{|a|\leq 2\sqrt{q}} O\,\left(a q^{-1}H(a,q)B^{5/6-1/3d}\right)\\ &\ll \max_{|a|\leq 2\sqrt{q}}\{H(a,q)\} B^{5/6-1/3d}\\ &\ll_{\epsilon} q^{3/2+\epsilon} B^{5/6-1/3d}. \end{align*} $$

This, together with the bound (17), implies the lemma.

Lemma 6.3.2. Let ${\mathfrak p}\subset {\mathcal O}_K$ be a prime ideal with norm $N_{K/{\mathbb Q}}({\mathfrak p})=q$ such that $2\nmid q$ and $3\nmid q$ . Then, we have the following estimate:

$$\begin{align*}\sum_{E\in {\mathcal E}_K(B)} \widehat{a_E}({\mathfrak p}^2)=-\kappa q B^{5/6}+O\,\left(q^{1/2+\epsilon}B^{\frac{5}{6}}+q^2 B^{\frac{5}{6}-\frac{1}{3d}}\right), \end{align*}$$

where the implied constant does not depend on q.

Proof. Let ${\mathcal E}_K^{good}(B)$ (resp. ${\mathcal E}_K^{\text {mult}}(B)$ ) denote the set of elliptic curves over K with good reduction (resp. multiplicative reduction) at ${\mathfrak p}$ and height bounded by B. In this case, we have that

$$\begin{align*}\sum_{E\in {\mathcal E}_K(B)} \widehat{a_E}({\mathfrak p}^2)=\sum_{|a|\leq 2\sqrt{q}} \sum_{\substack{E\in{\mathcal E}_K^{good}(B)\\ a_{\mathfrak p}(E)=a}} \widehat{a_E}({\mathfrak p}^2) + \sum_{E\in {\mathcal E}_K^{\text{mult}}(B)} \widehat{a_E}({\mathfrak p}^2). \end{align*}$$

Since $\widehat {a_E}({\mathfrak p}^2)=\widehat {a_E}({\mathfrak p})^2=1$ for all $E\in {\mathcal E}_K^{\text {mult}}(B)$ , we have, by Theorem 1.1.2, that

(18) $$ \begin{align} \left|\sum_{E\in {\mathcal E}_K^{\text{mult}}(B)} \widehat{a_E}({\mathfrak p}^2)\right|=\#{\mathcal E}_K^{\text{mult}}(B)=\kappa \frac{q-1}{q^2}\frac{q^{10}}{q^{10}-1} B^{5/6}+O(B^{5/6-1/3d})=O\,\left(B^{5/6}\right). \end{align} $$

In the case that E has good reduction at ${\mathfrak p}$ , a straightforward computation shows that $\widehat {a_E}({\mathfrak p}^2)$ equals $a_{\mathfrak p}(E)^2-2q$ . Therefore, by Proposition 6.2.2 and the proof of Theorem 1.1.3, we have that

$$ \begin{align*} \sum_{|a|\leq 2\sqrt{q}} \sum_{\substack{E\in{\mathcal E}_K^{good}(B)\\ a_{\mathfrak p}(E)=a}} \widehat{a_E}({\mathfrak p}^2) &=\sum_{|a|\leq 2\sqrt{q}} \sum_{\substack{E\in{\mathcal E}_K^{good}(B)\\ a_{\mathfrak p}(E)=a}} (a_{\mathfrak p}(E)^2-2q)\\&=\sum_{|a|\leq 2\sqrt{q}} (a^2-2q)\left(\kappa\frac{H(a,q)}{q^2}\frac{q^{10}}{q^{10}-1}B^{5/6}+O\,\left(q^{-1}H(a,q)B^{5/6-1/3d}\right)\right)\\&=\frac{\kappa q^{8}}{q^{10}-1}\left(\sum_{|a|\leq 2\sqrt{q}} a^2 H(a,q)-2q\sum_{|a|\leq 2\sqrt{q}} H(a,q)\right) B^{\frac{5}{6}}+O\,\left(q^2 B^{\frac{5}{6}-\frac{1}{3d}}\right)\\&=\frac{-\kappa q^{11}}{q^{10}-1} B^{5/6}+O\,\left(q^{1/2+\epsilon}B^{\frac{5}{6}}+q^2 B^{\frac{5}{6}-\frac{1}{3d}}\right). \end{align*} $$

