$\int\widetilde{f}_{2}{\mathrm{d}}g_{2}=0$,
$\int\widetilde{f}_{1}^{2}{\mathrm{d}}g_{1}=\chi^{2}(f_{1}\,\|\,g_{1})$, and
$\int\widetilde{f}_{2}^{2}{\mathrm{d}}g_{2}=\chi^{2}(f_{2}\,\|\,g_{2})$, we
have

$\displaystyle\chi^{2}(f_{1}\times f_{2}\,\|\,g_{1}\times g_{2})$
$\displaystyle\quad=\int\big{(}{\widetilde{f}}_{1}^{2}\times{\widetilde{f}}_{2}^{2}+{\widetilde{f}}_{1}^{2}+{\widetilde{f}}_{2}^{2}+2(\widetilde{f}_{1})^{2}\times\widetilde{f}_{2}+2(\widetilde{f}_{2})^{2}\times\widetilde{f}_{1}+2\widetilde{f}_{1}\times\widetilde{f}_{2}\big{)}{\mathrm{d}}(g_{1}\times
g_{2})$
$\displaystyle\quad=\chi^{2}(f_{1}\,\|\,g_{1})\cdot\chi^{2}(f_{2}\,\|\,g_{2})+\chi^{2}(f_{1}\,\|\,g_{1})+\chi^{2}(f_{2}\,\|\,g_{2}).$

Thus, we complete the proof of Property (ii) of Lemma D.3. Proof of Property
(iii) of Lemma D.3. Let $\widetilde{f}_{1}=f_{1}/g_{1}-1$ and
$\widetilde{f}_{2}=f_{2}/g_{2}-1$. It then holds that

$\displaystyle\chi^{2}(f_{1}+f_{2}\,\|\,g)=\int\Big{(}\frac{f_{1}+f_{2}}{g}-1\Big{)}^{2}{\mathrm{d}}g=\int(\widetilde{f}_{1}+\widetilde{f}_{2}+1)^{2}{\mathrm{d}}g.$

By further noting that
$\int\widetilde{f}_{1}^{2}{\mathrm{d}}g_{1}=\chi^{2}(f_{1}\,\|\,g_{1})$ and
$\int\widetilde{f}_{2}^{2}{\mathrm{d}}g_{2}=\chi^{2}(f_{2}\,\|\,g_{2})$, we
have

$\displaystyle\chi^{2}(f_{1}+f_{2}\,\|\,g)$ $\displaystyle\leq
3\int\big{(}(\widetilde{f}_{1})^{2}+(\widetilde{f}_{2})^{2}+1\big{)}{\mathrm{d}}g$
$\displaystyle=3\big{(}\chi^{2}(f_{1}\,\|\,g)+\chi^{2}(f_{2}\,\|\,g)+1\big{)},$

where the inequality follows from the Cauchy-Schwarz inequality. Thus, we
complete the proof of Property (iii) of Lemma D.3. Proof of Property (iv) of
Lemma D.3. Let $\widetilde{f}(x,\cdot)=f(x,\cdot)/g(\cdot)-1$. It then holds
that

$\displaystyle\chi^{2}\Big{(}\int\alpha(x)f(x,\cdot){\mathrm{d}}x\,\big{\|}\,g\Big{)}$
$\displaystyle=\int\Big{(}\int\alpha(x)\widetilde{f}(x,\theta){\mathrm{d}}x\Big{)}^{2}g({\mathrm{d}}\theta)+\Big{(}\int\alpha(x){\mathrm{d}}x-1\Big{)}^{2}$
$\displaystyle\leq\int\Big{(}\int\alpha(x)^{2}{\mathrm{d}}x\cdot\int{\widetilde{f}}(x,\theta)^{2}{\mathrm{d}}x\Big{)}g({\mathrm{d}}\theta)+\Big{(}\int\alpha(x){\mathrm{d}}x-1\Big{)}^{2}$
$\displaystyle=\int\alpha(x)^{2}{\mathrm{d}}x\cdot\int\chi^{2}(f(x,\cdot)\,\|\,g){\mathrm{d}}x+\Big{(}\int\alpha(x){\mathrm{d}}x-1\Big{)}^{2},$

where the first equality holds by noting that
$\int\widetilde{f}(x,\cdot){\mathrm{d}}g=0$ and the inequality follows from
the Cauchy-Schwarz inequality. Thus, we complete the proof of Property (iv) of
Lemma D.3. ∎

######  Lemma D.4.

For weighted homogeneous Sobolev norm defined by

$\displaystyle\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu)}=\sup\big{\\{}|\langle
f,\nu_{1}-\nu_{2}\rangle|\big{|}\|f\|_{\dot{H}^{1}(\mu)}\leq 1\big{\\}}.$

we have the following properties.

  * (i)

For a group of probability measures
$\mu_{x}:\mathcal{X}\rightarrow\mathscr{P}_{2}(\Theta)$ and
$\nu_{1},\nu_{2}\in\mathscr{P}_{2}(\Theta)$, if $\mu$ is in the convex hull of
$\mu_{x}$, i.e., there exists $\alpha_{x}\geq 0$ such that both
$\int\alpha_{x}{\mathrm{d}}x=1$ and $\mu=\int\alpha_{x}\mu_{x}{\mathrm{d}}x$
hold, it then holds that

$\displaystyle\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu)}\leq\sup_{x\in\mathcal{X}}\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu_{x})}.$

  * (ii)

Assume that we have measures $\mu\in\mathscr{P}_{2}(\Theta)$ and
$\nu_{x}:\mathcal{X}\rightarrow\mathscr{P}_{2}(\Theta)$. Let
$\beta_{1},\beta_{2}:\mathcal{X}\rightarrow\mathbb{R}$ be two functions on
$\mathcal{X}$ such that
$\int\beta_{1}(x){\mathrm{d}}x=\int\beta_{2}(x){\mathrm{d}}x$. Then, by
letting $\nu_{1}=\int\nu_{x}\beta_{1}(x){\mathrm{d}}x$ and
$\nu_{2}=\int\nu_{x}\beta_{2}(x){\mathrm{d}}x$, we have

$\displaystyle\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu)}\leq\frac{1}{2}\sup_{(x^{\prime},x^{\prime\prime})\in\mathcal{X}\times\mathcal{X}}\|\nu_{x^{\prime}}-\nu_{x^{\prime\prime}}\|_{\dot{H}^{-1}(\mu)}\cdot\int\big{|}\beta_{1}(x)-\beta_{2}(x)\big{|}{\mathrm{d}}x.$

###### Proof.

Proof of Property (i) of Lemma D.4. By definition of the weighted homogeneous
Sobolev norm, we have

$\displaystyle\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu)}$
$\displaystyle=\sup_{f}\bigg{\\{}\big{|}\langle
f,\nu_{1}-\nu_{2}\rangle\big{|}\bigg{|}\int|\nabla
f|^{2}\alpha_{x}{\mathrm{d}}\mu_{x}{\mathrm{d}}x\leq 1\bigg{\\}}$
$\displaystyle=\sup_{f,\lambda_{x}\geq 0}\bigg{\\{}\big{|}\langle
f,\nu_{1}-\nu_{2}\rangle\big{|}\bigg{|}\int|\nabla
f|^{2}{\mathrm{d}}\mu_{x}\leq\lambda_{x},\forall
x;\int\alpha_{x}\lambda_{x}{\mathrm{d}}x=1\bigg{\\}}$ (D.5)
$\displaystyle\leq\sup_{\begin{subarray}{c}\int\lambda_{x}\alpha_{x}{\mathrm{d}}x=1,\\\
\lambda_{x}\geq 0\end{subarray}}\inf_{x}\sup_{f_{x}}\bigg{\\{}\big{|}\langle
f_{x},\nu_{1}-\nu_{2}\rangle\big{|}\bigg{|}\int|\nabla
f_{x}|^{2}{\mathrm{d}}\mu_{x}\leq\lambda_{x}\bigg{\\}},$ (D.6)

where the first equality holds by noting that
$\mu=\int\alpha_{x}\mu_{x}{\mathrm{d}}x$. To illustrate the last equality, we
denote by $\mathscr{F}$ and $\mathscr{F}_{x}$ the allowed function class for
$f$ in (D.5) and $f_{x}$ in (D.6) to choose from, respectively. Note that we
have $\mathscr{F}\subseteq\mathscr{F}_{x}$ for any $x$, which is because the
constraints for each $f_{x}$ in (D.6) are relaxation of the constraints for
$f$ in (D.5). And so, the supremum taken over $\mathscr{F}$ is no larger than
the supremum over $\mathscr{F}_{x}$ for any $x$. Therefore, the supremum over
$\mathscr{F}$ is no larger than the the smallest supremum over
$\mathscr{F}_{x}$. Thus, (D.6) holds. Furthermore, we have

