# Reversal in Thermally Driven Rotation of Chiral Liquid Crystal Droplets

Shunsuke Takano<EMAIL_ADDRESS>Graduate School of Science
and Engineering, Waseda University, TWIns, 2-2 Wakamatsu-cho, Shinjuku, Tokyo,
162-8480, Japan  Takuya Nakanishi  Comprehensive Research Organization, Waseda
University, TWIns, 2-2 Wakamatsu-cho, Shinjuku, Tokyo, 162-8480, Japan  Kenta
Nakagawa  Comprehensive Research Organization, Waseda University, TWIns, 2-2
Wakamatsu-cho, Shinjuku, Tokyo, 162-8480, Japan  Toru Asahi<EMAIL_ADDRESS>Graduate School of Science and Engineering, Waseda University, TWIns, 2-2
Wakamatsu-cho, Shinjuku, Tokyo, 162-8480, Japan  Comprehensive Research
Organization, Waseda University, TWIns, 2-2 Wakamatsu-cho, Shinjuku, Tokyo,
162-8480, Japan

(28 August 2024)

###### Abstract

For chiral liquid crystals that express topologically protected defects and
thermally driven mechanical rotation, the size- and temperature-driven
reversal of the rotational direction of their droplets was demonstrated even
under a fixed temperature gradient. This unconventional reversal indicates the
dependence of thermomechanical coupling on the molecular orientational order,
this dependence is justified through an examination of the size, temperature,
and molecular structure as well as by phenomenological arguments on the order
parameter.

††preprint: APS/123-QED

Chirality is an abundant frontier for the emergence of topological structures
and transport dynamics in condensed matter. In liquid crystals (LCs),
chirality allows a pseudoscalar (Lifshitz invariant)
$\mathrm{\mathbf{n}}\cdot\nabla\times\mathrm{\mathbf{n}}=n_{i}\epsilon_{ijk}\nabla_{j}n_{k}\
(i,j,k=1,2,3)$ to be nonzero [1, 2], where $\mathrm{\mathbf{n}}$ is the
director, a locally defined unit vector along which LC molecules tend to
orient. The pseudoscalar stabilizes a spontaneous helical director field,
similar to Dzyaloshinsky-Moriya interaction in magnets [3, 4]. The helical
field is utilized for displays, thermometers, and asymmetric reactions for
helical polymerization [5, 6].

When chiral nematic (N*) LCs are confined in a spherical droplet, the periodic
layers of the helical director field deform into concentric shells under
degenerate planar anchoring (Fig. 1). Through these concentrically layered
shells, a line-like topological defect stretches from the center to the
surface of the droplet due to geometrical frustration prohibiting the director
from continuously covering the entire sphere. The defect has a winding number
(or the topological charge) of $+2$, equal to the Euler characteristic of the
sphere [7]. This orientational structure is known as the radial spherical
structure (RSS) and displays distinct textures, namely concentric circles with
a radial line and a double spiral [see Figs. 1(a) and (b), respectively] [8,
9, 10, 11, 12]. Within the defect, the molecular orientational order is more
violated than in the surroundings, but the loss of free energy is restricted,
thus providing a net stabilization of the system. Some models of the director
field have been proposed to suppress the reduction of the order in the defect
[13, 14]. Skyrmions, vortex-like structures formed in blue phases [15, 16] and
in confined spaces [17], have been noted to contribute to the stabilization of
the defect [12]. Also, in defects appearing as vortices in superfluid and in
Type II superconductors, the amplitude of the order parameter (OP)
representing the density of Bose-Einstein condensates decreases and usually
reaches zero [18, 19]. The vortices in superfluid $\mathrm{{}^{4}He}$
capturing solid nanoparticles have been optically visualized [20], and
superconductors permit the magnetic field to penetrate only through the
latticed vortices [21, 22]. In LCs, defects are considered to be analogues to
Dirac magnetic monopoles [7] and cosmic strings [23].

Figure 1:  RSS is distinct orientational structure composed of concentrically
layered shells and a single radial topological defect line (filament). (a) and
(b) Micrographs of spherical droplets of N* LCs in glycerol, which were taken
in absence of temperature gradients by detaching chamber from temperature
controllers. (c) and (d) Drawings of textures when observed from two different
orthogonal angles. (e)–(h) Director field under Ansatz that constrains
director at droplet surface to tangent plane, derived in Ref. [11]. Directors
are visualized as cylinders. Red region represents filament. (e) Concentric
director field near plane where filament lies. (f) Spiral director field on
sections orthogonal to filament. (g) Cylindrical region around the filament.
(h) Pyramid region opposite filament across center of the droplet. Helical
director field forms layers with their normal along radial axis.

The chirality of LCs induces their dynamical rotation under temperature
gradients [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. This rotation is
attributed to the coupling between translation and rotation by the helical
director field [32] or the conversion of thermal energy into torque by
thermomechanical (thermohydrodynamic) coupling [36, 25, 33]. Thermomechanical
coupling is characterized by a pseudoscalar coupling constant $\nu$, which is
nonzero if and only if the LC is locally chiral. Recently, the role of
topology has been discussed for rotations caused by the coupling of
translation and rotation [31, 35] or by chemomechanical (chemohydrodynamic)
coupling [37]. Nevertheless, topological involvement in thermomechanical
coupling is still unclear due to the experimental difficulties.

Here, we investigate the role of topological defects in thermomechanical
coupling by focusing on RSS. Hereinafter, we refer to the defect as a filament
to remind the reader of its finite thickness. A recently developed method [34]
enables weak thermomechanical coupling to be detected.

Experimental system.—N* LC mixtures $n$CB(X) were prepared by mixing nematic
LCs 4′-alkyl-4-cyanobiphenyls $n$CB ($n=5,6,7$), with chiral dopants
(X)-2-octyl 4-[4-(hexyloxy)benzoyloxy]benzoate ($\mathrm{X=R,S}$) at a
concentration of approximately $1.5\mathrm{\%\,w/w}$. $n$CB(X) was dispersed
as spherical droplets with a non-uniform radius in immiscible glycerol, where
Marangoni convection and mass diffusion fluxes are effectively suppressed [31,
32], and then introduced into a glass chamber. A temperature gradient was
applied to the $n$CB(X) droplets vertically upward (defined as the $z$-axis)
by heating the top and cooling the bottom of the chamber, and Rayleigh-Bénard
convection, in which heated fluid floats [38, 39], was prevented. Details on
the methods are available in the Supplemental Material [40].

Reversal with radius.—Figure 2(a) shows the rotational motion of 7CB(R)
droplets under a steady temperature gradient. As the clean chamber prevented
the cyanobipheyl LCs from adhering to the walls, the rotational motion of the
LC droplets was successfully demonstrated [40]. Remarkably, the rotational
direction of the droplets reversed depending on their radii. Figure 2(c) shows
the temporal change in the rotational angle of a large droplet $\alpha$ (with
radius of $13.0\,$\mathrm{\SIUnitSymbolMicro m}$$) and a small droplet $\beta$
(with radius of $9.1\,$\mathrm{\SIUnitSymbolMicro m}$$). Droplets $\alpha$ and
$\beta$ rotated in the opposite directions due to the difference in their
radii. As conventionally known, inverting the chirality of sample or the
direction of the temperature gradient causes the reversal of rotational
direction [26, 56, 34]. However, the present reversal with size (droplet
radius) was not predicted and is presented in this Letter for the first time.

Figure 2:  Rotation of droplets. (a) and (b) Micrographs of rotating droplets
of different sizes, $\alpha$ and $\beta$, under temperature gradient
$\nabla_{z}T=+1.00\times 10^{4}\,$\mathrm{K}\,\mathrm{m}^{-1}$$ (target symbol
in (b) is perpendicular to this figure in the direction of the reader).
Filament is oriented parallel to optical axis. Initial and current directions
of droplets are represented by colored and white circles, respectively. See
also Supplemental Material, Movie S4 [40]. (c) Rotation angle of droplets
$\alpha$ (blue) and $\beta$ (red) against time. Positive in angle axis
represents anticlockwise rotation. Measured rotation angles from initial
position are plotted with regression lines. Shaded areas indicate
uncertainties of doubled standard deviations due to thermal fluctuations. For
smaller droplets, violation of angle was more significant due to thermal
fluctuations. The contrast of the images was adjusted.

The reversal of the rotational direction is attributed to the reversal of
torque exerting on the droplet. A droplet with RSS comprises two orientational
structures (Fig. 1): concentric shells consisting of the periodic layers of a
helical director field and the radial filament of a topological defect [7],
each of which has a different dimension in terms of extensive property.

Figure 3(a) schematically shows the dimension of these structures in RSS. The
shells and the filament are three- and quasi-one-dimensional with the volume
increasing in proportion to the cubic radius $R^{3}$ and to the radius
$R^{1}$, respectively. Thermomechanical torques must occur in opposite
directions between them; torques in $n$CB(R) are induced anticlockwise and
clockwise in the shells and the filament, respectively. The torques
proportional to each volume compete with each other inside the droplet, and
the sum of conflicting torques determines the rotational direction of the
droplet. A droplet with a radius large enough, in which the shells are
volumetrically dominant, experiences a net anticlockwise torque, whereas a
droplet whose radius is so small that the filament transcends the shells
experiences a net clockwise torque. Note that, heat is converted into torque
differently depending on the microscopic orientational order of the molecules
in each structure.

The torque competition between these two types of structures in the droplets
is formulated by the dimensional analysis below. A minimal expression of the
dissipation function $\Phi$ up to numerical factors is a quadratic form of the
variables of angular velocity $\omega_{0}$ and temperature gradient
$\nabla_{z}T$, as

$\Phi=-2\nu\left(R^{3}-q_{0}^{-2}R^{1}\right)\omega_{0}\nabla_{z}T+\mu
R^{3}\omega_{0}^{2}+\kappa T^{-1}R^{3}(\nabla_{z}T)^{2}.$ (1)

The first term represents the energy converted per unit time through
thermomechanical coupling. The coefficients with dimensions of volume $R^{3}$
and $q_{0}^{-2}R^{1}$ describe the contributions of the three-dimensional
shells and the quasi-one-dimensional filament, respectively. Adopting
Onsager’s variational principle [57], we obtain the angular velocity in the
steady state,

$\omega_{0}(R)=\left(1-\frac{1}{(q_{0}R)^{2}}\right)\frac{\nu\nabla_{z}T}{\mu},$
(2)

which minimizes the dissipation function in Eq. (1) under a fixed temperature
gradient $\nabla_{z}T$. The contribution of the filament, represented by the
factor $(q_{0}R)^{-2}$, is responsible for the radial dependence and reversal
of the droplets’ rotation. Eq. (2) derives a switching radius $R^{\ast}$ that
is almost equal to the period of the helical director field
$|q_{0}|^{-1}\simeq 10\,$\mathrm{\SIUnitSymbolMicro m}$$. As shown in Fig.
3(b), Eq. (2) describes the dependence of the angular velocity on the radius
with the reversal of the rotational direction well.