Observe that

$$\begin{align*}\frac{q^{10}}{q^{10}-1}=\sum_{k=0}^{\infty} q^{-10k}=1+O(q^{-10}). \end{align*}$$

Therefore,

$$\begin{align*}\sum_{|a|\leq 2\sqrt{q}} \sum_{\substack{E\in{\mathcal E}_K^{good}(B)\\ a_{\mathfrak p}(E)=a}}\widehat{a_E}({\mathfrak p}^2)=-\kappa q B^{5/6}+O\,\left(q^{1/2+\epsilon}B^{\frac{5}{6}}+q^2 B^{\frac{5}{6}-\frac{1}{3d}}\right). \end{align*}$$

This, together with the bound (18), implies the lemma.

6.4. Bounding the average analytic rank of elliptic curves

We now prove our main result.

Proof. By Lemma 6.3.1 and the observation that $\widehat {\phi }$ is supported on the interval $[-\nu ,\nu ]$ , we have that

$$ \begin{align*} S_1(\phi,B)&=\frac{2}{\log(B) \#{\mathcal E}_{K}(B)} \sum_{\substack{ v\in \text{Val}_0(K)\\ 2\nmid q_v,\ 3\nmid q_v}} \frac{\log(q_v)}{q_v} \cdot \widehat{\phi}\left( \frac{\log(q_v)}{\log(B)}\right)\sum_{E\in{\mathcal E}_{K}(B)} \widehat{a_E}({\mathfrak p}_v)\\ &\ll_{\epsilon} \frac{2}{\log(B) \#{\mathcal E}_{K}(B)} \sum_{\substack{v\in \text{Val}_0(K)\\ 2\nmid q_v,\ 3\nmid q_v}} \frac{\log(q_v)}{q_v} \cdot \widehat{\phi}\left( \frac{\log(q_v)}{\log(B)}\right)\left(\frac{1}{q_v} B^{5/6}+q_v^{3/2+\epsilon}B^{5/6-1/3d}\right)\\ &\ll \frac{2}{\log(B) \#{\mathcal E}_{K}(B)} \sum_{\substack{v\in \text{Val}_0(K) \\ q_v\leq B^\nu \\ 2\nmid q_v,\ 3\nmid q_v}} \frac{\log(q_v)}{q_v} \left(\frac{1}{q_v} B^{5/6}+q_v^{3/2+\epsilon}B^{5/6-1/3d}\right). \end{align*} $$

By Corollary 5.1.1, we know that $\#{\mathcal E}_K(B)=\kappa B^{5/6}+O(B^{5/6-1/3d})$ . Using this and the prime ideal theorem, we have that

(19) $$ \begin{align} S_1(\phi,B) \ll_{\epsilon} \frac{2}{\log(B)} \sum_{\substack{v\in \text{Val}_0(K) \\ q_v\leq B^\nu \\ 2\nmid q_v,\ 3\nmid q_v}} \frac{\log(q_v)}{q_v} \left(\frac{1}{q_v}+q_v^{3/2+\epsilon}B^{-1/3d}\right) \ll\frac{1+B^{(3/2+\epsilon)\nu-1/3d}}{\log(B)}. \end{align} $$

By Lemma 6.3.2, we have

$$ \begin{align*} S_2(\phi,B) &=\frac{2}{\log(B) \#{\mathcal E}_{K}(B)} \sum_{\substack{v\in \text{Val}_0(K) \\ 2\nmid q_v,\ 3\nmid q_v}} \frac{\log(q_v)}{q_v^2} \cdot \widehat{\phi}\left(\frac{2\log(q_v)}{\log(B)}\right)\sum_{E\in{\mathcal E}_{K}(B)} \widehat{a_E}({\mathfrak p}_v^2)\\&=\frac{2}{\log(B) \#{\mathcal E}_{K}(B)} \sum_{\substack{v\in \text{Val}_0(K) \\ 2\nmid q_v,\ 3\nmid q_v}} \frac{\log(q_v)}{q_v^2} \cdot \widehat{\phi}\left(\frac{2\log(q_v)}{\log(B)}\right)\left(-\kappa q_v B^{\frac{5}{6}}+O(q_v^{\frac{1}{2}+\epsilon}B^{\frac{5}{6}}+q_v^2 B^{\frac{5}{6}-\frac{1}{3d}})\right). \end{align*} $$