$\displaystyle\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu)}$
$\displaystyle\leq\sup_{\begin{subarray}{c}\int\lambda_{x}\alpha_{x}{\mathrm{d}}x=1,\\\
\lambda_{x}\geq 0\end{subarray}}\inf_{x}\sup_{f_{x}}\bigg{\\{}\big{|}\langle
f_{x},\nu_{1}-\nu_{2}\rangle\big{|}\bigg{|}\int|\nabla
f_{x}|^{2}{\mathrm{d}}\mu_{x}\leq\lambda_{x}\bigg{\\}}$ (D.7)
$\displaystyle=\sup_{\begin{subarray}{c}\int\lambda_{x}\alpha_{x}{\mathrm{d}}x=1,\\\
\lambda_{x}\geq
0\end{subarray}}\inf_{x}\big{\\{}\sqrt{\lambda_{x}}\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu_{x})}\big{\\}}$
(D.8)
$\displaystyle\leq\sup_{\begin{subarray}{c}\int\lambda_{x}\alpha_{x}{\mathrm{d}}x=1,\\\
\lambda_{x}\geq
0\end{subarray}}\inf_{x}\big{\\{}\sqrt{\lambda_{x}}\big{\\}}\cdot\sup_{x}\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu_{x})},$

where the first equality holds by noting that (D.8) is a rescaling of (D.7)
with respect to $\lambda_{x}$ in the constraints of (D.7). Here, we let

$y=\sup_{\begin{subarray}{c}\int\lambda_{x}\alpha_{x}{\mathrm{d}}x=1,\\\
\lambda_{x}\geq 0\end{subarray}}\inf_{x}\big{\\{}\sqrt{\lambda_{x}}\big{\\}}.$

Then, it holds that $y\leq 1$. Otherwise, there must exists $\lambda_{x}$ such
that $\int\lambda_{x}\alpha_{x}{\mathrm{d}}x=1$ and that $\lambda_{x}>1$ holds
for any $x\in\mathcal{X}$, which contradicts with our conditions that
$\int\alpha_{x}{\mathrm{d}}x=1$ and $\alpha_{x}\geq 0$. Therefore, it further
holds that

$\displaystyle\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu)}\leq
y\cdot\sup_{x}\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu_{x})}\leq\sup_{x}\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu_{x})}.$

Thus, we complete the proof of Property (i) of Lemma D.4. Proof of Property
(ii) of Lemma D.4. Let $\alpha=\beta_{1}-\beta_{2}$. Then, we have
$\int\alpha(x){\mathrm{d}}x=0$. Let $\alpha^{+}=\max\\{0,\alpha\\}$ and
$\alpha^{-}=-\min\\{0,\alpha\\}$. Then, we have
$\alpha^{+}-\alpha^{-}=\alpha=\beta_{1}-\beta_{2}$ and that
$\alpha^{+}+\alpha^{-}=|\alpha|=|\beta_{1}-\beta_{2}|$. Since $\int\alpha=0$,
it holds that
$\int\alpha^{+}(x){\mathrm{d}}x=\int\alpha^{-}(x){\mathrm{d}}x=A$, where
$A\geq 0$. We further let
$\lambda(x,x^{\prime})=A^{-1}\alpha^{+}(x)\alpha^{-}(x^{\prime})$. Then, it
holds that $\alpha^{+}(x)=\int\lambda(x,x^{\prime}){\mathrm{d}}x^{\prime}$ and
that $\alpha^{-}(x^{\prime})=\int\lambda(x,x^{\prime}){\mathrm{d}}x$.
Therefore, we have

$\displaystyle\|\nu_{1}-\nu_{2}\|_{\dot{H}^{-1}(\mu)}=\Big{\|}\int_{\mathcal{X}}(\alpha^{+}-\alpha^{-})\nu_{x}{\mathrm{d}}x\Big{\|}_{\dot{H}^{-1}(\mu)}=\Big{\|}\int_{\mathcal{X}\times\mathcal{X}}\lambda(x,x^{\prime})(\nu_{x}-\nu_{x^{\prime}}){\mathrm{d}}x{\mathrm{d}}x^{\prime}\Big{\|}_{\dot{H}^{-1}(\mu)}.$

By defintion of the weighted homogeneous Sobolev norm, it further holds that

$\displaystyle\Big{\|}\int_{\mathcal{X}\times\mathcal{X}}\lambda(x,x^{\prime})(\nu_{x}-\nu_{x^{\prime}}){\mathrm{d}}x{\mathrm{d}}x^{\prime}\Big{\|}_{\dot{H}^{-1}(\mu)}$
$\displaystyle\quad=\sup_{f}\bigg{\\{}\Big{|}\big{\langle}\int\lambda(x,x^{\prime})(\nu_{x}-\nu_{x^{\prime}}){\mathrm{d}}x{\mathrm{d}}x^{\prime},f\big{\rangle}\Big{|}\,\bigg{|}\,\int|\nabla
f|^{2}\leq 1\bigg{\\}}.$ (D.9)

We assume the supremum in (D) is reached at $f^{*}$. Then, we have
$\int|\nabla f^{*}|^{2}{\mathrm{d}}\mu\leq 1$ and that

$\displaystyle\Big{\|}\int_{\mathcal{X}\times\mathcal{X}}\lambda(x,x^{\prime})(\nu_{x}-\nu_{x^{\prime}}){\mathrm{d}}x{\mathrm{d}}x^{\prime}\Big{\|}_{\dot{H}^{-1}(\mu)}$
$\displaystyle=\Big{|}\big{\langle}\int\lambda(x,x^{\prime})(\nu_{x}-\nu_{x^{\prime}}){\mathrm{d}}x{\mathrm{d}}x^{\prime},f^{*}\big{\rangle}\Big{|}$
$\displaystyle\leq\sup_{(x,x^{\prime})\in\mathcal{X}\times\mathcal{X}}\big{|}\langle\nu_{x}-\nu_{x^{\prime}},f^{*}\rangle\big{|}\cdot\int\lambda(x,x^{\prime}){\mathrm{d}}x{\mathrm{d}}x^{\prime}$
$\displaystyle\leq\frac{1}{2}\sup_{(x,x^{\prime})\in\mathcal{X}\times\mathcal{X}}\|\nu_{x}-\nu_{x^{\prime}}\|_{\dot{H}^{-1}(\mu)}\cdot\int|\beta_{1}-\beta_{2}|{\mathrm{d}}x,$

where the last inequality holds by noting that
$\int\lambda(x,x^{\prime}){\mathrm{d}}x{\mathrm{d}}x^{\prime}=A=1/2\cdot\int|\alpha|{\mathrm{d}}x=1/2\cdot\int|\beta_{1}-\beta_{2}|{\mathrm{d}}x$.
Thus, we complete the proof of Property (ii) of Lemma D.4. ∎

######  Lemma D.5.

For probability measures $\nu_{1},\nu_{2}\in\mathscr{P}_{2}(\Theta_{1})$ and
$p_{1},p_{2}\in\mathscr{P}_{2}(\Theta_{2})$, it holds that

$\displaystyle W_{2}^{2}(\nu_{1}\times p_{1},\nu_{2}\times p_{2})\leq
W_{2}^{2}(\nu_{1},\nu_{2})+W_{2}^{2}(p_{1},p_{2}).$

###### Proof.

By the property of optimal transport (Ambrosio et al., 2008), there exists
mapping $T_{\nu}$ and $T_{p}$ such that

$\displaystyle
W_{2}^{2}(\nu_{1},\nu_{2})=\int\|\theta_{1}-T_{\nu}(\theta_{1})\|\nu_{1}({\mathrm{d}}\theta_{1}),$
$\displaystyle
W_{2}^{2}(p_{1},p_{2})=\int\|\theta_{1}-T_{p}(\theta_{1})\|p_{1}({\mathrm{d}}\theta_{1}),$

and that $(T_{\nu})_{\sharp}\nu_{1}=\nu_{2}$ and
$(T_{p})_{\sharp}p_{1}=p_{2}$. Note that we have $(T_{\nu}\times
T_{p})_{\sharp}(\nu_{1}\times p_{1})=\nu_{2}\times p_{2}$. Thus, by definition
of $W_{2}$ distance, it holds that

$\displaystyle W_{2}^{2}(\nu_{1}\times p_{1},\nu_{2}\times p_{2})$
$\displaystyle\leq\int\big{\|}(\theta_{1},\theta_{2})-\big{(}T_{\nu}(\theta_{1}),T_{p}(\theta_{1})\big{)}\big{\|}^{2}\nu_{1}({\mathrm{d}}\theta_{1})p_{1}({\mathrm{d}}\theta_{2})$
$\displaystyle=\int\Big{(}\big{\|}\theta_{1}-T_{\nu}(\theta_{1})\big{\|}^{2}+\big{\|}\theta_{2}-T_{p}(\theta_{2})\big{\|}^{2}\Big{)}\nu_{1}({\mathrm{d}}\theta_{1})p_{1}({\mathrm{d}}\theta_{2})$
$\displaystyle=W_{2}^{2}(\nu_{1},\nu_{2})+W_{2}^{2}(p_{1},p_{2}).$