Figure 3:  (a) Schematic diagram of torque in constituent structures. Blue and
red lines represent torque induced in shells (shaded in blue) and filament
(shaded in red), respectively. Each torque is proportional to volume and
opposite to each other. Rotational direction of droplets shifts at switching
radius $R^{\ast}$ where torques are in equilibrium. (b) Angular velocity
against radius. Angular velocity $\omega_{0}$ is scaled by viscosity of
glycerol $\mu$. Plots are measured values and curves are regression lines
obeying $\mu\omega_{0}=a_{3}-a_{1}R^{-2}$, complementing Eq. (2) with
temperature-dependent factors $a_{3}$ and $a_{1}$. Color of each plot and
curve indicates the temperature and corresponds to color bar above.

Molecular order.—In the analysis above, we assumed that the molecular order
leads to a difference in the direction of the induced torque [Fig. 3(a)],
which is natural because thermomechanical coupling would relate to certain
intermolecular correlations as an analogy to viscosity [5]. The intermolecular
correlation is represented by the director $\mathrm{\mathbf{n}}$ and the
scalar OP $S$, where the latter is the height of molecular order and reaches
zero in isotropic (I) phase. The significance of $S$ in physical properties of
LCs is widely recognized. In particular, the variation of $S$ explains the
temperature dependence of birefringence [58], the elastic constants [48, 59],
and Leslie viscosity coefficients [52].

Figure 4 schematically shows the relationship between $S$ and the coupling
constant $\nu$. In $n$CB(R), $\nu$ becomes positive and negative when $S$ is
high and low, respectively. Since $S$ is reduced in the filament similar to
the amplitudes of the macroscopic wavefunction at vortices of superconductors,
$\nu$ is negative in the filament and, in contrast, positive in the shells
with higher $S$. In $n$CB(S), the mirrored counterpart, $\nu$ has the opposite
sign.

Figure 4:  Dependence of coupling constant $\nu$ on scalar OP $S$ and
temperature $T$. (left to right) Gradation from highly ordered state (high
$S$, low $T$) with positive $\nu$ and torque of anticlockwise (ACW) rotation
to poorly ordered state (low $S$, high $T$) with negative $\nu$ and torque of
clockwise (CW) rotation.  Figure 5:  (a)–(d) Temperature dependence of
rotational direction of droplets of 7CB(R), 7CB(S), 6CB(R), and 5CB(R). All of
them showed reversal of rotational direction with temperature. Anticlockwise
(ACW) and clockwise (CW) rotations are plotted in blue and red, respectively.
Shaded areas indicate violin plots, showing the frequency distribution of
radius across rotational direction and temperature determined by kernel
density estimation. Since temperature is treated as a categorical variable,
its original scaling is not displayed. Black solid and dotted lines show
switching radius $R^{\ast}$ and its confidence interval with doubled standard
deviations, respectively. (e) and (f) Switching radius against temperature $T$
and relative temperature $T-T^{\ast}$, respectively. Error bars indicate
confidence interval as shown in (a)–(d).

Reversal with temperature.—As temperature rises, $S$ declines due to thermal
fluctuation, and $\nu$ changes accordingly; thus, the reversal of the
rotational direction with temperature is also expected (Fig. 4). Figures
5(a)–(d) show the rotational direction of droplets at different temperatures,
clearly demonstrating the above-mentioned reversal. For instance, droplets of
7CB(R) with a radius of approximately $10\,$\mathrm{\SIUnitSymbolMicro m}$$
rotated anticlockwise at $33.1$\mathrm{\SIUnitSymbolCelsius}$$ but clockwise
at temperatures above $35.5$\mathrm{\SIUnitSymbolCelsius}$$ [Fig. 5(a)]. The
switching radius $R^{\ast}$ observed near $10\,$\mathrm{\SIUnitSymbolMicro
m}$$ coincides with the dimensional analysis in Eq. (2). The shift of
$R^{\ast}$ with temperature indicates that the coupling constant $\nu$ tends
to be negative due to the decrease in $S$ at high temperatures and, in
contrast, at a sufficiently low temperature, all droplets rotate anticlockwise
since $\nu$ is positive thanks to the higher molecular order even in the
filament. The spiral texture (namely, period and sense of helix) was identical
at high and low temperatures. This is consistent with the independence of the
helical field and thermomechanical coupling [27].

This reversal with temperature was also confirmed in 7CB(S). Droplets with a
radius of approximately $10\,$\mathrm{\SIUnitSymbolMicro m}$$ rotated
clockwise and anticlockwise at $33.1$\mathrm{\SIUnitSymbolCelsius}$$ and above
$35.5$\mathrm{\SIUnitSymbolCelsius}$$, respectively [Fig. 5(b)]. The
rotational direction at a given temperature and radius was opposite for 7CB(R)
and 7CB(S). Similar reversal of the rotational direction was observed for all
the alkyl chain lengths considered, including $n=5$ and $6$ [Figs. 5(c) and
(d)].

Figures 5(e) and (f) show the switching radius against the temperature. The
temperature dependence of $R^{\ast}$ varied for every alkyl chain length $n$
[Fig. 5(e)]. Notably, however, a common dependence was found among n on the
relative temperature $T-T^{\ast}$, where $T^{\ast}$ is the N*-I transition
temperature, indicating that the reversal with temperature was driven by $S$
(rather than $T$ itself).

A theoretical insight.—We recast the theory of thermomechanical coupling
conventionally formulated with the director $\mathrm{\mathbf{n}}$. Omitting
the contributions of the electromagnetic field, hydrodynamic flow and elastic
distortion, the evolution of the director, which is purely driven by
thermomechanical coupling, is [36]

$\dot{n}_{i}=-\nu\epsilon_{ijk}n_{j}\nabla_{k}T,$ (3)

where $\nu$ is the coupling constant renormalized by division by the viscosity
against the background flow. The director $\mathrm{\mathbf{n}}$ specifies only
the mean orientation direction of the molecules, and $\mathrm{\mathbf{n}}$
ignores the degree of molecular orientation, which is represented by the
scalar OP $S$. The complete OP for describing the microscopic molecular order
is a second-order symmetric traceless tensor (the Q-tensor)

$Q_{ij}=\frac{3}{2}S\left(n_{i}n_{j}-\frac{1}{3}\delta_{ij}\right),$ (4)

whose non-degenerate eigenvalue is equal to $S$ and its eigenvector is
$\mathrm{\mathbf{n}}$ [48]. The evolution of the orientational order is re-
described by replacing $\mathrm{\mathbf{n}}$ with $Q$. Consistent with the
existing theory shown in Eq. (3), assume that the evolution of $Q$ is
proportional to the temperature gradient $\nabla T$:

$\dot{Q}_{ij}=-L_{ijk}\nabla_{k}T,$ (5)

where $L_{ijk}$ is a third-order tensor coefficient being invariant with
respect to $i\leftrightarrow j$, inheriting the symmetry of $\dot{Q}_{ij}$.
Since $L$ has the same symmetry as the N* phase, it must be a function of $Q$.
Since elastic distortion [48] and anisotropic viscosity [52] are successfully
described by tensorial arguments, an analogy of them, $L$ is expanded by terms
of the powers of $Q$ [40]. These considerations on the symmetry derive the
$S$-dependent coupling constant

$\nu(S)=\nu_{1}+\nu_{2}S,$ (6)

where $\nu_{1}$ and $\nu_{2}$ are pseudoscalars hardly varying with
temperature. Eq. (6) indicates that the strength and the direction of
thermomechanical torque are influenced by the molecular order parametrized by
$S$. As reported for chiral ferroelectric LCs, spontaneous polarization
reverses its direction with temperature because of a shift in molecular order,
especially conformation [60, 61]. Thus, the sign reversal of $\nu$ is probably
induced by changes in molecular order due to temperature or frustrations.
Though the sign reversal of $\nu$ indicates the opposites signs between
$\nu_{1}$ and $\nu_{2}$, its necessity is not derived within the present
phenomenological argument. Instead, molecular simulations [62] or statistical
mechanics would be valid for estimating $\nu_{1}$ and $\nu_{2}$. The present
reconstruction adopting $Q$, a suitable OP for expressing the microscopic
molecular orientational order, will support the future elucidation of the
molecular mechanisms of thermomechanical coupling.

Conclusion.—We have found that droplets of chiral LCs with topological defects
show reversal of rotational motion under a temperature gradient. This
nontrivial reversal depending on size and temperature has not yet been
expected but is ascribed to the dependence of thermomechanical coupling on the
degree of molecular orientational order, which is justified by our
phenomenological argument. In the context of topology of materials, the phase
or direction components of the order parameter is usually taken up [63]. We
propose, however, that the amplitude component is also noteworthy for
emergence of distinct properties. Future studies will explore superconductors
and magnets, as well as LCs, for analogous behaviors.

###### Acknowledgements.

S.T. thanks Y. Tabe and Y. Maruyama for their support at the early stage. This
work was supported by JST SPRING, Grant Number JPMJSP2128 and Waseda Research
Institute for Science and Engineering, Grant-in-Aid for Young Scientists
(Early Bird).

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# On the distribution of analytic ranks of elliptic curves

Peter J. Cho† Department of Mathematical Sciences, Ulsan National Institute of
Science and Technology, UNIST-gil 50, Ulsan 44919, Republic of Korea
<EMAIL_ADDRESS>and  Keunyoung Jeong⋆ Department of Mathematical
Sciences, Ulsan National Institute of Science and Technology, UNIST-gil 50,
Ulsan 44919, Republic of Korea<EMAIL_ADDRESS>
###### Abstract.

In this paper, under GRH for elliptic $L$-functions, we give an upper bound
for the probability for an elliptic curve with analytic rank $\leq a$ for
$a\geq 11$, and also give an upper bound of $n$-th moments of analytic ranks
of elliptic curves. These are applications of counting elliptic curves with
local conditions, for example, having good reduction at $p$.

###### Key words and phrases:

elliptic curve, rank, trace formula, GRH, BSD conjecture

###### 2010 Mathematics Subject Classification:

Primary 11G05, Secondary 11G40, 11M26

† This work was supported by Basic Science Research Program through the
National Research Foundation of Korea(NRF) funded by the Ministry of
Education(2019R1F1A1062599).

⋆ The author was supported by the National Research Foundation of Korea(NRF)
funded by the Ministry of Education, under the Basic Science Research
Program(2019R1C1C1004264).

##  1\. Introduction

In this article, we study the distribution of analytic ranks of elliptic
curves over the rational numbers. Let $P(r_{E}\geq a)$ be the proportion of
elliptic curves with analytic rank $r_{E}\geq a$. We give a numerical bound
for $P(r_{E}\geq a)$.

######  Theorem 1.1.

Assume GRH for elliptic $L$-functions. Let $C$ be a positive constant, let $n$
a positive integer. We have

$\displaystyle
P\left(r_{E}\geq(1+C)9n\right)\leq\frac{\sum_{k=0}^{n}{{2n}\choose{2k}}\left(\frac{1}{2}\right)^{2n-2k}(2k)!\left(\frac{1}{6}\right)^{k}}{(C\cdot
9n)^{2n}}.$

Now, we can give a numerical evidence that there are at most a small
proportion of elliptic curves with large analytic ranks.