Assuming $\epsilon <1/4$ and using $\#{\mathcal E}_K(B)=\kappa B^{5/6}+O(B^{5/6-1/3d})$ , we have

$$ \begin{align*} S_2(\phi,B) &=\frac{2}{\log(B)} \sum_{\substack{v\in \text{Val}_0(K) \\ 2\nmid q_v,\ 3\nmid q_v}} \frac{\log(q_v)}{q_v^2} \cdot \widehat{\phi}\left( \frac{2\log(q_v)}{\log(B)}\right)\left(-q_v +O(q_v^{1/2+\epsilon}+q_v^2 B^{-1/3d})\right)\\&=\frac{-2}{\log(B)} \sum_{\substack{v\in \text{Val}_0(K) \\ 2\nmid q_v,\ 3\nmid q_v}} \frac{\log(q_v)}{q_v} \cdot \widehat{\phi}\left( \frac{2\log(q_v)}{\log(B)}\right) +O\,\left(\frac{1}{\log{B}}\sum_{\substack{v\in \text{Val}_0(K) \\ q_v\leq B^{\nu/2}}}\left(\frac{1}{q_v^{3/2-\epsilon}}+\log(q_v)B^{-1/3d}\right)\right)\\ &=\frac{-1}{\log(B)} \sum_{\substack{ v\in \text{Val}_0(K) \\ q_v\leq B^{\nu/2} \\ 2\nmid q_v,\ 3\nmid q_v}} \left(\frac{\nu \log(q_v)}{2q_v} - \frac{\log(q_v)^2}{q_v\log(B)}\right) +O\,\left(\frac{1+B^{\nu/2-1/3d}}{\log(B)}\right). \end{align*} $$

Arguments using the prime ideal theorem show that

$$\begin{align*}\sum_{\substack{ v\in \text{Val}_0(K) \\ q_v\leq X}} \frac{\log(q_v)}{q_v}=\log(X)+O(1) \end{align*}$$

and

$$\begin{align*}\sum_{\substack{ v\in \text{Val}_0(K) \\ q_v\leq X}} \frac{\log(q_v)^2}{q_v}=\frac{\log(X)^2}{2}+O(\log\log(X)). \end{align*}$$

From this, it follows that

$$ \begin{align*} &\sum_{\substack{ v\in \text{Val}_0(K) \\ q_v\leq B^{\nu/2} \\ 2\nmid q_v,\ 3\nmid q_v}} \left(\frac{\nu \log(q_v)}{2q_v} - \frac{\log(q_v)^2}{q_v\log(B)}\right)\\& \quad = \frac{\nu}{2} \left(\log(B^{\nu/2})+O(1)\right)-\frac{1}{\log(B)}\left(\frac{\log(B^{\nu/2})^2}{2}+O(\log\log(B^{\nu/2}))\right) \\& \quad =\left(\frac{\nu^2}{4}\log(B)+O(1)\right)-\left(\frac{\nu^2}{8}\log(B)+O(1)\right)\\& \quad =\frac{\nu^2}{8}\log(B)+O(1). \end{align*} $$

Therefore,

$$\begin{align*}S_2(\phi,B)=\frac{-\phi(0)}{2}+O\,\left(\frac{1+B^{\nu/2-1/3d}}{\log(B)}\right). \end{align*}$$

Taking $\nu =(3d(3/2+\epsilon ))^{-1}$ and combining the above estimate for $S_2$ with our bound (19) for $S_1$ , we obtain

$$\begin{align*}-S_1(\phi,B)-S_2(\phi,B)=\frac{\phi(0)}{2}+o(1). \end{align*}$$

Substituting this into (15) and taking the limit superior as B goes to infinity gives the following upper bound for the average analytic rank of elliptic curves over K:

$$\begin{align*}\frac{1}{\nu}+\frac{1}{2}=3d(3/2+\epsilon)+\frac{1}{2}, \end{align*}$$

for any $0<\epsilon <1/4$ . Since the limit superior exists, and since $\epsilon $ can be taken arbitrarily small, we obtain the bound $(9d+1)/2$ .