Thus, we complete the proof of Lemma D.5. ∎

######  Lemma D.6.

For probability density function
$\rho,\rho_{1},\rho_{2}\in\mathscr{P}_{2}(\Theta)$, let
$\widetilde{\rho}_{1}=\alpha^{-1}\rho_{1}+(1-\alpha^{-1})\rho$ and
$\widetilde{\rho}_{2}=\alpha^{-1}\rho_{2}+(1-\alpha^{-1})\rho$. Then, We have

$W_{2}(\widetilde{\rho}_{1},\widetilde{\rho}_{2})\leq\alpha^{-1/2}W_{2}(\rho_{1},\rho_{2}).$

###### Proof.

Recall the definition of Wasserstain-2 distance that

$\displaystyle
W_{2}(\rho_{1},\rho_{2})=\Big{(}\inf_{\gamma\in\Gamma(\rho_{1},\rho_{2})}\int\|x-y\|^{2}{\mathrm{d}}\gamma(x,y)\Big{)}^{1/2},$

where $\Gamma(\rho_{1},\rho_{2})$ is the set of all couplings of $\rho_{1}$
and $\rho_{2}$. We assume that the infimum is reached by
$\gamma^{*}(x,y)\in\Gamma(\rho_{1},\rho_{2})$, i.e.,

$\displaystyle
W_{2}(\rho_{1},\rho_{2})=\Big{(}\int\|x-y\|^{2}{\mathrm{d}}\gamma^{*}(x,y)\Big{)}^{1/2}.$

We denote by $\gamma^{\prime}(x,y)$ the distribution such that

$\displaystyle\gamma^{\prime}(x,y)=\alpha^{-1}\gamma^{*}(x,y)+(1-\alpha^{-1})\rho(x)\delta(x-y),$

where $\delta(x,y)$ is dirac delta function and it holds that
$\gamma^{\prime}(x,y)\in\Gamma(\widetilde{\rho}_{1},\widetilde{\rho}_{2})$.
Hence, it follows that

$\displaystyle W_{2}(\widetilde{\rho}_{1},\widetilde{\rho}_{2})$
$\displaystyle\leq\Big{(}\int\|x-y\|^{2}{\mathrm{d}}\gamma^{\prime}(x,y)\Big{)}^{1/2}$
$\displaystyle=\Big{(}\int\|x-y\|^{2}\big{(}\alpha^{-1}\gamma^{*}(x,y)+(1-\alpha^{-1})\rho(x)\delta(x-y)\big{)}{\mathrm{d}}(x,y)\Big{)}^{1/2}$
$\displaystyle=\alpha^{-1/2}W_{2}(\rho_{1},\rho_{2}).$

Thus, we complete the proof of Lemma D.6. ∎

##  Appendix E Auxiliary Lemmas

We use the definition of absolutely continuous curves in
$\mathscr{P}_{2}(\mathbb{R}^{D})$ in Ambrosio et al. (2008).

######  Definition E.1 (Absolutely Continuous Curve).

Let $\beta:[a,b]\rightarrow\mathscr{P}_{2}(\mathbb{R}^{D})$ be a curve. Then,
we say $\beta$ is an absolutely continuous curve if there exists a square-
integrable function $f:[a,b]\rightarrow\mathbb{R}$ such that

$\displaystyle
W_{2}(\beta_{s},\beta_{t})\leq\int_{s}^{t}f(\tau)\,{\mathrm{d}}\tau$

for any $a\leq s<t\leq b$.

Then, we have the following first variation formula.

######  Lemma E.2 (First Variation Formula, Theorem 8.4.7 in Ambrosio et al.
(2008)).

Given $\nu\in\mathscr{P}_{2}(\mathbb{R}^{D})$ and an absolutely continuous
curve $\mu:[0,T]\rightarrow\mathscr{P}_{2}(\mathbb{R}^{D})$, let
$\beta:[0,1]\rightarrow\mathscr{P}_{2}(\mathbb{R}^{D})$ be the geodesic
connecting $\mu_{t}$ and $\nu$. It holds that

$\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}\frac{W_{2}(\mu_{t},\nu)^{2}}{2}=-\langle\dot{\mu}_{t},\dot{\beta}_{0}\rangle_{\mu_{t},W_{2}},$

where $\dot{\mu}_{t}=\partial_{t}\mu_{t}$,
$\dot{\beta}_{0}=\partial_{s}\beta_{s}{\,|\,}_{s=0}$, and the inner product is
defined in (A.2).

######  Lemma E.3 (Eulerian Representation of Geodesics, Proposition 5.38 in
Villani (2003)).

Let $\beta:[0,1]\rightarrow\mathscr{P}_{2}(\mathbb{R}^{D})$ be a geodesic and
$v$ be the corresponding vector field such that
$\partial_{t}\beta_{t}=-\mathop{\mathrm{div}}(\beta_{t}\cdot v_{t})$. It holds
that

$\displaystyle\frac{{\mathrm{d}}}{{\mathrm{d}}t}(\beta_{t}\cdot
v_{t})=-\mathop{\mathrm{div}}(\beta_{t}\cdot v_{t}\otimes v_{t}),$

where $\otimes$ is the outer product between two vectors.

######  Lemma E.4 (Talagrand’s Inequality, Corollary 2.1 in Otto and Villani
(2000)).

Let $\nu$ be $N(0,\kappa\cdot I_{D})$. It holds for any
$\mu\in\mathscr{P}_{2}(\mathbb{R}^{D})$ that

$\displaystyle W_{2}(\mu,\nu)^{2}\leq 2D_{\rm KL}(\mu\,\|\,\nu)/\kappa.$

######  Lemma E.5 (Theorem 1 in Peyre (2011)).

Let $\mu,\nu$ be two probability measures in $\mathscr{P}_{2}(\theta)$. Then,
it holds that

$\displaystyle W_{2}(\mu,\nu)\leq 2\|\mu-\nu\|_{\dot{H}^{-1}(\nu)}.$

##  Appendix F Conclusions and Limitations

In this work, we study the time envolution of a two-timescale AC represented
by a two-layer neural network in the mean-field limit. Specifically, the actor
updates its policy via proximal policy optimization, which is closely related
to the replicator dynamics, while the critic updates by temporal-difference
learning, which is captured by a semigradient flow in the Wasserstein space.
By introducing a restarting mechanism, we establish the convergence and
optimality of AC with two-layer overparameterized neural network. However, the
study has potential limitations. In this work we only study the continuous-
time limiting regime, which is an idealized setting with infinitesimal
learning rates, and establish finite-time convergence and optimality
guarantees. Finite-time results for the more realistic discrete-time setting
is left for future research.


# Statistics for Iwasawa invariants of elliptic curves, III

Anwesh Ray  Chennai Mathematical Institute, H1, SIPCOT IT Park, Kelambakkam,
Siruseri, Tamil Nadu 603103, India<EMAIL_ADDRESS>
###### Abstract.

Given a prime $p\geq 5$, a conjecture of Greenberg predicts that the
$\mu$-invariant of the $p$-primary Selmer group should vanish for most
elliptic curves with good ordinary reduction at $p$. In support of this
conjecture, I show that the $5$-primary Iwasawa $\mu$\- and
$\lambda$-invariants simultaneously vanish for an explicit positive density of
elliptic curves $E_{/\mathbb{Q}}$. The elliptic curves in question have good
ordinary reduction at $5$, and are ordered by their _height_. The results are
proven by leveraging work of Bhargava and Shankar on the distribution of
$5$-Selmer groups of elliptic curves defined over $\mathbb{Q}$.

###### Key words and phrases:

arithmetic statistics, Iwasawa theory, Selmer groups of elliptic curves

###### 2020 Mathematics Subject Classification:

11R23, 11R45, 11G05

##  1\. Introduction

###  1.1. Motivation and historical context

Let $p$ be a prime number and $K$ be a number field. Denote by
$\mathbb{Z}_{p}\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}\mathbb{Z}/p^{n}\mathbb{Z}$ the ring of $p$-adic
integers. Set $\operatorname{Cl}_{p}(K)$ to denote the $p$-primary part of the
class group of $K$, and $h_{p}(K)\mathrel{\mathop{\mathchar
58\relax}}=\\#\operatorname{Cl}_{p}(K)$. Fix an algebraic closure $\bar{K}$ of
$K$, and let $\mu_{p^{n}}\subset\bar{K}$ be the $p^{n}$-th roots of unity. Let
$K(\mu_{p^{\infty}})$ denote the infinite cyclotomic extension $\bigcup_{n\geq
1}K(\mu_{p^{n}})$. The _cyclotomic $\mathbb{Z}_{p}$-extension_ of $K$ is the
unique Galois extension $K_{\infty}/K$ contained in $K(\mu_{p^{\infty}})$ such
that $\operatorname{Gal}(K_{\infty}/K)$ is isomorphic to $\mathbb{Z}_{p}$ (as
a topological group). The _$n$ -th layer_ $K_{n}/K$ is the degree
$p^{n}$-extension of $K$ which is contained in $K_{\infty}$. There is a
beautiful asymptotic growth formula for the $p$-primary class numbers
$h_{p}(K_{n})$, as $n$ increases. In greater detail, setting
$p^{e_{n}}=h_{p}(K_{n})$, Iwasawa [Iwa73] showed that there exist invariants
$\mu_{p}(K),\lambda_{p}(K)\in\mathbb{Z}_{\geq 0}$ and
$\nu_{p}(K)\in\mathbb{Z}$ such that for $n\gg 0$, one has that

$e_{n}=p^{n}\mu_{p}(K)+n\lambda_{p}(K)+\nu_{p}(K).$

It is a longstanding conjecture due to Iwasawa that $\mu_{p}(K)=0$ for any
number field $K$ and any prime number $p$. When $F/\mathbb{Q}$ is an abelian
Galois extension, the conjecture was proven by Ferrero and Washington [FW79].