######  Corollary 1.2.

Assume GRH for elliptic $L$-functions. Let
$f(t)=\sum_{k=0}^{t}{{2t}\choose{2k}}\left(\frac{1}{2}\right)^{2t-2k}(2k)!\left(\frac{1}{6}\right)^{k}$.
Then,

  1. (1)

$P(r_{E}\leq a)\geq 1-\frac{f(1)}{(a-8)^{2}}$ for $11\leq a\leq 17$.

  2. (2)

$P(r_{E}\leq a)\geq 1-\min_{1\leq l\leq
n}\left\\{\frac{f(l)}{(a+1-9l)^{2l}}\right\\}$ for $9n\leq a\leq 9n+8$, $n\geq
2$.

For small $a$’s, $P(r_{E}\leq a)$ is recorded in the following table.

$a$ | $P(r_{E}\leq a)$ | $a$ | $P(r_{E}\leq a)$ | $a$ | $P(r_{E}\leq a)$ | $a$ | $P(r_{E}\leq a)$
---|---|---|---|---|---|---|---
$11$ | $\geq 0.935185$ | $16$ | $\geq 0.990885$ | $21$ | $\geq 0.996548$ | $26$ | $\geq 0.999812$
$12$ | $\geq 0.963541$ | $17$ | $\geq 0.992798$ | $22$ | $\geq 0.998033$ | $27$ | $\geq 0.999877$
$13$ | $\geq 0.976666$ | $18$ | $\geq 0.994166$ | $23$ | $\geq 0.999051$ | $28$ | $\geq 0.999916$
$14$ | $\geq 0.983796$ | $19$ | $\geq 0.995179$ | $24$ | $\geq 0.999488$ | $34$ | $\geq 0.999985$
$15$ | $\geq 0.988095$ | $20$ | $\geq 0.995949$ | $25$ | $\geq 0.999699$ | $35$ | $\geq 0.999988$

######  Remark.

(1) The model of [PPVW19] conjectures that there are only finitely many
elliptic curves with rank $>21$, and we show that the proportion of elliptic
curves with analytic rank $>21$ is less than 0.0035 under the GRH for elliptic
$L$-functions.

(2) Heath-Brown’s result [H-B04, Theorem 2] seems stronger than ours for a
very large rank $a$. However, our emphasis is on a small rank $a$ not a large
rank $a$. Theorem 1.1 is not an explicitation of the implicit constant in
[H-B04, Theorem 2], and our method of the proof is different. For the proof,
we establish the Frobenius trace formula for elliptic curves (Theorem 3.1),
which is a new tool for elliptic curves. In the process of proof, we encounter
the following inequality:

$\displaystyle\left|\\{E\in\mathcal{E}(X)|r_{E}\geq\frac{1+C}{\sigma_{2n}}\\}\right|$
$\displaystyle\left(\frac{C\sigma_{2n}}{4}\right)^{2n}\leq\sum_{E\in\mathcal{E}(X)}\left(-\frac{2}{\log
X}\sum_{m_{i}}\frac{\widehat{a}_{E}(m_{i})\Lambda(m_{i})}{\sqrt{m_{i}}}\widehat{\phi}_{2n}\left(\frac{\log
m_{i}}{\log X}\right)\right)^{2n}.$

Since, using the Frobenius trace formula, we have the equality

$\displaystyle\sum_{E\in\mathcal{E}(X)}\left(-\frac{2}{\log
X}\sum_{m_{i}}\frac{\widehat{a}_{E}(m_{i})\Lambda(m_{i})}{\sqrt{m_{i}}}\widehat{\phi}_{2n}\left(\frac{\log
m_{i}}{\log X}\right)\right)^{2n}$
$\displaystyle=\left(\frac{\sigma_{2n}^{2}}{4}\right)^{2n}\sum_{S_{2}\subset\\{1,2,3,\dots,2n\\}}\left(\frac{1}{2}\right)^{|S_{2}^{c}|}|S_{2}|!\left(\frac{1}{6}\right)^{|S_{2}|/2}|\mathcal{E}(X)|+O\left(\frac{X^{\frac{5}{6}}}{\log
X}\right),$

we don’t lose any information for the inequality. See [H-B04, Sec. 7]. Our
approach is in the same spirit with Katz and Sarnak’s $n$-level density
conjecture. For the introduction to the $n$-level density conjecture and some
partial results, we refer to [Rub01, Mil, CK15].

(3) In the proof of the Frobenius trace formula, Theorem 3.1, we use similar
arguments in [CK15]. The new feature here is the Eichler-Selberg trace formula
which replaces the role of orthogonality of characters in [CK15].

(4) For algebraic ranks, we remark that the recent development of [BS15, BS]
gives better bounds than ours. Since the average of the order of
$\operatorname{Sel}_{5}(E/\mathbb{Q})$ is 6 by [BS], we have

$P(r_{E}\geq a)5^{a}\leq 6$

by the exact sequence

$0\to\frac{E(\mathbb{Q})}{5E(\mathbb{Q})}\to\operatorname{Sel}_{5}(E/\mathbb{Q})\to\Sha(E/\mathbb{Q})[5]\to
0.$

Hence for example, $P(r_{E}<20)$ is bounded by $1-6/5^{20}\approx
0.99999999999993708544$. However, for the average of analytic ranks, Young’s
bound $\frac{25}{14}$ [You06] under GRH is the best record. It gives a bound
for $P(r_{E}\geq a)$, which is $\frac{25}{14a}$.

The second one is regarding the moments of the analytic ranks of elliptic
curves. One can show that the limsup of $n$-th moments of analytic ranks of
elliptic curves exist, by applying the result of Heath-Brown [H-B04, Theorem
2]. Using an approach of Miller [Mil], we propose an explicit upper bound on
the $n$-th moment of analytic ranks for every positive integer $n$.

######  Theorem 1.3.

Assume GRH for elliptic $L$-functions. Let $r_{E}$ be the analytic rank of an
elliptic curve $E$. Let $\sigma_{n}=\frac{2}{9n}$ for a positive integer $n$.
For every positive integer $n$, we have

$\displaystyle\limsup_{X\rightarrow\infty}\frac{1}{|\mathcal{E}(X)|}\sum_{E\in\mathcal{E}(X)}r_{E}^{n}\leq\sum_{S}\left(\frac{1}{\sigma_{n}}\right)^{|S^{c}|}\sum_{\begin{subarray}{c}S_{2}\subset
S\\\
|S_{2}|\text{even}\end{subarray}}\left(\frac{1}{2}\right)^{|S_{2}^{c}|}\left|S_{2}\right|!\left(\frac{1}{6}\right)^{|S_{2}|/2}.$

where $S$ runs over subsets of $\\{1,2,3,\dots,n\\}$, and $S_{2}$ runs over
subsets of even cardinality of the set $S$. In particular, the limit sup of
the 2nd moment of analytic ranks is bounded by $90.584$ and the limit sup of
the 3rd moment of analytic ranks is bounded by $2758$.

These results are obtained by accurate count of elliptic curves with local
conditions, which is of its own interest. Here a local condition at $p$ means
one of good reduction, bad reduction, multiplicative reduction, additive
reduction, $a_{p}$ in the Weil bound[1][1][1]An elliptic curve $E$ satisfies a
local condition $a_{p}$ if $a_{p}(E)=a_{p}$., split and non-split reduction,
or one of the Kodaira–Néron type.

Before stating the results, we explain the model of elliptic curves in our
consideration. We treat elliptic curves of the form:

$E_{A,B}:y^{2}=x^{3}+Ax+B,\,\,A,B\in\mathbb{Z}\qquad\textrm{if a prime $q$
satisfies $q^{4}\mid A$, then $q^{6}\nmid B$}.$

We denote the prime condition by $(M)$. Elliptic curves in this model might be
not minimal at $2$ and $3$ but are minimal at all primes $q\geq 5$. Also, each
isomorphism class of elliptic curves appears in this model exactly one time.
We define a naive height of $E_{A,B}$ by

$H(E_{A,B}):=\max(|A|^{3},|B|^{2}).$

The family of our interest is

$\mathcal{E}(X):=\\{(A,B)\in\mathbb{Z}^{2}:4A^{3}+27B^{2}\neq 0,H(E_{A,B})\leq
X,\textrm{ and $(A,B)$ satisfies $(M)$}\\},$

and let

$\displaystyle\mathcal{E}_{p}^{\mathcal{LC}}(X)=\\{(A,B)\in\mathcal{E}(X):E_{A,B}\textrm{
satisfies $\mathcal{LC}$ at the prime }p\\},$

where $\mathcal{LC}$ is one of the local conditions. Then, we have

######  Theorem 1.4.

For $5\leq p\leq X^{\frac{1}{3m_{\mathcal{LC}}}}$,

$\displaystyle\left|\mathcal{E}_{p}^{\mathcal{LC}}(X)\right|=c_{\mathcal{LC}}(p)\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O(h_{\mathcal{LC},p}(X)),$

where $m_{\mathcal{LC}},c_{\mathcal{LC}}(p),h_{\mathcal{LC},p}(X)$ are given
by the following table.

Local conditions | $c_{\mathcal{LC}}(p)$ | $h_{\mathcal{LC},p}(X)$ | $m_{\mathcal{LC}}$
---|---|---|---
good | $\frac{p^{2}-p}{p^{2}}\frac{p^{10}}{p^{10}-1}$ | $pX^{\frac{1}{2}}$ | $1$
bad | $\frac{p^{10}-p}{p^{11}}\frac{p^{10}}{p^{10}-1}$ | $pX^{\frac{1}{2}}$ | $1$
mult | $\frac{p-1}{p^{2}}\frac{p^{10}}{p^{10}-1}$ | $X^{\frac{1}{2}}$ | $1$
split, non-split | $\frac{p-1}{2p^{2}}\frac{p^{10}}{p^{10}-1}$ | $X^{\frac{1}{2}}$ | $1$
addi | $\frac{p^{9}-p}{p^{11}}\frac{p^{10}}{p^{10}-1}$ | $pX^{\frac{1}{2}}$ | $1$
$a$ | $\frac{(p-1)H(a^{2}-4p)}{2p^{2}}\frac{p^{10}}{p^{10}-1}$ | $H(a^{2}-4p)X^{\frac{1}{2}}$ | $1$
$\mathrm{I}_{m}$ | $\frac{1}{p^{m}}\left(1-\frac{1}{p}\right)^{2}\frac{p^{10}}{p^{10}-1}$ | $pX^{\frac{1}{2}}$ | $m+1$
$\mathrm{II}$ | $\frac{1}{p^{2}}\left(1-\frac{1}{p}\right)\frac{p^{10}}{p^{10}-1}$ | $X^{\frac{1}{2}}$ | $2$
$\mathrm{III}$ | $\frac{1}{p^{3}}\left(1-\frac{1}{p}\right)\frac{p^{10}}{p^{10}-1}$ | $X^{\frac{1}{2}}$ | $3$
$\mathrm{IV}$ | $\frac{1}{p^{4}}\left(1-\frac{1}{p}\right)\frac{p^{10}}{p^{10}-1}$ | $X^{\frac{1}{2}}$ | $4$
$\mathrm{IV}^{*}$ | $\frac{1}{p^{7}}\left(1-\frac{1}{p}\right)\frac{p^{10}}{p^{10}-1}$ | $p^{-1}X^{\frac{1}{2}}$ | $6$
$\mathrm{III}^{*}$ | $\frac{1}{p^{8}}\left(1-\frac{1}{p}\right)\frac{p^{10}}{p^{10}-1}$ | $p^{-3}X^{\frac{1}{2}}$ | $5$
$\mathrm{II}^{*}$ | $\frac{1}{p^{9}}\left(1-\frac{1}{p}\right)\frac{p^{10}}{p^{10}-1}$ | $p^{-2}X^{\frac{1}{2}}$ | $7$
$\mathrm{I}_{m}^{*}$ | $\frac{1}{p^{m+5}}\left(1-\frac{1}{p}\right)^{2}\frac{p^{10}}{p^{10}-1}$ | $pX^{\frac{1}{2}}$ | $m+6$
$\mathrm{I}_{0}^{*}$ | $\frac{1}{p^{5}}\left(1-\frac{1}{p}\right)\frac{p^{10}}{p^{10}-1}$ | $pX^{\frac{1}{2}}$ | $5$

Here $H(\cdot)$ is the Hurwitz class number.

We can count elliptic curve not only with a single local condition but also
with finitely many local conditions. Let
$\mathcal{S}=\\{\mathcal{LC}_{p_{i}}\\}$ be a finite set of local conditions,
where $\mathcal{LC}_{p}$ can be good, bad, split, non-split, additive,
$a_{p}$, or $T$, one of the Kodaira–Néron type. Let $|\mathcal{LC}_{p}|$ be
the probability given in Theorems 1.4 and let us define
$\left|\mathcal{S}\right|=\prod_{i}\left|\mathcal{LC}_{p_{i}}\right|$. We
simply denote $m_{\mathcal{LC}_{p_{i}}}$ by $m_{i}$. Let

$\mathcal{E}^{\mathcal{S}}(X)=\left\\{(A,B)\in\mathcal{E}(X):E_{A,B}\mbox{
satisfies }\mathcal{S}\right\\}.$

In the following theorem, we show that these conditions are independent.

######  Theorem 1.5.

Let $\mathcal{LC}_{i}$ be a local condition at $p_{i}$ where $p_{i}$ is a
prime greater than $3$ and $\prod_{i=1}^{n}p_{i}^{m_{i}}\leq
X^{\frac{1}{3}-\epsilon}$ for some $\epsilon>0$. Then, we have

$\displaystyle\mathcal{E}^{\mathcal{S}}(X)=|\mathcal{S}|\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\left(\prod_{i=1}^{n}c_{p_{i}}\right)X^{\frac{1}{2}}\right),$

where
$c_{p_{i}}=|\mathcal{LC}_{p_{i}}|\left(\frac{p_{i}^{10}-1}{p_{i}^{10}}\right)p_{i}^{m_{i}}$.
We note that when $\mathcal{LC}_{p}=a$, then
$c_{p}=\frac{p-1}{2p}H(a^{2}-4p)$.

In [Wat08, §3.4], Watkins predicted probabilities for Kodaira–Néron types at a
prime $p\geq 5$ with a heuristic approach. Our results agree with those in
[Wat08]. Also, we found a result [BCD11] of counting elliptic curves with a
local condition $a_{p}$. In [BCD11] they consider the family of elliptic
curves $y^{2}=x^{3}+Ax+B$ without the minimality condition $(M)$, hence it
contains many isomorphic elliptic curves. Their dominant error term is
$O(X^{\frac{5}{6}}/p)$, which is much weaker than
$O(H(a^{2}-4p)X^{\frac{1}{2}})$ in Theorem 1.4 for a small prime $p$.

In Cremona and Sadek’s recent preprint [CS], the authors consider similar
problems. They uses the general Weierstrass model, so their approach gives a
natural probabilities for each local conditions at $p=2$ or 3. In their model,
they give the probability for semi-stability, which is 60.85%. This result
seems more natural than Wong’s probability for semistability 17.9% [Won01].
Even though we can count elliptic curves with finitely many local conditions,
we have an explicit error term. There will be potential applications in
elliptic $L$-functions.

In Section 2, we give a proof for Theorem 1.4 and 1.5. In Section 3, we state
and give a proof for the Frobenius trace formula for elliptic curves. Section
4 is devoted for the proof for Theorem 1.1 and 1.3.

##  2\. Counting Elliptic Curves

###  2.1. With a single local condition

In this section, we give proofs of Theorems 1.4. For positive integers $m$ and
$n$, we say $\alpha\in\mathbb{Z}/p^{m}\mathbb{Z}$ is divided by $p^{n}$ if
$n<m$ and $\alpha=ap^{n}$ for some $a\in\mathbb{Z}/p^{m}\mathbb{Z}$, and
exactly divided by $p^{n}$ if there exists
$u\in(\mathbb{Z}/p^{m}\mathbb{Z})^{\times}$ such that $\alpha=up^{n}$. Let

$\mathcal{D}_{p^{m},\alpha,\beta}(X):=\\{(A,B)\in\mathcal{D}(X):(A,B)\equiv(\alpha,\beta)\pmod{p^{m}}\\},$

and

$\mathcal{M}_{p^{m},\alpha,\beta}(X):=\\{(A,B)\in\mathcal{D}_{p^{m},\alpha,\beta}(X):(A,B)\textrm{
satisfies }(M)\\},$

for $(\alpha,\beta)\in(\mathbb{Z}/p^{m}\mathbb{Z})^{2}$ such that
$p^{4}\nmid\alpha$ or $p^{6}\nmid\beta$. For nonzero
$(\alpha,\beta)\in(\mathbb{Z}/p^{m}\mathbb{Z})^{2}$ such that
$p^{4}\nmid\alpha$ or $p^{6}\nmid\beta$, we define

$\mathcal{E}_{p^{m},\alpha,\beta}(X):=\\{(A,B)\in\mathcal{M}_{p^{m},\alpha,\beta}(X):4A^{3}+27B^{2}\neq
0\\},$

and $\mathcal{S}_{p^{m},\alpha,\beta}(X)$ as the unique set satisfying
$\mathcal{M}_{p^{m},\alpha,\beta}(X)=\mathcal{E}_{p^{m},\alpha,\beta}(X)\bigsqcup\mathcal{S}_{p^{m},\alpha,\beta}(X)$.

######  Lemma 2.1.

Let $m\leq C,n\leq D$ and
$\alpha\in\mathbb{Z}/m\mathbb{Z},\beta\in\mathbb{Z}/n\mathbb{Z}$. The
cardinality of a set

$\\{(c,d)\in\mathbb{Z}^{2}:|c|\leq C,|d|\leq
D,(c,d)\equiv(\alpha,\beta)\pmod{\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z}}\\},$

is

$4\cdot\lfloor\frac{C+1}{m}\rfloor\cdot\lfloor\frac{D+1}{n}\rfloor+O\left(\lfloor\frac{C+1}{m}\rfloor+\lfloor\frac{D+1}{n}\rfloor+1\right).$

We define $*$-operator on a pair by $d*(a,b):=(d^{4}a,d^{6}b)$.

######  Lemma 2.2.

For a nonzero $(\alpha,\beta)\in(\mathbb{Z}/p^{m}\mathbb{Z})^{2}$ such that
$p^{4}\nmid\alpha$ or $p^{6}\nmid\beta$,

$\mathcal{D}_{p^{m},\alpha,\beta}(X)=\bigsqcup_{\begin{subarray}{c}d\leq
X^{\frac{1}{12}}\\\ p\nmid
d\end{subarray}}d*\mathcal{M}_{p^{m},d^{-4}\alpha,d^{-6}\beta}(d^{-12}X).$

###### Proof.

This is an analogue of [Bru92], but note that the condition $p^{4}\nmid\alpha$
or $p^{6}\nmid\beta$ gives a restriction on $d$. ∎

We note that Lemma 2.2 gives

$|\mathcal{M}_{p^{m},\alpha,\beta}(X)|=\sum_{\begin{subarray}{c}d\leq
X^{\frac{1}{12}}\\\ p\nmid
d\end{subarray}}\mu(d)|\mathcal{D}_{p^{m},d^{-4}\alpha,d^{-6}\beta}(d^{-12}X)|.$

######  Proposition 2.3.

For a prime $5\leq p\leq X^{\frac{1}{3m}}$ and nonzero
$(\alpha,\beta)\in(\mathbb{Z}/p^{m}\mathbb{Z})^{2}$ such that
$p^{4}\nmid\alpha$ or $p^{6}\nmid\beta$,

$|\mathcal{E}_{p^{m},\alpha,\beta}(X)|=\frac{1}{p^{2m}}\frac{p^{10}}{p^{10}-1}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\frac{X^{\frac{1}{2}}}{p^{m}}\right).$

Furthermore,

$|\mathcal{E}_{p,0,0}(X)|=\frac{p^{8}-1}{p^{10}-1}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O(pX^{\frac{1}{2}}).$

###### Proof.

We have
$|\mathcal{E}_{p^{m},\alpha,\beta}(X)|=|\mathcal{M}_{p^{m},\alpha,\beta}(X)|+O(|\mathcal{S}_{p^{m},\alpha,\beta}(X)|)$,
and $|\mathcal{S}_{p^{m},\alpha,\beta}(X)|\ll O(X^{\frac{1}{6}}/p^{m})$. On
the other hand

$|\mathcal{D}_{p^{m},d^{-4}\alpha,d^{-6}\beta}(\frac{X}{d^{12}})|=\frac{4X^{\frac{5}{6}}}{p^{2m}d^{10}}+O\left(\frac{X^{\frac{1}{2}}}{p^{m}d^{6}}+1\right)$

by Lemma 2.1. Together with Lemma 2.2 for $(\alpha,\beta)$ such that
$p^{4}\nmid\alpha$ or $p^{6}\nmid\beta$, we have