It is natural to investigate generalizations of Iwasawa theory to motives and
their Selmer groups. Mazur [Maz72] initiated the Iwasawa theory of elliptic
curves (and abelian varieties) with good ordinary reduction at a prime $p>2$.
Following seminal work of Rubin, Kato, Greenberg and others, the Iwasawa
theory of Galois representations associated to motives came into further
prominence. The main algebraic objects of study are the $p$-primary Selmer
groups associated to elliptic curves, considered over the cyclotomic
$\mathbb{Z}_{p}$-extension of a number field. When the elliptic curve in
question has good ordinary reduction at an odd prime $p$, Kato [Kat04] and
Rubin [Rub81] showed that the associated Selmer groups are cotorsion over the
Iwasawa algebra. This allows one to associated Iwasawa $\mu$ and
$\lambda$-invariants $\mu_{p}(E)$ and $\lambda_{p}(E)$ respectively to these
Selmer groups. These invariants encode the growth properties of $p$-primary
Selmer groups, Mordell–Weil groups and Tate-Shafarevich groups in cyclotomic
towers. In this context, there is an outstanding conjecture due to Greenberg.
Given an elliptic curve $E_{/\mathbb{Q}}$ and a prime number $p$, set
$\bar{\rho}_{E,p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
to denote the Galois representation on the $p$-torsion of
$E(\bar{\mathbb{Q}})$.

######  Conjecture 1.1 (Greenberg).

Let $p$ be an odd prime number and $E_{/\mathbb{Q}}$ be an elliptic curve with
good ordinary reduction at $p$. Suppose that the Galois representation
$\rho_{E,p}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{GL}_{2}(\mathbb{F}_{p})$
is irreducible. Then, the $\mu$-invariant associated to the $p$-primary Selmer
group of $E$ over $\mathbb{Q}_{\infty}$ is equal to $0$.

Duke [Duk97] showed that the condition that $\bar{\rho}_{E,p}$ is irreducible
is satisfied for a density $1$ set of elliptic curves $E_{/\mathbb{Q}}$, when
ordered by _height_. Thus, if the set of elliptic curves with good ordinary
reduction at $p$ has positive density, the conjecture of Greenberg predicts
that $\mu_{p}(E)=0$ for a positive density of elliptic curves. I prove this
expectation for the prime $p=5$. It is indeed the case that many results in
the Iwasawa theory of elliptic curves are valid for $p\geq 5$, and the case
$p=3$ is altogether more problematic. This is precisely the reason why the
case $p=5$ is studied in this article.

###  1.2. Main result

Before stating our main result, I introduce some notation. Any elliptic curve
$E_{/\mathbb{Q}}$ admits a unique globally minimal Weierstrass equation. In
other words, $E$ is isomorphic to a unique elliptic curve $E_{A,B}$ with
minimal Weierstrass equation

$E_{A,B}\mathrel{\mathop{\mathchar 58\relax}}y^{2}=x^{3}+Ax+B,$

where $A,B$ are integers such that $\operatorname{gcd}(A^{3},B^{2})$ is not
divisible by $d^{12}$ for any integer $d>1$. Let $\mathcal{C}$ denote the set
of all globally minimal Weierstrass equations over $\mathbb{Z}$. The _height
of $E$_ is defined as $\operatorname{ht}(E)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{max}\\{|A|^{3},B^{2}\\}$. Given a real number $X>0$,
set $\mathcal{C}(X)$ to denote the set of all globally minimal Weierstrass
equations $E_{A,B}$ such that $\operatorname{ht}(E_{A,B})\leq X$. One
identifies $\mathcal{C}(X)$ with the set

$\begin{Bmatrix}&\mathinner{\\!\left\lvert A\right\rvert}\leq\sqrt[3]{X},\
\mathinner{\\!\left\lvert B\right\rvert}\leq\sqrt{X}\\\
(A,B)\in\mathbb{Z}\times\mathbb{Z}\mathrel{\mathop{\mathchar 58\relax}}&\quad
4A^{3}+27B^{2}\neq 0\\\ &\textrm{for all primes }\ell\textrm{ if
}\ell^{4}|A,\textrm{then }\ell^{6}\nmid B\end{Bmatrix}.$

######  Theorem A.

There is a set of elliptic curves $E_{/\mathbb{Q}}$ of _positive density_ such
that have good ordinary reduction at $5$ for which $\mu_{5}(E)$ and
$\lambda_{5}(E)$ are $0$. In fact, the lower density of this set is explicitly
bounded below as follows

$\liminf_{X\rightarrow\infty}\frac{\\#\\{E\in\mathcal{C}(X)\mid\mu_{5}(E)=\lambda_{5}(E)=0\\}}{\\#\mathcal{C}(X)}\geq\frac{\zeta(10)\times\prod_{\ell\geq
7}\left(1-2\ell^{-2}+\ell^{-3}\right)}{157286400}.$

For an elliptic curve $E_{/\mathbb{Q}}$ for which the Selmer group
$\operatorname{Sel}_{5}(E/\mathbb{Q})=0$, an explicit criterion due to
Greenberg shows that the $\mu_{5}$\- and $\lambda_{5}$-invariants vanish when
certain local invariants vanish, along with certain local invariants, cf.
Proposition 2.1. On the other hand, Bhargava and Shankar [BS13] study the
distribution of $5$-Selmer groups. In particular, it is shown that a positive
density of such curves $E_{/\mathbb{Q}}$ have $5$-Selmer rank $0$. Our result
is proven by adapting their methods to show that Greenberg’s criterion is
satisfied for an explicit positive density of elliptic curves.

###  1.3. Related work

In [KR21], Kundu and I studied related questions for a general prime number
$p\geq 5$. The density bounds asserted by Theorems 4.3, 4.4 and 4.5 in _loc.
cit._ are expressed in terms of a certain quantity $\mathfrak{d}_{p}^{(1)}$,
which is the proportion of elliptic curves $E/\mathbb{Q}$, ordered by height
such that $\Sh(E/\mathbb{Q})[p^{\infty}]=0$. There are no general upper bounds
known for $\mathfrak{d}_{p}^{(1)}$, however, there are precise predictions for
this quantity due to Delaunay [Del01]. Thus, in the work [KR21], it is shown
that for $p\geq 5$, the Iwasawa invariants $\mu_{p}(E)$ and $\lambda_{p}(E)$
vanish for a positive density of elliptic curves, provided one assumes
Delaunay’s heuristic. The main feature of the present work is that although
the results are only proven only for $p=5$, they are unconditional.

###  1.4. Organization and methodology

Including the introduction, the article consists of three sections. Section 2
is devoted to establishing relevant notation and for developing background on
the Iwasawa theory of elliptic curves. In section 3, I prove the main result
of this article. First, I introduce what it means for a subset of elliptic
curves to be defined by local conditions at all primes, and techniques for
computing the densities of such subsets. Then, I describe a suitable subset
$\mathcal{E}$ of $\mathcal{C}$ defined by local conditions such that if the
$5$-Selmer group of an elliptic curve $E$ in $\mathcal{E}$ vanishes, then
$\mu_{5}(E)=0$ and $\lambda_{5}(E)=0$. This set has the crucial property that
exactly half of the elliptic curves $E\in\mathcal{E}$ has root number $+1$.
Furthermore, it is _large_ in the sense of [BS15, BS13]. The final step is
then to use the strategy of Bhargava and Shankar to prove that
$\operatorname{Sel}_{5}(E/\mathbb{Q})=0$ for at least $3/8$-th of the curves
in $\mathcal{E}$. This step crucially makes use of a result of Dokchitser and
Dokchitser [DD10] (cf. [BS13, Theorem 39]). The density of $\mathcal{E}$ is
computed and thus one obtains the explicit lower bound of Theorem A.

##  2\. Iwasawa theory of elliptic curves

In this section, I discuss preliminaries on the Iwasawa theory of elliptic
curves $E_{/\mathbb{Q}}$ with good ordinary reduction at a prime number $p$. I
begin by discussing the properties of Selmer groups associated to elliptic
curves defined over number fields. Then I introduce Selmer groups considered
over the cyclotomic $\mathbb{Z}_{p}$-extension of $\mathbb{Q}$, and consider
these as modules over the Iwasawa algebra. The Iwasawa $\mu$\- and
$\lambda$-invariants are then defined. I conclude the section with Proposition
2.1, which is a numerical criterion that gives conditions for the vanishing of
these invariants. For a more comprehensive account on the Iwasawa theory of
elliptic curves, see for instance [Gre99, Gre01].