$\displaystyle|\mathcal{E}_{p^{m},\alpha,\beta}(X)|$
$\displaystyle=|\mathcal{M}_{p^{m},\alpha,\beta}(X)|+O(|\mathcal{S}_{p^{m},\alpha,\beta}(X)|)=\sum_{\begin{subarray}{c}d\leq
X^{\frac{1}{12}}\\\ p\nmid
d\end{subarray}}\mu(d)|\mathcal{D}_{p^{m},d^{-4}\alpha,d^{-6}\beta}(\frac{X}{d^{12}})|+O\left(\frac{X^{\frac{1}{6}}}{p^{m}}\right),$

which is equal to

$\displaystyle=\sum_{\begin{subarray}{c}d\leq X^{\frac{1}{12}}\\\ p\nmid
d\end{subarray}}\mu(d)\left(\frac{4X^{\frac{5}{6}}}{p^{2m}d^{10}}+O\left(\frac{X^{\frac{1}{2}}}{p^{m}d^{6}}+1\right)\right)+O\left(\frac{X^{\frac{1}{6}}}{p^{m}}\right)$
$\displaystyle=\sum_{\begin{subarray}{c}d=1\\\ p\nmid
d\end{subarray}}^{\infty}\mu(d)\left(\frac{4X^{\frac{5}{6}}}{p^{2m}d^{10}}+O\left(\frac{X^{\frac{1}{2}}}{p^{m}d^{6}}\right)\right)+O\left(X^{\frac{1}{12}}+\frac{X^{\frac{1}{6}}}{p^{m}}\right)-\sum_{\begin{subarray}{c}d\geq
X^{\frac{1}{12}}\\\ p\nmid
d\end{subarray}}\mu(d)\left(\frac{4X^{\frac{5}{6}}}{p^{2m}d^{10}}+O\left(\frac{X^{\frac{1}{2}}}{p^{m}d^{6}}\right)\right)$
$\displaystyle=\frac{4X^{\frac{5}{6}}}{p^{2m}}\left(\sum_{\begin{subarray}{c}d=1\\\
p\nmid
d\end{subarray}}^{\infty}\frac{\mu(d)}{d^{10}}\right)+O\left(\frac{X^{\frac{1}{12}}}{p^{m}}+\frac{X^{\frac{1}{2}}}{p^{m}}\sum_{d=1}^{\infty}\frac{1}{d^{6}}+X^{\frac{1}{12}}+\frac{X^{\frac{1}{6}}}{p^{m}}\right)$
$\displaystyle=\frac{1}{p^{2m}}\frac{p^{10}}{p^{10}-1}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\frac{X^{\frac{1}{2}}}{p^{m}}\right).$

Hence we have the first one. On the other hand, the computation of
$|\mathcal{E}(X)|$ from [Bru92, Lemma 4.3] and

$|\mathcal{E}(X)|=|\mathcal{E}_{p,0,0}(X)|+\sum_{\alpha,\beta\in(\mathbb{Z}/p\mathbb{Z})^{\times}}|\mathcal{E}_{p,\alpha,\beta}(X)|,$

give the result for $|\mathcal{E}_{p,0,0}(X)|$. ∎

We collect auxiliary lemmas on the number of elliptic curves corresponding to
each local condition. For the good and bad reduction conditions, we use the
cardinality of

$\\{(\alpha,\beta)\in(\mathbb{Z}/p\mathbb{Z})^{2}:4\alpha^{3}+27\beta^{2}\equiv
0\pmod{p}\\},$

which is $p$. Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an
order of $K$. We define the Hurwitz class number $H(\mathcal{O})$ by

$H(\mathcal{O}):=\sum_{\mathcal{O}\subset\mathcal{O}^{\prime}\subset\mathcal{O}_{K}}\frac{2}{|\mathcal{O}^{\prime\times}|}h(\mathcal{O}^{\prime}),$

where $h(\mathcal{O}^{\prime})$ is the class number of an order
$\mathcal{O}^{\prime}$. We also write $H(\mathcal{O})$ as $H(D)$, when $D$ is
the discriminant of $\mathcal{O}$. By [Cox13, Theorem 14.18], we have

(2.1)
$\left|\\{(\alpha,\beta)\in(\mathbb{Z}/p\mathbb{Z})^{2}:a_{p}(E_{\alpha,\beta})=a\\}\right|=\frac{p-1}{2}H(a^{2}-4p).$

For multiplicative reduction, we need

######  Lemma 2.4.

Let $p\geq 5$ be a prime and $m\geq 1$ be an integer. Then,

$\left|\\{(\alpha,\beta)\in\left((\mathbb{Z}/p^{m}\mathbb{Z})^{\times}\right)^{2}:4\alpha^{3}+27\beta^{2}\equiv
0\pmod{p^{m}}\\}\right|=p^{m}\left(1-\frac{1}{p}\right),$

and

$\left|\\{(\alpha,\beta)\in\left((\mathbb{Z}/p\mathbb{Z})^{\times}\right)^{2}:4\alpha^{3}+27\beta^{2}\equiv
0\pmod{p},\beta=-\beta^{\prime 2}\\}\right|=\frac{p-1}{2}.$

The following is a result of Tate’s algorithm for $E_{A,B}$.

######  Lemma 2.5.

Let $p\geq 5$ and $E_{A,B}:y^{2}=x^{3}+Ax+B$ be a minimal model over
$\mathbb{Q}_{p}$. Then, $E$ has

$\left\\{\begin{array}[]{cccc}\textrm{Type $\mathrm{II}$ if and only if }p\mid
A,p\|B&\\\ \textrm{Type $\mathrm{III}$ if and only if }p\|A,p^{2}\mid B&\\\
\textrm{Type $\mathrm{IV}$ if and only if }p^{2}\mid A,p^{2}\|B&\\\
\end{array}\right.\qquad\left\\{\begin{array}[]{ccc}\textrm{Type
$\mathrm{II}^{*}$ if and only if }p^{4}\mid A,p^{5}\|B\\\ \textrm{Type
$\mathrm{III}^{*}$ if and only if }p^{3}\|A,p^{5}\mid B\\\ \textrm{Type
$\mathrm{IV}^{*}$ if and only if }p^{3}\mid A,p^{4}\|B\end{array}\right.$

Furthermore, for the fixed positive integer $m$, an elliptic curve $E$ has
reduction type $\mathrm{I}_{m}^{*}$ at $p$ if and only if $p^{2}\|A$ or
$p^{3}\|B$, and $v_{p}(4(A/p^{2})^{3}+27(B/p^{3})^{2})=m$.

Finally for the local conditions $\mathrm{I}_{m}^{*}$, we need

######  Lemma 2.6.

For $m>0$, we have

$|\\{(\alpha,\beta)\in(\mathbb{Z}/p^{m+6}\mathbb{Z})^{2}:p^{2}\|\alpha,p^{3}\|\beta,p^{m+6}||\left(4\alpha^{3}+27\beta^{2}\right)\\}|=p^{m+5}(p-1)^{2}.$

For $m=0$, we have

$|\\{(\alpha,\beta)\in(\mathbb{Z}/p^{6}\mathbb{Z})^{2}:p^{2}\|\alpha,p^{3}\|\beta,p^{6}\|\left(4\alpha^{3}+27\beta^{2}\right)\\}|=p^{6}(p-1).$

###### Proof of Theorem 1.4..

We give a proof for $\mathcal{LC}$ = good, $a_{p}(E),\mathrm{II}^{*}$, and
$\mathrm{I}_{m}^{*}$. Let
$A:=\\{(\alpha,\beta)\in(\mathbb{Z}/p\mathbb{Z})^{2}:4\alpha^{3}+27\beta^{2}\equiv
0\pmod{p}\\}$. Now, the number of elliptic curves whose height is less than
$X$, and have good reduction at $p\geq 5$ is

$|\mathcal{E}_{p}^{\mathrm{good}}(X)|=\sum_{(\alpha,\beta)\not\in
A}|\mathcal{E}_{p,\alpha,\beta}(X)|.$

By Proposition 2.3, we have

$\displaystyle|\mathcal{E}_{p}^{\mathrm{good}}(X)|$ $\displaystyle=$
$\displaystyle\frac{(p^{2}-p)}{p^{2}}\frac{p^{10}}{p^{10}-1}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O((p-1)X^{\frac{1}{2}}).$

For $a$ in a Weil bound,

$\frac{(p-1)H(a^{2}-4p)}{2p^{2}}\frac{p^{10}}{p^{10}-1}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left((p-1)\frac{H(a^{2}-4p)}{2p}X^{\frac{1}{2}}\right),$

by (2.1). Similarly, by Lemma 2.5, an elliptic curve $E_{A,B}$ has type
$\mathrm{II}^{*}$ at $p$ if and only if $(A,B)$ satisfies $p^{4}\mid
A,p^{5}\|B$ modulo $p^{7}$. Since there are $p^{3}(p^{2}-p)$ such pairs, we
have

$|\mathcal{E}_{p}^{\mathrm{II}^{*}}(X)|=\frac{p^{3}(p^{2}-p)}{p^{14}}\frac{p^{10}}{p^{10}-1}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(p^{3}(p^{2}-p)\frac{X^{\frac{1}{2}}}{p^{7}}\right).$

Now, $\mathrm{I}_{m}^{*}$-case directly followed by Lemma 2.6. More
concretely,

$|\mathcal{E}_{p}^{\mathrm{I}_{m}^{*}}(X)|=\sum_{\alpha,\beta}|\mathcal{E}_{p^{m+6},\alpha,\beta}(X)|=\frac{p^{m+5}(p-1)^{2}}{p^{2m+12}}\frac{p^{10}}{p^{10}-1}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\frac{p^{m+5}(p-1)^{2}}{p^{m+6}}X^{\frac{1}{2}}\right).$

Other cases can be proven similarly. ∎

###  2.2. With finitely many local conditions

In this section, we give a proof of Theorem 1.5. Let $P=\\{p_{i}^{m_{i}}\\}$
be a finite set of powers of primes such that $p_{i}\geq 5$, $I=I_{P}$ be a
set of nonzero
$(\alpha_{i},\beta_{i})\in(\mathbb{Z}/p_{i}^{m_{i}}\mathbb{Z})^{2}$. We also
define a convolution on $I$ by $d*I:=\\{(d^{4}\alpha_{i},d^{6}\beta_{i})\\}.$
Let

$\mathcal{D}_{P,I}(X):=\\{(A,B)\in\mathcal{D}(X):(A,B)\equiv(\alpha_{i},\beta_{i})\pmod{p_{i}^{m_{i}}}\textrm{
for all }p_{i}^{m_{i}}\in P\\},$

and

$\mathcal{M}_{P,I}(X):=\\{(A,B)\in\mathcal{D}_{P,I}(X):(A,B)\textrm{ satisfies
}(M)\\}.$

Then, the analogue of Lemma 2.2 is as follows.

######  Lemma 2.7.

Assume that all pairs in $I$ satisfy $p_{i}^{4}\nmid\alpha_{i}$ or
$p_{i}^{6}\nmid\beta_{i}$. Then,

$\mathcal{D}_{P,I}(X)=\bigsqcup_{\begin{subarray}{c}d\leq X^{\frac{1}{12}}\\\
p_{i}\nmid d\end{subarray}}d*\mathcal{M}_{P,d^{-1}*I}(d^{-12}X).$

We also define

$\mathcal{E}_{P,I}(X):=\\{(A,B)\in\mathcal{M}_{P,I}(X):4A^{3}+27B^{2}\neq
0\\},$

and $\mathcal{S}_{P,I}(X)$ as the unique set satisfying
$\mathcal{M}_{P,I}(X)=\mathcal{E}_{P,I}(X)\bigsqcup\mathcal{S}_{P,I}(X)$. The
bound of error term $\mathcal{S}_{P,I}(X)$ can be obtained by following lemma.