###  2.1. Selmer groups associated to elliptic curves

Let $p\geq 5$ be a prime number and $E$ be an elliptic curve defined over
$\mathbb{Q}$. Let $F/\mathbb{Q}$ be a number field, and choose an algebraic
closure $\bar{F}$ of $F$. Denote by $E(F)$ (resp. $\Sh(E/F)$) the Mordell–Weil
group (Tate–Shafarevich group) of $E$ over $F$. Set $\Sigma_{F}$ to be the set
of nonarchimedian primes of $F$; for $v\in F$, let $\bar{F}_{v}/F_{v}$ be an
algebraic closure. Given a natural number $n$, I set $E[n]$ to be the
$n$-torsion subgroup of $E(\bar{F})$. Set $\operatorname{G}_{F}$ (resp.
$\operatorname{G}_{F_{v}}$) to denote the absolute Galois group
$\operatorname{Gal}(\bar{\mathbb{Q}}/F)$ (resp.
$\operatorname{Gal}(\bar{F}_{v}/F_{v})$). For each prime
$\ell\in\Sigma_{\mathbb{Q}}$, I choose an embedding
$\iota_{\ell}\mathrel{\mathop{\mathchar
58\relax}}\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{\ell}$. This
induces an injection of Galois groups $j_{\ell}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{\mathbb{Q}_{\ell}}\hookrightarrow\operatorname{G}_{\mathbb{Q}}$.
Given a prime $v\in\Sigma_{F}$, let $\ell$ be the rational prime for which
$v|\ell$ and identify $\bar{F}_{v}$ with $\bar{\mathbb{Q}}_{\ell}$. For
$v\in\Sigma_{F}$, let $j_{v}\mathrel{\mathop{\mathchar
58\relax}}\operatorname{G}_{F_{v}}\hookrightarrow\operatorname{G}_{F}$ be the
inclusion through which $j_{\ell}$ factors. This induces a restriction map in
cohomology

$\operatorname{res}_{v}\mathrel{\mathop{\mathchar
58\relax}}H^{1}(F,\cdot)\rightarrow H^{1}(F_{v},\cdot).$

Setting $E[p^{\infty}]\mathrel{\mathop{\mathchar 58\relax}}=\bigcup_{n\geq
1}E[p^{n}]$, denote by
$\operatorname{res}_{v}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}H^{1}(F,E[p^{\infty}])\rightarrow H^{1}(F_{v},E)[p^{\infty}]$ the
composite

$H^{1}(F,E[p^{\infty}])\xrightarrow{\operatorname{res}_{v}}H^{1}(F_{v},E[p^{\infty}])\rightarrow
H^{1}(F_{v},E)[p^{\infty}].$

Here, the second map arises from the Kummer sequence. Setting $\Phi_{F}$ to
denote the product $\prod_{v\in\Sigma_{F}}\operatorname{res}_{v}^{\prime}$,
the $p$-primary Selmer group of $E$ over $F$ is defined as follows

$\operatorname{Sel}_{p^{\infty}}(E/F)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(F,E[p^{\infty}])\xrightarrow{\Phi_{F}}\prod_{v}H^{1}(F_{v},E)[p^{\infty}]\right).$

On the other hand, the $p$-Selmer group of $E$ is defined as follows

$\operatorname{Sel}_{p}(E/F)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{ker}\left(H^{1}(F,E[p])\rightarrow\prod_{v}H^{1}(F_{v},E)[p]\right).$

The $p$-primary Selmer group fits into a natural short exact sequence

$0\rightarrow
E(F)\otimes\mathbb{Q}_{p}/\mathbb{Z}_{p}\rightarrow\operatorname{Sel}_{p^{\infty}}(E/F)\rightarrow\Sh(E/F)[p^{\infty}]\rightarrow
0.$

The Shafarevich–Tate conjecture predicts that $\Sh(E/F)[p^{\infty}]$ is
finite. In particular, this conjecture implies that

$\operatorname{corank}_{\mathbb{Z}_{p}}\operatorname{Sel}_{p^{\infty}}(E/F)=\operatorname{rank}E(F).$

###  2.2. Iwasawa theory of Selmer groups

Set $\mu_{p^{\infty}}\mathrel{\mathop{\mathchar
58\relax}}=\bigcup_{n}\mu_{p^{n}}$, and
$\mathbb{Q}(\mu_{p^{\infty}})\mathrel{\mathop{\mathchar
58\relax}}=\bigcup_{n}\mathbb{Q}(\mu_{p^{n}})$. The _cyclotomic
$\mathbb{Z}_{p}$-extension_ $\mathbb{Q}_{\infty}/\mathbb{Q}$ is the unique
$\mathbb{Z}_{p}$-extension of $\mathbb{Q}$ which is contained in
$\mathbb{Q}(\mu_{p^{\infty}})$. For $n\geq 0$, the degree $p^{n}$-extension of
$\mathbb{Q}$ contained in $\mathbb{Q}_{\infty}$ is denoted $\mathbb{Q}_{n}$,
and referred to as the _$n$ -th layer_ in $\mathbb{Q}_{\infty}$. Thus one has
an infinite tower of extensions

$\mathbb{Q}=\mathbb{Q}_{0}\subset\mathbb{Q}_{1}\subset\mathbb{Q}_{2}\subset\cdots\mathbb{Q}_{n}\subset\mathbb{Q}_{n+1}\subset\cdots\subset\mathbb{Q}_{\infty}.$

Set $\Gamma\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Gal}(\mathbb{Q}_{\infty}/\mathbb{Q})$ and for $n\geq
0$, identify $\Gamma/\Gamma^{p^{n}}$ with the Galois group
$\operatorname{Gal}(\mathbb{Q}_{n}/\mathbb{Q})$. The _Iwasawa algebra_
$\Lambda$ is the pro-completed group algebra

$\Lambda\mathrel{\mathop{\mathchar
58\relax}}=\varprojlim_{n}\mathbb{Z}_{p}[\Gamma/\Gamma^{p^{n}}].$

The main object of study is the $p$-primary Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})$, defined to be the
direct limit

$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})\mathrel{\mathop{\mathchar
58\relax}}=\varinjlim_{n}\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{n})$

with respect to natural restriction maps.

Let $\gamma\in\Gamma$ be a topological generator and set
$T\mathrel{\mathop{\mathchar 58\relax}}=(\gamma-1)\in\Lambda$. With respect to
the choice of $\gamma$, I identify $\Lambda$ with the formal power series ring
$\mathbb{Z}_{p}\llbracket T\rrbracket$. A monic polynomial
$f(T)\in\mathbb{Z}_{p}\llbracket T\rrbracket$ is said to be a _distinguished
polynomial_ if its nonleading coefficients are all divisible by $p$. Let $M$
be a module over $\Lambda$ and set $M^{\vee}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Hom}_{\mathbb{Z}_{p}}\left(M,\mathbb{Q}_{p}/\mathbb{Z}_{p}\right)$.
Say that $M$ is cofinitely generated (resp. cotorsion) as a $\Lambda$-module
if $M^{\vee}$ is finitely generated (resp. torsion) as a module over
$\Lambda$. Suppose that $M$ is cofinitely generated and cotorsion over
$\Lambda$. Let $M$ and $M^{\prime}$ be cofinitely generated and cotorsion
$\Lambda$-modules. Then, $M$ and $M^{\prime}$ are said to be _pseudo-
isomorphic_ if there is a map $\phi\mathrel{\mathop{\mathchar
58\relax}}M\rightarrow M^{\prime}$ of $\Lambda$-modules, whose kernel and
cokernel are finite. By the structure theory of finitely generated and torsion
$\Lambda$-modules, any cofinitely generated and cotorsion $\Lambda$-module $M$
is pseudo-isomorphic to a module $M^{\prime}$ whose Pontryagin dual is given
as a direct sum of cyclic torsion $\Lambda$-modules

(2.1)
$(M^{\prime})^{\vee}\simeq\left(\bigoplus_{i=1}^{s}\frac{\Lambda}{(p^{n_{i}})}\right)\oplus\left(\bigoplus_{j=1}^{t}\frac{\Lambda}{(f_{j}(T))}\right).$

Here, $s$ and $t$ are natural numbers (possibly $0$),
$n_{i}\in\mathbb{Z}_{\geq 1}$ and $f_{j}(T)$ are distinguished polynomials.
Then, one defines the Iwasawa $\mu$\- and $\lambda$-invariants as follows

$\mu_{p}(M)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{i=1}^{s}n_{i}\text{ and
}\lambda_{p}(M)\mathrel{\mathop{\mathchar
58\relax}}=\sum_{j=1}^{t}\operatorname{deg}f_{j}.$

Here, if $s=0$ (resp. $t=0$), the sum $\sum_{i=1}^{s}n_{i}$ (resp.
$\sum_{j=1}^{t}\operatorname{deg}f_{j}$) is interpreted as being equal to $0$.
These invariants are independent of the choice of $M^{\prime}$ and the
isomorphism (2.1); I refer to [Was97, Chapter 13] for further details.

Assume that $E$ has good ordinary reduction at $p$. It follows from work of
Kato [Kat04] and Rubin [Rub81] that
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})$ is cofinitely
generated and cotorsion over $\Lambda$. Set $\mu_{p}(E)$ (resps.
$\lambda_{p}(E)$) to denote the Iwasawa $\mu$-invariant (resp.
$\lambda$-invariant) of the dual Selmer group
$\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})^{\vee}$. Denote by
$\widetilde{E}$ the reduced curve at $p$, and $\widetilde{E}(\mathbb{F}_{p})$
its group of $\mathbb{F}_{p}$-rational points. The prime $p$ is said to
_anomalous_ if $p$ divides $\\#\widetilde{E}(\mathbb{F}_{p})$. For a prime
$\ell$, set $c_{\ell}(E)$ to denote the _Tamagawa number at $\ell$_.

######  Proposition 2.1 (Greenberg).

Let $E$ be an elliptic curve and $p\geq 5$ be a prime number of good ordinary
reduction for $E$. Assume that the following conditions are satisfied.

  1. (1)

The Selmer group $\operatorname{Sel}_{p}(E/\mathbb{Q})=0$.

  2. (2)

The prime $p$ is non-anomalous, i.e., does not divide
$\\#\widetilde{E}(\mathbb{F}_{p})$.

  3. (3)

The prime $p$ does not divide $c_{\ell}(E)$ for all primes $\ell\neq p$.

Then, one has that $\operatorname{Sel}_{p^{\infty}}(E/\mathbb{Q}_{\infty})=0$.
In particular, one has that

$\mu_{p}(E)=0\text{ and }\lambda_{p}(E)=0.$

###### Proof.

The result follows from [Gre99, remark following Proposition 3.8, p.80, ll.
19–26]. ∎

##  3\. Density results

In this section I prove the main result of the article. I begin by recalling a
density result due to Brumer.

######  Lemma 3.1.

With respect to notation above, one has the following asymptotic

$\\#\mathcal{C}(X)=\frac{4X^{5/6}}{\zeta(10)}+O\left(\sqrt{X}\right).$

###### Proof.

The above result is [Bru92, Lemma 4.3]. ∎

I discuss the notion of what it means for a subset $\Phi\subseteq\mathcal{C}$
to be defined by _congruence conditions_. Given a prime $\ell$, a subset
$\Phi_{\ell}\subseteq\mathcal{C}$ is defined by a congruence condition if
there is

  * •

an integer $n_{\ell}\geq 6$,

  * •

a set of residue classes

$\bar{\Phi}_{\ell}\subseteq\\{(\bar{A},\bar{B})\in\mathbb{Z}/\ell^{m_{\ell}}\times\mathbb{Z}/\ell^{m_{\ell}}\mid\ell^{4}\nmid\bar{A}\text{
or }\ell^{6}\nmid\bar{B}\\},$

such that $\Phi_{\ell}$ consists of all pairs $(A,B)\in\mathcal{C}$ such that

$\left(A\mod{\ell^{m_{\ell}}},B\mod{\ell^{m_{\ell}}}\right)\in\bar{\Phi}_{\ell}.$

The _local density_ of $\Phi_{\ell}$ is defined as follows

$\mathfrak{d}(\Phi_{\ell})\mathrel{\mathop{\mathchar
58\relax}}=\frac{\\#\bar{\Phi}_{\ell}}{\\#\\{(\bar{A},\bar{B})\in\mathbb{Z}/\ell^{m_{\ell}}\times\mathbb{Z}/\ell^{m_{\ell}}\mid\ell^{4}\nmid\bar{A}\text{
or
}\ell^{6}\nmid\bar{B}\\}}=\frac{\\#\bar{\Phi}_{\ell}}{(1-\ell^{-10})\ell^{2m_{\ell}}}.$

Let $\ell_{1},\dots,\ell_{k}$ be distinct prime numbers for which there are
local conditions $\Phi_{\ell_{i}}$ defined. Let $m\mathrel{\mathop{\mathchar
58\relax}}=\prod_{i=1}^{k}\ell_{i}$, set $\Phi_{m}\mathrel{\mathop{\mathchar
58\relax}}=\bigcap_{i=1}^{k}\Phi_{\ell_{i}}$ and

$\mathfrak{d}(\Phi_{m})\mathrel{\mathop{\mathchar
58\relax}}=\prod_{i=1}^{k}\mathfrak{d}(\Phi_{\ell_{i}}).$

A set $\Phi\subseteq\mathcal{C}$ is said to be _defined by finitely many
congruence conditions_ if $\Phi=\Phi_{m}=\bigcap_{\ell|m}\Phi_{\ell}$ for some
squarefree number $m$. More generally, $\Phi\subset\mathcal{C}$ is simply
defined by local conditions if $\Phi=\bigcap_{\ell}\Phi_{\ell}$, where the
intersection runs over all primes $\ell$ and $\Phi_{\ell}$ is a local
condition at $\ell$. Given a real number $X>0$, set

$\Phi(X)\mathrel{\mathop{\mathchar
58\relax}}=\Phi\cap\mathcal{C}(X)=\\{(A,B)\in\Phi\mid\operatorname{ht}(E_{A,B})\leq
X\\}.$

If $\Phi$ is any subset of $\mathcal{C}$, the density of $\Phi$ is defined to
be the following limit

$\mathfrak{d}(\Phi)\mathrel{\mathop{\mathchar
58\relax}}=\lim_{X\rightarrow\infty}\frac{\\#\Phi(X)}{\\#\mathcal{C}(X)},$

provided the limit exists. The upper and lower densities are defined as
follows

$\begin{split}&\overline{\mathfrak{d}}(\Phi)\mathrel{\mathop{\mathchar
58\relax}}=\limsup_{X\rightarrow\infty}\frac{\\#\Phi(X)}{\\#\mathcal{C}(X)},\\\
&\underline{\mathfrak{d}}(\Phi)\mathrel{\mathop{\mathchar
58\relax}}=\liminf_{X\rightarrow\infty}\frac{\\#\Phi(X)}{\\#\mathcal{C}(X)}.\end{split}$

######  Theorem 3.2 (Bhargava–Shankar).

In any family $\Phi$ defined by finitely many congruence conditions on the
coefficients $A$ and $B$, the average size of
$\operatorname{Sel}_{5}(E/\mathbb{Q})$ is $6$. In greater detail,

$\lim_{X\rightarrow\infty}\frac{\left(\sum_{E\in\Phi(X)}\\#\operatorname{Sel}_{5}(E/\mathbb{Q})\right)}{\\#\Phi(X)}=6.$

More generally, it is conjectured that the average size of
$\operatorname{Sel}_{n}(E/\mathbb{Q})$ is $\sigma(n)=\sum_{d|n}d$, cf. [PR12].

######  Proposition 3.3.

Consider a subset $\Phi=\bigcap_{\ell}\Phi_{\ell}$ of $\mathcal{C}$ is defined
by congruence conditions $\Phi_{\ell}$ of density $\mathfrak{d}(\Phi_{\ell})$.
With respect to notation above, suppose that

  1. (1)

$\sum_{\ell}\left(1-\mathfrak{d}(\Phi_{\ell})\right)<\infty$.

  2. (2)

The product $\prod_{\ell}\mathfrak{d}(\Phi_{\ell})$ converges.

  3. (3)

Let $\Phi_{\ell}^{\prime}$ be the complement
$\mathcal{C}\setminus\Phi_{\ell}$. Then, there is an absolute constant $C>0$
independent of $\ell$ such that

$\Phi_{\ell}^{\prime}(X)\leq
C\left(1-\mathfrak{d}(\Phi_{\ell})\right)X^{\frac{5}{6}}.$

Then, I have that

$\\#\Phi(X)\sim\left(\prod_{\ell}\mathfrak{d}(\Phi_{\ell})\right)\frac{4X^{5/6}}{\zeta(10)}.$

###### Proof.

Given a positive real number $Z$, set $\Phi_{Z}\mathrel{\mathop{\mathchar
58\relax}}=\bigcap_{\ell\leq Z}\Phi_{\ell}$, where $\ell$ ranges over the
prime numbers $\leq Z$. Recall from Lemma 3.1 that

$\\#\mathcal{C}(X)\sim\frac{4X^{5/6}}{\zeta(10)}.$

Since there are only finitely many local conditions defining $\Phi_{Z}$, I
have that

$\begin{split}\\#\Phi_{Z}(X)&\sim\left(\prod_{\ell\leq
Z}\mathfrak{d}(\Phi_{\ell})\right)\times\\#\mathcal{C}(X)\\\
&\sim\left(\prod_{\ell\leq
Z}\mathfrak{d}(\Phi_{\ell})\right)\times\frac{4X^{5/6}}{\zeta(10)}.\end{split}$