######  Lemma 2.8.

For $P,I$, we have

$|\mathcal{S}_{P,I}(X)|\leq\frac{4^{|I|}}{\prod_{i}p_{i}^{m_{i}}}X^{\frac{1}{6}}+O(1).$

######  Proposition 2.9.

Assume that all pairs in $I$ satisfy $p_{i}^{4}\nmid\alpha_{i}$ or
$p_{i}^{6}\nmid\beta_{i}$, and $\prod p_{i}^{m_{i}}\leq X^{\frac{1}{3}}$.
Then,

$|\mathcal{E}_{P,I}(X)|=\prod_{i}\left(\frac{1}{p_{i}^{2m_{i}}}\frac{p_{i}^{10}}{p_{i}^{10}-1}\right)\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\frac{X^{\frac{1}{2}}}{\prod
p_{i}^{m_{i}}}\right).$

###### Proof.

It can be derived the proof of Proposition 2.3, Lemma 2.8, and the Chinese
remainder theorem. ∎

We recall some notations in Section 1. Let $\mathcal{LC}_{p}$ be one of good,
bad, split, non-split, an integer $a_{p}$ in the Weil’s bound at $p$, or one
of the Kodaira–Néron types $T$. We define $|\mathcal{LC}_{p}|$ as a
$\displaystyle\lim_{X\to\infty}\frac{|\mathcal{E}_{p}^{\mathcal{LC}_{p}}(X)|}{|\mathcal{E}(X)|}$,
which is $c_{\mathcal{LC}_{p}}(p)$ in Theorem 1.4. For the finite set of local
conditions $\mathcal{S}=\\{\mathcal{LC}_{p_{i}}\\}$, we define
$|\mathcal{S}|=\prod_{i}|\mathcal{LC}_{p_{i}}|$.

Now we are ready to prove Theorem 1.5. We first assume that any
$\mathcal{LC}_{p}$ is neither bad nor additive. Let $P=(p_{i}^{m_{i}})$, then
in the proof of Theorem 1.4, there are

$|\mathcal{LC}_{p_{i}}|\left(\frac{p_{i}^{10}-1}{p_{i}^{10}}\right)p_{i}^{2m_{i}}$

pairs $(\alpha_{i,j},\beta_{i,j})$ in
$(\mathbb{Z}/p_{i}^{m_{i}}\mathbb{Z})^{2}$ such that $E_{A,B}$ satisfies
$\mathcal{LC}_{p_{i}}$ if and only if
$(A,B)\equiv(\alpha_{i,j},\beta_{i,j})\pmod{p_{i}^{m_{i}}}$ for some
$(\alpha_{i,j},\beta_{i,j})$. Let $I_{p_{i}}=\\{(\alpha_{i,j},\beta_{i,j})\\}$
be the set of all the forementioned pairs.

Let $I=\prod I_{p_{i}}$. We denote an element of $I$ by $(\alpha,\beta)$ whose
$I_{p_{i}}$ component is $(\alpha_{i,j},\beta_{i,j})$, and define
$(A,B)\equiv(\alpha,\beta)\pmod{P}$ if and only if
$(A,B)\equiv(\alpha_{i,j},\beta_{i,j})\pmod{p_{i}^{m_{i}}}$. By Chinese
remainder theorem, there are

$\prod_{i}|\mathcal{LC}_{p_{i}}|\left(\frac{p_{i}^{10}-1}{p_{i}^{10}}\right)p_{i}^{2m_{i}}$

elements $(\alpha,\beta)$ in $I$ such that $E_{A,B}$ satisfies all
$\mathcal{LC}_{p_{i}}$ if and only if $(A,B)\equiv(\alpha,\beta)\pmod{P}$ for
some $(\alpha,\beta)$ in $I$. Together with Proposition 2.9, we have

$\displaystyle|\mathcal{E}^{\mathcal{S}}(X)|$ $\displaystyle=$
$\displaystyle\sum_{I}|\mathcal{E}_{P,I}(X)|=\sum_{I}\left(\prod_{i}\left(\frac{1}{p_{i}^{2m_{i}}}\frac{p_{i}^{10}}{p_{i}^{10}-1}\right)\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\frac{X^{\frac{1}{2}}}{\prod
p_{i}^{m_{i}}}\right)\right)$ $\displaystyle=$
$\displaystyle\left(\prod_{i}|\mathcal{LC}_{p_{i}}|\right)\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\prod_{i}|\mathcal{LC}_{p_{i}}|\left(\frac{p_{i}^{10}-1}{p_{i}^{10}}\right)p_{i}^{m_{i}}X^{\frac{1}{2}}\right)$
$\displaystyle=$
$\displaystyle|\mathcal{S}|\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\prod_{i}|\mathcal{LC}_{p_{i}}|\left(\frac{p_{i}^{10}-1}{p_{i}^{10}}\right)p_{i}^{m_{i}}X^{\frac{1}{2}}\right).$

Hence we proved this theorem except when there is a local condition
$\mathcal{LC}_{p}$ which is either bad or additive.

We use an induction on the number of $\mathcal{LC}_{p}$’s that are bad. Let
$\mathcal{S}=\\{\mathcal{LC}_{p_{i}}\\}$ be a finite set of local conditions
such that there is an $\mathcal{LC}_{p_{n}}$ which is bad, and
$\mathcal{S}^{\prime}=\mathcal{S}-\\{\mathcal{LC}_{p_{n}}\\}$. By induction
hypothesis for $\mathcal{S}^{\prime}\cup\\{\mathcal{LC}^{\prime}_{p_{n}}\\}$
where $\mathcal{LC}^{\prime}_{p_{n}}$ is good, we have

$|\mathcal{E}^{\mathcal{S}^{\prime}\cup\\{\mathcal{LC}^{\prime}_{p_{n}}\\}}(X)|=|\mathcal{S}^{\prime}|\frac{p_{n}^{2}-p_{n}}{p_{n}^{2}}\frac{p_{n}^{10}}{p^{10}_{n}-1}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\left(\prod_{i=1}^{n-1}c_{p_{i}}\right)p_{n}X^{\frac{1}{2}}\right).$

Again by induction hypothesis for $\mathcal{S}^{\prime}$, we have

$|\mathcal{E}^{\mathcal{S}^{\prime}}(X)|=|\mathcal{S}^{\prime}|\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\left(\prod_{i=1}^{n-1}c_{p_{i}}\right)X^{\frac{1}{2}}\right).$

Since
$|\mathcal{E}^{\mathcal{S}^{\prime}}(X)|=|\mathcal{E}^{\mathcal{S}^{\prime}\cup\\{\mathcal{LC}^{\prime}_{p_{n}}\\}}(X)|+|\mathcal{E}^{\mathcal{S}}(X)|$,
we can count elliptic curves with bad reduction conditions.

We can do the same thing for additive condition because additive condition is
the complement of the union of the conditions good, split, and non-split.

##  3\. The Frobenius Trace formula for elliptic curves

Let $L(s,E)$ be the normalized elliptic $L$-function and for which we have

$\displaystyle
L(s,E)=\sum_{n=1}^{\infty}\frac{\lambda_{E}(n)}{n^{s}},\qquad-\frac{L^{\prime}}{L}(s,E)=\sum_{n=1}^{\infty}\frac{\widehat{a_{E}}(n)\Lambda(n)}{n^{s}}.$

Here is the Frobenius trace formula for elliptic curves.

######  Theorem 3.1.

[Frobenius Trace Formula for Elliptic Curves] Let $k$ be a fixed positive
integer. Let $p_{i}$ be distinct primes $\geq 5$ such that
$\prod_{i=1}^{k}p_{i}=O(X^{\frac{1}{3}-\epsilon})$ for a fixed positive
$\epsilon>0$. Assume $e_{i}=1$ or $2$, and $r_{i}$ is odd or $2$ if $e_{i}=1$,
$r_{i}=1$ if $e_{i}=2$ for $i=1,\dots,k$. Then,

$\displaystyle\sum_{E\in\mathcal{E}(X)}\widehat{a_{E}}(p_{1}^{e_{1}})^{r_{1}}\widehat{a_{E}}(p_{2}^{e_{2}})^{r_{2}}\cdots\widehat{a_{E}}(p_{k}^{e_{k}})^{r_{k}}$
$\displaystyle=c|\mathcal{E}(X)|+O_{k}\left(2^{r}\left(\prod_{i=1}^{k}p_{i}\right)X^{\frac{1}{2}}\right)+O_{k}\left(\left(\sum_{i=1}^{k}\frac{1}{p_{i}}\right)X^{\frac{5}{6}}\right)$

where

$\displaystyle c=\left\\{\begin{array}[]{rl}0&\mbox{ if $e_{j}=1$ and $r_{j}$
is odd for some $j$},\\\ -1&\mbox{if $r_{j}=2$ for all $j$ with $e_{j}=1$, and
the sum of $r_{j}$'s with $e_{j}=2$ is odd,}\\\
1&\mbox{otherwise.}\end{array}\right.$

and $r$ is either the smallest odd integer $r_{i}$ or $0$ if there is no odd
$r_{i}$, and the last error term exists only if $e_{i}=1$ and $r_{i}=2$ or
$e_{i}=2$ for all $i$.

Also, we have

$\displaystyle\sum_{E\in\mathcal{E}(X)}\lambda_{E}(p_{1}^{e_{1}})^{r_{1}}\lambda_{E}(p_{2}^{e_{2}})^{r_{2}}\cdots\lambda_{E}(p_{k}^{e_{k}})^{r_{k}}$
$\displaystyle=d|\mathcal{E}(X)|+O_{k}\left(2^{r}\left(\prod_{i=1}^{k}p_{i}\right)X^{\frac{1}{2}}\right)+O_{k}\left(\left(\sum_{i=1}^{k}\frac{1}{p_{i}}\right)X^{\frac{5}{6}}\right)$

where

$\displaystyle d=\left\\{\begin{array}[]{rl}0&\mbox{ if $e_{j}=2$ or $e_{j}=1$
and $r_{j}$ is odd for some $j$},\\\ 1&\mbox{if $e_{i}=1$ and $r_{i}=2$ for
all $i$.}\end{array}\right.$

and the last error term exists only if $e_{i}=1$ and $r_{i}=2$ or $e_{i}=2$
for all $i$.