Since $\Phi(X)$ is contained in $\Phi_{Z}(X)$, I find that

(3.1)
$\limsup_{X\rightarrow\infty}\frac{\\#\Phi(X)}{X^{5/6}}\leq\lim_{X\rightarrow\infty}\frac{\\#\Phi_{Z}(X)}{X^{5/6}}=\left(\prod_{\ell\leq
Z}\mathfrak{d}(\Phi_{\ell})\right)\times\frac{4}{\zeta(10)}.$

Thus, taking $Z\rightarrow\infty$, I find that

$\limsup_{X\rightarrow\infty}\frac{\\#\Phi(X)}{X^{5/6}}\leq\left(\prod_{\ell\leq
Z}\mathfrak{d}(\Phi_{\ell})\right)\times\frac{4}{\zeta(10)}.$

Since

$\Phi_{Z}(X)\subseteq\Phi(X)\cup\left(\bigcup_{\ell>Z}\Phi_{\ell}^{\prime}(X)\right),$

I find that

$\begin{split}\liminf_{X\rightarrow\infty}\frac{\\#\Phi(X)}{X^{5/6}}\geq&\lim_{X\rightarrow\infty}\frac{\\#\Phi_{Z}(X)}{X^{5/6}}-C\sum_{\ell>Z}\left(1-\mathfrak{d}(\Phi_{\ell})\right),\\\
=&\left(\prod_{\ell\leq
Z}\mathfrak{d}(\Phi_{\ell})\right)\times\frac{4X^{5/6}}{\zeta(10)}-C\sum_{\ell>Z}\left(1-\mathfrak{d}(\Phi_{\ell})\right).\end{split}$

Letting $Z\rightarrow\infty$, I find that

(3.2)
$\liminf_{X\rightarrow\infty}\frac{\\#\Phi(X)}{X^{5/6}}\geq\left(\prod_{\ell}\mathfrak{d}(\Phi_{\ell})\right)\times\frac{4X^{5/6}}{\zeta(10)}.$

From (3.1) and (3.2), I deduce that

$\\#\Phi(X)\sim\left(\prod_{\ell}\mathfrak{d}(\Phi_{\ell})\right)\frac{4X^{5/6}}{\zeta(10)}.$

∎

Given $E_{A,B}\in\mathcal{C}$, set $\Delta_{A,B}\mathrel{\mathop{\mathchar
58\relax}}=4A^{3}+27B^{2}$. For all primes $\ell$ define $\Pi_{\ell}$ to be
the set of all pairs $(A,B)\in\mathcal{C}$ such that
$\ell^{2}\nmid\Delta_{A,B}$.

######  Definition 3.4.

I define a subset $\mathcal{E}=\bigcap_{\ell}\mathcal{E}_{\ell}$ of
$\mathcal{C}$ defined by local conditions at all primes.

  * •

For all primes $\ell\neq 2,5$, I set
$\mathcal{E}_{\ell}\mathrel{\mathop{\mathchar 58\relax}}=\Pi_{\ell}$.

  * •

Let $\mathcal{E}_{2}$ be set of pairs $(A,B)\in\mathcal{C}$ such that $A>0$,
$A=4A^{\prime}$, $B=16B^{\prime}$, where $A^{\prime}\equiv 189\mod{256}$ and
$B^{\prime}\equiv\pm 1\mod{256}$.

  * •

Define $\mathcal{E}_{5}$ to consist of pairs $(A,B)\equiv(1,\pm 1)\mod{5}$.

For $E\in\mathcal{E}$, let $E_{-1}=E_{A,-B}$ be the quadratic twist of $E$ by
$-1$. It is easy to see that $E_{-1}\in\mathcal{E}$.

######  Lemma 3.5.

Let $E=E_{A,B}$ belong to $\mathcal{E}$. Then the following assertions hold.

  1. (1)

Then $E$ has additive reduction at $2$, and its j-invariant is a $2$-adic
unit.

  2. (2)

Let $\Delta^{\prime}$ denote the prime to $2$ part of the discriminant of $E$,
then, $\Delta^{\prime}>0$ and $\Delta^{\prime}\equiv 1\mod{4}$.

  3. (3)

The elliptic curves $E$ has good ordinary reduction at $5$. Moreover, the
reduction at $5$ is non-anomalous, i.e.,
$5\nmid\\#\widetilde{E}(\mathbb{F}_{5})$.

  4. (4)

The Tamagawa product $\prod_{\ell}c_{\ell}(E)$ is not divisible by $5$.

###### Proof.

Throughout this proof, I set $\Delta\mathrel{\mathop{\mathchar
58\relax}}=\Delta_{A,B}=4A^{3}+27B^{2}$. Part (1) is a direct consequence of
Tate’s algorithm, cf. [Sil94, step 7, p. 367]. The j-invariant is computed as
follows

$j(E)=1728\frac{4A^{3}}{4A^{3}+27B^{2}}=\frac{27\times 2^{6}\times A^{\prime
3}}{(A^{\prime 3}+27B^{\prime 2})}.$

One finds that

$A^{\prime 3}+27B^{\prime 2}\equiv 189^{3}+27\equiv 64\mod{256}.$

This implies in particular that $j(E)$ is a $2$-adic unit. For (2), note that

$\Delta=4A^{3}+27B^{2}=2^{8}(A^{\prime 3}+27B^{\prime 2}),$

and

$A^{\prime 3}+27B^{\prime 2}=2^{6}(4n+1),$

where $n$ is a positive natural number. Therefore, $\Delta^{\prime}>0$ and is
$\equiv 1\mod{4}$.

For part (3), consider the curves $\mathcal{A}\mathrel{\mathop{\mathchar
58\relax}}y^{2}=x^{3}+x+1$ and $\mathcal{A}^{\prime}\mathrel{\mathop{\mathchar
58\relax}}y^{2}=x^{3}+x-1$. Then, according to data on LMFDB, the Fourier
coefficients at $5$ for the associated modular form are given by
$a_{5}(\mathcal{A})=-3$, $a_{5}(\mathcal{A}^{\prime})=-3$. Thus, I find that

$\\#\widetilde{\mathcal{A}}(\mathbb{F}_{5}),\\#\widetilde{\mathcal{A}^{\prime}}(\mathbb{F}_{5})=9.$

Thus, given any elliptic curve $E\in\mathcal{E}$, one finds that
$\\#\widetilde{E}(\mathbb{F}_{5})=9$. Thus, $E$ has good ordinary reduction
and $5\nmid\\#\widetilde{E}(\mathbb{F}_{5})$.

I now prove part (4). Note that for all primes $\ell\neq 2$, one has that
$\ell^{2}\nmid\Delta$. This implies that $5\nmid c_{\ell}(E)$ for all primes
$\ell\neq 2$, cf. [Sil09, 2nd column of Table on p. 448]. In fact, the
assertion only requires that $\ell^{5}\nmid\Delta$. Note that since $E$ has
additive reduction at $2$, one deduces that $c_{2}(E)\leq 4$. This implies
that $5\nmid\prod_{\ell}c_{\ell}(E)$. ∎

######  Proposition 3.6.

Let $E\in\mathcal{E}$ be an elliptic curve such that
$\operatorname{Sel}_{5}(E/\mathbb{Q})=0$. Then, it follows that
$\operatorname{Sel}_{5^{\infty}}(E/\mathbb{Q}_{\infty})=0$. In particular,
$\mu_{5}(E)=\lambda_{5}(E)=0$.

###### Proof.

Setting $p\mathrel{\mathop{\mathchar 58\relax}}=5$, the result follows from
Lemma 3.5 and Proposition 2.1. ∎

I shall now show that the set of elliptic curves $E\in\mathcal{E}$ for which
$\operatorname{Sel}_{5}(E/\mathbb{Q})$ has positive density, and moreover,
give a lower bound for this density. Given an elliptic curve $E$, set
$\omega(E)$ to denote its root number. Let $\mathcal{E}^{+}$ (resp.
$\mathcal{E}^{-}$) be the subset of curves in $\mathcal{E}$ for which
$\omega(E)=1$ (resp. $\omega(E)=-1$).

######  Proposition 3.7.

Let $E\in\mathcal{E}$, the following assertions hold

  1. (1)

$\omega(E)=-\omega(E_{-1})$.

  2. (2)

Exactly half of the curves in $\mathcal{E}$ have root number $+1$ (resp.
$-1$). In other words,

$\lim_{X\rightarrow\infty}\left(\frac{\\#\mathcal{E}^{\pm}(X)}{\\#\mathcal{E}(X)}\right)=\frac{1}{2}.$

###### Proof.

For $E\in\mathcal{E}$, it is easy to see that $E_{-1}\in\mathcal{E}$ as well.
Recall from Lemma 3.5 and the conditions defining $\mathcal{E}_{\ell}$, that
for $E\in\mathcal{E}$, the following conditions are satisfied

  * •

$E$ and $E_{-1}$ both have additive reduction at $2$ and their $j$-invariants
are $2$-adic units.

  * •

$E$ has squarefree discriminant away from $2$,

  * •

$\Delta^{\prime}\equiv 1\mod{4}$, where $\Delta^{\prime}$ is the positive odd
part of the discriminant of $E$.

It then follows from the above properties that $\omega(E_{-1})=-\omega(E)$,
see [BS15, p. 618] for further details. Since the height of $E$ is the same as
that of $E_{-1}$, part (2) is a direct consequence of (1). ∎

######  Lemma 3.8.

The following assertions hold.

  1. (1)

For $\ell>5$, one has that
$(1-\ell^{-10})\mathfrak{d}(\mathcal{E}_{\ell})=(1-2\ell^{-2}+\ell^{-3})$.

  2. (2)

One has that $\\#\mathcal{E}_{\ell}^{\prime}(X)<C\ell^{-2}X^{5/6}$ for some
absolute constant $C>0$.

  3. (3)

One has that $(1-3^{-10})\mathfrak{d}(\mathcal{E}_{3})=(1-3^{-2})$.

  4. (4)

$(1-2^{-10})\mathfrak{d}(\mathcal{E}_{2})=1/4194304$ and
$(1-5^{-10})\mathfrak{d}(\mathcal{E}_{5})=2/25$.

###### Proof.

For part (1), I estimate the number of pairs
$(a,b)\in\mathbb{Z}/\ell^{6}\times\mathbb{Z}/\ell^{6}$ such that
$4a^{3}\not\equiv 27b^{2}\mod{\ell^{2}}$. Since $\ell\neq 2,3$, one can
replace $a$ with $3a$ and $b$ with $2b$ and thus are required to estimate the
number of such pairs for which $a^{3}\neq b^{2}$. Setting
$m_{\ell}\mathrel{\mathop{\mathchar 58\relax}}=6$, it follows from [CS23,
Lemma 3.6] that
$\\#\bar{\mathcal{E}_{\ell}}=\ell^{12}\left(1-2\ell^{-2}+\ell^{-3}\right)$,
and therefore,

$\mathfrak{d}(\mathcal{E}_{\ell})=\frac{(1-2\ell^{-2}+\ell^{-3})}{1-\ell^{-10}}.$

Part (2) proven in _loc. cit._ immediately following Lemma 3.6.

The proofs of parts (3) and (4) are easy and are thus omitted. ∎

######  Proposition 3.9.

With respect to notation above, the following assertions hold.

  1. (1)

The set $\mathcal{E}$ has density equal to

$\mathfrak{d}(\mathcal{E})=\zeta(10)\times\prod_{\ell\geq
7}\left(1-2\ell^{-2}+\ell^{-3}\right)\times(1-3^{-2})\times\frac{1}{52428800}.$

  2. (2)

The densities of $\mathcal{E}^{+}$ and $\mathcal{E}^{-}$ are given by

$\mathfrak{d}(\mathcal{E}^{+})=\mathfrak{d}(\mathcal{E}^{-})=\zeta(10)\times\prod_{\ell\geq
7}\left(1-2\ell^{-2}+\ell^{-3}\right)\times(1-3^{-2})\times\frac{1}{104857600}.$

In particular, $\mathcal{E}^{+}$ and $\mathcal{E}^{-}$ have positive density.

###### Proof.

Part (1) of the result follows from Proposition 3.3 and Lemma 3.8. Part (2) is
a direct consequence of part (1) and Proposition 3.7. ∎

######  Theorem 3.10 (Bhargava–Shankar).

When all elliptic curves $E\in\mathcal{E}$ are ordered according to height,
the average size of $\operatorname{Sel}_{5}(E/\mathbb{Q})$ is $6$. In other
words, one has the following limit

$\lim_{X\rightarrow\infty}\frac{\sum_{E\in\mathcal{E}(X)}\\#\operatorname{Sel}_{5}(E/\mathbb{Q})}{\\#\mathcal{E}(X)}=6.$

###### Proof.

For all but finitely many primes, the condition $\mathcal{E}_{\ell}$ coincides
with $\Pi_{\ell}$. Thus, the family $\mathcal{E}$ is _large_ in the sense
[BS13, p.21, ll.-6 to -4]. The result then follows from Theorem 31 of _loc.
cit._ ∎

I conclude this article by proving the main result.

###### Proof of Theorem A.

As explained in the proof of Theorem 3.10, the set $\mathcal{E}$ is _large_ ,
and the average size of the $5$-Selmer group in this set is $6$. It then
follows from [BS13, Proposition 40] that
$\operatorname{Sel}_{5}(E/\mathbb{Q})=0$ for at least $3/8$-th of the curves
in $\mathcal{E}$. Proposition 3.6 asserts that for each $E\in\mathcal{E}$ such
that $\operatorname{Sel}_{5}(E/\mathbb{Q})=0$, one has that $\mu_{5}(E)=0$ and
$\lambda_{5}(E)=0$. Therefore, I conclude that

$\liminf_{X\rightarrow\infty}\frac{\\#\\{E\in\mathcal{C}(X)\mid\mu_{5}(E)=\lambda_{5}(E)=0\\}}{\\#\\#\mathcal{C}(X)}\geq\frac{3}{8}\mathfrak{d}(\mathcal{E}).$

According to Proposition 3.9,

$\mathfrak{d}(\mathcal{E})=\zeta(10)\times\prod_{\ell\geq
7}\left(1-2\ell^{-2}+\ell^{-3}\right)\times(1-3^{-2})\times\frac{1}{52428800},$

from which the result follows. ∎

## References

  * [Bru92] Armand Brumer.  The average rank of elliptic curves I.  Invent. Math., 109(1):445–472, 1992.
  * [BS13] Manjul Bhargava and Arul Shankar.  The average size of the 5-selmer group of elliptic curves is 6, and the average rank is less than 1.  arXiv preprint arXiv:1312.7859, 2013.
  * [BS15] Manjul Bhargava and Arul Shankar.  Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0.  Ann. of Math. (2), 181(2):587–621, 2015.
  * [CS23] John E. Cremona and Mohammad Sadek.  Local and global densities for Weierstrass models of elliptic curves.  Math. Res. Lett., 30(2):413–461, 2023.
  * [DD10] Tim Dokchitser and Vladimir Dokchitser.  On the Birch-Swinnerton-Dyer quotients modulo squares.  Ann. of Math. (2), 172(1):567–596, 2010.
  * [Del01] Christophe Delaunay.  Heuristics on Tate-Shafarevitch groups of elliptic curves defined over $\mathbb{Q}$.  Experiment. Math., 10(2):191–196, 2001.
  * [Duk97] William Duke.  Elliptic curves with no exceptional primes.  C. R. Acad. Sci. Paris Sér. I Math., 325(8):813–818, 1997.
  * [FW79] Bruce Ferrero and Lawrence C. Washington.  The Iwasawa invariant $\mu_{p}$ vanishes for abelian number fields.  Ann. of Math. (2), 109(2):377–395, 1979.
  * [Gre99] Ralph Greenberg.  Iwasawa theory for elliptic curves.  In Arithmetic theory of elliptic curves (Cetraro, 1997), volume 1716 of Lecture Notes in Math., pages 51–144. Springer, Berlin, 1999.
  * [Gre01] Ralph Greenberg.  Introduction to Iwasawa theory for elliptic curves.  In Arithmetic algebraic geometry (Park City, UT, 1999), volume 9 of IAS/Park City Math. Ser., pages 407–464. Amer. Math. Soc., Providence, RI, 2001.
  * [Iwa73] Kenkichi Iwasawa.  On ${\bf Z}_{l}$-extensions of algebraic number fields.  Ann. of Math. (2), 98:246–326, 1973.
  * [Kat04] Kazuya Kato.  $p$-adic Hodge theory and values of zeta functions of modular forms.  Number 295, pages ix, 117–290. 2004.  Cohomologies $p$-adiques et applications arithmétiques. III.
  * [KR21] Debanjana Kundu and Anwesh Ray.  Statistics for Iwasawa invariants of elliptic curves.  Trans. Amer. Math. Soc., 374(11):7945–7965, 2021.
  * [Maz72] Barry Mazur.  Rational points of abelian varieties with values in towers of number fields.  Invent. Math., 18:183–266, 1972.
  * [PR12] Bjorn Poonen and Eric Rains.  Random maximal isotropic subspaces and Selmer groups.  J. Amer. Math. Soc., 25(1):245–269, 2012.
  * [Rub81] Karl Rubin.  Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer.  Invent. Math., 64(3):455–470, 1981.
  * [Sil94] Joseph H. Silverman.  Advanced topics in the arithmetic of elliptic curves, volume 151 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1994.
  * [Sil09] Joseph H. Silverman.  The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics.  Springer, Dordrecht, second edition, 2009.
  * [Was97] Lawrence C. Washington.  Introduction to cyclotomic fields, volume 83 of Graduate Texts in Mathematics.  Springer-Verlag, New York, second edition, 1997.