We give a proof of Theorem 3.1 for one prime and two primes. First, we
consider

$\displaystyle\sum_{E\in\mathcal{E}(X)}\widehat{a}_{E}(p)^{r}=\sum_{|a|\leq\lfloor
2\sqrt{p}\rfloor}\sum_{\begin{subarray}{c}E\in\mathcal{E}(X)\\\
a_{p}(E)=a\end{subarray}}\widehat{a}_{E}(p)^{r}+\sum_{\begin{subarray}{c}E\in\mathcal{E}(X)\\\
E\textrm{ mult red at
}p\end{subarray}}\widehat{a}_{E}(p)^{r}+\sum_{\begin{subarray}{c}E\in\mathcal{E}(X)\\\
E\textrm{ addi red at }p\end{subarray}}\widehat{a}_{E}(p)^{r}$

for an odd $r$. By Theorem 1.4 for $|a|\leq\lfloor 2\sqrt{p}\rfloor$, there
are

$\frac{(p-1)H(a^{2}-4p)}{2p^{2}}\frac{p^{10}}{p^{10}-1}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left((p-1)\frac{H(a^{2}-4p)}{2p}X^{\frac{1}{2}}\right)$

elliptic curves in $\mathcal{E}(X)$ with $\widehat{a}_{E}(p)=a/\sqrt{p}$.
Since $\sum_{|a|\leq\lfloor 2\sqrt{p}\rfloor}a^{r}H(a^{2}-4p)=0$, only the
contribution from the error term survives and it is at most
$O(2^{r}pX^{\frac{1}{2}})$ because

$\frac{p-1}{2p}\sum_{|a|\leq\lfloor
2\sqrt{p}\rfloor}\left(\frac{|a|}{\sqrt{p}}\right)^{r}H(a^{2}-4p)\leq
2^{r-1}\frac{p-1}{p}\sum_{|a|\leq\lfloor
2\sqrt{p}\rfloor}H(a^{2}-4p)=2^{r}(p-1).$

If an elliptic curve $E$ has additive reduction at a prime $p$, then
$\widehat{a}_{E}(p)=0$. So there is no contribution from the additive
reduction case. If $E$ has split (resp. non-split) multiplicative reduction,
then $\widehat{a}_{E}(p)=1/\sqrt{p}$ (resp. $\widehat{a}_{E}(p)=-1/\sqrt{p}$).
There are

$\frac{p-1}{2p^{2}}\frac{p^{10}}{p^{10}-1}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O(X^{\frac{1}{2}})$

elliptic curves in $\mathcal{E}(X)$ with the split prime $p$, and also there
are the same number of elliptic curves with the non-split prime $p$ up to
error term $O(X^{\frac{1}{2}})$, by Theorem 1.4. The contribution from
multiplicative reduction is $O(X^{\frac{1}{2}}/p^{\frac{r}{2}})$, hence we
have

$\displaystyle\sum_{E\in\mathcal{E}(X)}\widehat{a}_{E}(p)^{r}=O(2^{r}pX^{\frac{1}{2}}).$

The next case is

$\displaystyle\sum_{E\in\mathcal{E}(X)}\widehat{a}_{E}(p)^{2}.$

From the identity $\sum_{|a|\leq\lfloor
2\sqrt{p}\rfloor}a^{2}H(a^{2}-4p)=2p^{2}-2$ which is an application of the
Eichler–Selberg trace formula [Bir68],[2][2][2]There is a typo in [Bir68,
Theorem 2]. the contribution for good reductions at $p$ is

$\displaystyle\sum_{|a|\leq\lfloor
2\sqrt{p}\rfloor}\sum_{\begin{subarray}{c}E\in\mathcal{E}(X)\\\
a_{E}(p)=a\end{subarray}}\widehat{a}_{E}(p)^{2}=\left(1+O\left(\frac{1}{p}\right)\right)\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O(pX^{\frac{1}{2}}),$

and the contribution from the multiplicative reduction is

$\displaystyle\sum_{\begin{subarray}{c}E\in\mathcal{E}(X)\\\ E\textrm{ mult
red at
}p\end{subarray}}\widehat{a}_{E}(p)^{2}=\frac{p-1}{p^{3}}\frac{p^{10}}{p^{10}-1}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\frac{X^{\frac{1}{2}}}{p}\right).$

Hence we have

$\displaystyle\sum_{E\in\mathcal{E}(X)}\widehat{a}_{E}(p)^{2}=\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\frac{X^{\frac{5}{6}}}{p}+pX^{\frac{1}{2}}\right).$

The last case for a single prime $p$ is

$\displaystyle\sum_{E\in\mathcal{E}(X)}\widehat{a}_{E}(p^{2}).$

Note that $\widehat{a}_{E}(p^{2})=\widehat{a}_{E}(p)^{2}-2$ when $E$ has good
reduction at $p$, and $\widehat{a}_{E}(p^{2})=\widehat{a}_{E}(p)^{2}$ when $E$
has bad reduction at $p$. Then, from the computation for the previous case, we
can see easily that

$\displaystyle\sum_{E\in\mathcal{E}(X)}\widehat{a}_{E}(p^{2})=-\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\frac{X^{\frac{5}{6}}}{p}+pX^{\frac{1}{2}}\right),$

which is exactly the trace formula for one prime in Theorem 3.1.

Now, we consider the trace formula for two primes

$\displaystyle\sum_{E\in\mathcal{E}(X)}\widehat{a}_{E}(p_{1}^{e_{1}})^{r_{1}}\widehat{a}_{E}(p_{2}^{e_{2}})^{r_{2}}.$

Assume that $e_{1}=1$ and $r_{1}$ is odd. Fix a local condition for the second
prime $p_{2}$. We vary the local conditions for the first prime $p_{1}$, and
this gives $O(2^{r_{1}}p_{1}c_{p_{2}}X^{\frac{1}{2}})$. Then we sum up this
term over all local conditions for $p_{2}$, the error term
$O(2^{r_{1}}p_{1}p_{2}X^{\frac{1}{2}})$ follows.

Now, we need to deal with the cases $e_{i}=1$ and $r_{i}=2$ or $e_{i}=2$ and
$r_{i}=1$ for $i=1,2$. First, we consider the cases
$\mathcal{LC}_{p_{1}}=a_{1}$, and $\mathcal{LC}_{p_{2}}=a_{2}$ for
$|a_{1}|\leq\lfloor\sqrt{p_{1}}\rfloor$,
$|a_{2}|\leq\lfloor\sqrt{p_{2}}\rfloor$. Their contribution is, for example
$r_{1}=r_{2}=2$,

$\displaystyle\left(\prod_{i=1}^{2}\frac{(p_{i}-1)p_{i}^{10}}{2p_{i}^{3}(p_{i}^{10}-1)}\right)\frac{4}{\zeta(10)}X^{\frac{5}{6}}\cdot\sum_{|a_{1}|\leq\lfloor\sqrt{p_{1}}\rfloor,|a_{2}|\leq\lfloor\sqrt{p_{2}}\rfloor}a_{1}^{2}H(a_{1}^{2}-4p_{1})a_{2}^{2}H(a_{2}^{2}-4p_{2})$
$\displaystyle+O\left(\sum_{\begin{subarray}{c}|a_{1}|\leq\lfloor\sqrt{p_{1}}\rfloor,|a_{2}|\leq\lfloor\sqrt{p_{2}}\rfloor\end{subarray}}\frac{a_{1}^{2}H(a_{1}^{2}-4p_{1})a_{2}^{2}H(a_{2}^{2}-4p_{2})}{(2p_{1})(2p_{2})}X^{\frac{1}{2}}\right),$

which is, by the identity
$\sum_{|a|\leq\lfloor\sqrt{p}\rfloor}a^{2}H(a^{2}-4p)=2p^{2}-2$,

$\displaystyle\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\left(\frac{1}{p_{1}}+\frac{1}{p_{2}}\right)X^{\frac{5}{6}}+p_{1}p_{2}X^{\frac{1}{2}}\right).$

Since the case that $p_{1}$ or $p_{2}$ has multiplicative reduction, using the
trivial bound for
$\widehat{a}_{E}(p_{1}^{e_{1}})^{r_{1}}\widehat{a}_{E}(p_{2}^{e_{2}})^{r_{2}}$,
gives
$O\left(\left(\frac{1}{p_{1}^{2}}+\frac{1}{p_{2}^{2}}\right)X^{\frac{5}{6}}\right)$,
we verify the trace formula for $r_{1}=r_{2}=2$ case. The other three cases
can be handled similarly and we have

$\sum_{E\in\mathcal{E}(X)}\widehat{a}_{E}(p_{1}^{e_{1}})^{r_{1}}\widehat{a}_{E}(p_{2}^{e_{2}})^{r_{2}}=(-1)^{r_{1}+r_{2}}\frac{4}{\zeta(10)}X^{\frac{5}{6}}+O\left(\left(\frac{1}{p_{1}}+\frac{1}{p_{2}}\right)X^{\frac{5}{6}}+p_{1}p_{2}X^{\frac{1}{2}}\right).$

For a general $k$ primes, we can prove the trace formula in the same way.

For $\lambda_{E}(p^{e})$, we note that $\lambda_{E}(p)=\widehat{a}_{E}(p)$ and
$\lambda_{E}(p^{2})=\widehat{a}_{E}(p)^{2}-1$ if $E$ has good reduction at $p$
and $\lambda_{E}(p^{2})=\widehat{a}_{E}(p)^{2}$ otherwise. Hence, we can prove
the trace formula for $\lambda_{E}(n)$ similarly.

##  4\. The distribution of analytic ranks of elliptic curves

From now on, assume that every elliptic $L$-function satisfies Generalized
Riemann Hypothesis. Let $\gamma_{E}$ denote the imaginary part of a non-
trivial zero of $L(s,E)$. We index them using the natural order in real
numbers:

$\displaystyle\cdots\gamma_{E,-3}\leq\gamma_{E,-2}\leq\gamma_{E,-1}\leq\gamma_{E,0}\leq\gamma_{E,1}\leq\gamma_{E,2}\leq\gamma_{E,3}\cdots$

if analytic rank $r_{E}$ is odd,

$\displaystyle\cdots\gamma_{E,-3}\leq\gamma_{E,-2}\leq\gamma_{E,-1}\leq
0\leq\gamma_{E,1}\leq\gamma_{E,2}\leq\gamma_{E,3}\cdots$

otherwise.

In this section, first we obtain an upper bound on every $n$-th moment of
analytic ranks of elliptic curves and obtain a bound on the proportion of
elliptic curves with analytic rank $r_{E}\geq(1+C)9n$ for a positive constant
$C$ and a positive integer $n$. For this purpose, we compute an $n$-level
density with multiplicity. See [Mil, Part VI].

For $\sigma_{n}=\frac{2}{9n}$, we choose the following test function.

$\displaystyle\widehat{\phi}_{n}(u)=\frac{1}{2}\left(\frac{1}{2}\sigma_{n}-\frac{1}{2}|u|\right)\textrm{
for }|u|\leq\sigma_{n},\qquad\textrm{and
}\phi_{n}(x)=\frac{\sin^{2}(2\pi\frac{1}{2}\sigma_{n}x)}{(2\pi x)^{2}}.$

Note that $\phi_{n}(0)=\frac{\sigma_{n}^{2}}{4}$ and
$\widehat{\phi}_{n}(0)=\frac{\sigma_{n}}{4}$. We can understand the constant
$\frac{2}{9}$ as the limit of our trace formula. Then, easily we can check

(4.1)
$\displaystyle\int_{\mathbb{R}}|u|\widehat{\phi}_{n}(u)^{2}=\frac{1}{6}\phi_{n}(0)^{2}.$

The $n$-level density with multiplicity is

$\displaystyle
D_{n}^{*}(\mathcal{E},\Phi)=\frac{1}{|\mathcal{E}(X)|}\sum_{E\in\mathcal{E}(X)}\sum_{j_{1},j_{2},\dots,j_{n}}\phi_{n}\left(\gamma_{E,j_{1}}\frac{\log
X}{2\pi}\right)\phi_{n}\left(\gamma_{E,j_{2}}\frac{\log
X}{2\pi}\right)\cdots\phi_{n}\left(\gamma_{E,j_{n}}\frac{\log
X}{2\pi}\right),$

where $\gamma_{E,j_{k}}$ is an imaginary part of $j_{k}$-th zero of $L(s,E)$.
Then, trivially we have

(4.2)
$\displaystyle\frac{1}{|\mathcal{E}(X)|}\sum_{E\in\mathcal{E}(X)}r_{E}^{n}\leq\frac{1}{\phi_{n}(0)^{n}}D_{n}^{*}(\mathcal{E},\Phi).$

We show that for any $n$, $D_{n}^{*}(\mathcal{E},\Phi)$ has a closed
expression. By Weil’s explicit formula,

$\displaystyle\sum_{j}\phi_{n}\left(\gamma_{E,j}\frac{\log X}{2\pi}\right)$
$\displaystyle=\widehat{\phi}_{n}(0)\frac{\log N_{E}}{\log
X}-2\sum_{k=1}^{2}\sum_{p}\frac{\widehat{a_{E}}(p^{k})\log p}{p^{k/2}\log
X}\widehat{\phi}_{n}\left(\frac{k\log p}{\log X}\right)+O\left(\frac{1}{\log
X}\right)$
$\displaystyle\leq\widehat{\phi}_{n}(0)-2\sum_{k=1}^{2}\sum_{p}\frac{\widehat{a_{E}}(p^{k})\log
p}{p^{k/2}\log X}\widehat{\phi}_{n}\left(\frac{k\log p}{\log
X}\right)+O\left(\frac{1}{\log X}\right),$

where the inequality holds due to $\log N_{E}/\log X\leq 1+O(1/\log X)$ by
Ogg’s formula [Sil, §11].

Hence, we have

$\displaystyle\frac{1}{|\mathcal{E}(X)|}\sum_{E\in\mathcal{E}(X)}\sum_{j_{1},j_{2},\dots,j_{n}}\phi_{n}\left(\gamma_{E,j_{1}}\frac{\log
X}{2\pi}\right)\phi_{n}\left(\gamma_{E,j_{2}}\frac{\log
X}{2\pi}\right)\cdots\phi_{n}\left(\gamma_{E,j_{n}}\frac{\log X}{2\pi}\right)$
$\displaystyle\leq\frac{1}{|\mathcal{E}(X)|}\sum_{E\in\mathcal{E}(X)}\left(\widehat{\phi}_{n}(0)-\frac{2}{\log
X}\sum_{m<X^{\frac{2}{9n}}}\frac{\widehat{a}_{E}(m)\log
m}{\sqrt{m}}\widehat{\phi}_{n}\left(\frac{\log m}{\log
X}\right)+O\left(\frac{1}{\log X}\right)\right)^{n},$

where $m$’s are primes or squares of a prime. By the standard argument such as
[Rub01, Lemma 2] or [CK15, Lemma 4.4], we can take the term $O(1/\log X)$ out
of the bracket, and we have

$\displaystyle D_{n}^{*}(\mathcal{E},\Phi)$
$\displaystyle\leq\frac{1}{|\mathcal{E}(X)|}\sum_{S}\widehat{\phi}_{n}(0)^{|S^{c}|}\left(-\frac{2}{\log
X}\right)^{|S|}$
$\displaystyle\times\sum_{m_{i_{1}},m_{i_{2}},\dots,m_{i_{k}}}\frac{\Lambda(m_{i_{1}})\Lambda(m_{i_{2}})\cdots\Lambda(m_{i_{k}})}{\sqrt{m_{i_{1}}m_{i_{2}}\cdots
m_{i_{k}}}}\widehat{\phi}_{n}\left(\frac{\log m_{i_{1}}}{\log
X}\right)\cdots\widehat{\phi}_{n}\left(\frac{\log m_{i_{k}}}{\log X}\right)$
$\displaystyle\times\sum_{E\in\mathcal{E}(X)}\widehat{a}_{E}(m_{i_{1}})\widehat{a}_{E}(m_{i_{2}})\cdots\widehat{a}_{E}(m_{i_{k}})+O\left(\frac{1}{\log
X}\right),$

where $m_{i}$’s are primes or squares of a prime with $m_{i}\leq
X^{\frac{2}{9n}}$ and $S=\\{i_{i},i_{2},\dots,i_{k}\\}$ runs over every subset
of $\\{1,2,3,\cdots,n\\}$. In next two propositions, we show that only the
terms $m_{i_{1}}m_{i_{2}}\cdots m_{i_{k}}=\square$ give the main term and the
contribution of the other terms is $O(1/\log X)$.

######  Proposition 4.1.

$\displaystyle\sum_{E\in\mathcal{E}(X)}\sum_{m_{i_{1}}m_{i_{2}}\dots
m_{i_{k}}\neq\square}\frac{\Lambda(m_{i_{1}})\cdots\Lambda(m_{i_{k}})\widehat{a}_{E}(m_{i_{1}})\cdots\widehat{a}_{E}(m_{i_{k}})}{\sqrt{m_{i_{1}}m_{i_{2}}\cdots
m_{i_{k}}}}\widehat{\phi}_{n}\left(\frac{\log m_{i_{1}}}{\log
X}\right)\cdots\widehat{\phi}_{n}\left(\frac{\log m_{i_{k}}}{\log X}\right)$
$\displaystyle\ll|\mathcal{E}(X)|.$

###### Proof.

Note that
$\widehat{a}_{E}(m_{i_{1}})\widehat{a}_{E}(m_{i_{2}})\cdots\widehat{a}_{E}(m_{i_{k}})$
is of the form

$\widehat{a}_{E}(p_{1})^{e_{1}}\widehat{a}_{E}(p_{2})^{e_{2}}\cdots\widehat{a}_{E}(p_{t})^{e_{t}}\widehat{a}_{E}(q_{1}^{2})^{l_{1}}\widehat{a}_{E}(q_{2}^{2})^{l_{2}}\cdots\widehat{a}_{E}(q_{s}^{2})^{l_{s}},$

with with $e_{1}+\cdots+e_{t}+l_{1}+\cdots+l_{s}=k$. Here
$p_{1},p_{2},\dots,p_{t}$ are distinct primes and $q_{1},q_{2},\dots,q_{s}$
are distinct primes, but some $q_{j}$ might be equal to some $p_{i}$. For a
while we assume that the primes $p_{1},\dots,p_{t},q_{1},\dots,q_{s}$ are all
distinct.

Since one of $e_{i}$’s is odd by our assumption, then by the Frobenius trace
formula 3.1,

$\displaystyle\sum_{E\in\mathcal{E}(X)}\widehat{a}_{E}(p_{1})^{e_{1}}\widehat{a}_{E}(p_{2})^{e_{2}}\cdots\widehat{a}_{E}(p_{t})^{e_{t}}\widehat{a}_{E}(q_{1}^{2})^{l_{1}}\widehat{a}_{E}(q_{2}^{2})^{l_{2}}\cdots\widehat{a}_{E}(q_{s}^{2})^{l_{s}}=O(p_{1}p_{2}\cdots
p_{t}q_{1}q_{2}\cdots q_{s}X^{\frac{1}{2}}).$

The contribution of this case in the worst situation is at most

$\displaystyle\ll
X^{\frac{1}{2}}\left(\sum_{p<X^{\frac{2}{9n}}}p^{\frac{1}{2}}\log
p\right)^{k}\ll X^{\frac{1}{2}}(X^{\frac{1}{3n}})^{n}\ll X^{\frac{5}{6}}.$

Now, we assume that some $p_{i}$ is equal to some $q_{j}$. Since
$\widehat{a}_{E}(q^{2})^{l}=(\widehat{a}_{E}(q)^{2}-2)^{l}$ if $E$ has good
reduction at $q$ and $\widehat{a}_{E}(q^{2})^{l}=\widehat{a}_{E}(q)^{2l}$
otherwise, still we can use the Frobenius trace formula. ∎

######  Proposition 4.2.

For a subset $S=\\{i_{1},i_{2},\dots,i_{k}\\}$ of $\\{1,2,\dots,n\\}$,

$\displaystyle\frac{1}{|\mathcal{E}(X)|}\left(\frac{-2}{\log
X}\right)^{|S|}\sum_{E\in\mathcal{E}(X)}\sum_{m_{i_{1}}m_{i_{2}}\dots
m_{i_{k}}=\square}\left(\prod_{j=1}^{|S|}\frac{\Lambda(m_{i_{j}})\widehat{a}_{E}(m_{i_{j}})}{\sqrt{m_{i_{j}}}}\widehat{\phi}_{n}\left(\frac{\log
m_{i_{j}}}{\log X}\right)\right)$
$\displaystyle=\sum_{\begin{subarray}{c}S_{2}\subset S\\\
|S_{2}|\text{even}\end{subarray}}\left(\frac{1}{2}\phi_{n}(0)\right)^{|S_{2}^{c}|}\left|S_{2}\right|!\left(\int_{\mathbb{R}}|u|\widehat{\phi}_{n}(u)^{2}du\right)^{\frac{|S_{2}|}{2}}+O\left(\frac{1}{\log
X}\right).$

###### Proof.

In this proof, we compute the double sum not considering the term
$\frac{1}{|\mathcal{E}(X)|}\left(\frac{-2}{\log X}\right)^{k}$. We show that
every contribution except one is $\ll X^{\frac{5}{6}}(\log X)^{k-1}$, hence
they become the error term $O(1/\log X)$ in the end.

Note that
$\widehat{a}_{E}(m_{i_{1}})\widehat{a}_{E}(m_{i_{2}})\cdots\widehat{a}_{E}(m_{i_{k}})$
is of the form

$\widehat{a}_{E}(p_{1})^{e_{1}}\widehat{a}_{E}(p_{2})^{e_{2}}\cdots\widehat{a}_{E}(p_{t})^{e_{t}}\widehat{a}_{E}(q_{1}^{2})^{l_{1}}\widehat{a}_{E}(q_{2}^{2})^{l_{2}}\cdots\widehat{a}_{E}(q_{s}^{2})^{l_{s}},$

with with $e_{1}+\cdots+e_{t}+l_{1}+\cdots+l_{s}=k$ and $e_{i}$’s are all
even. If $e_{i}\geq 4$ for some $i$ or $l_{j}\geq 2$ for some $j$, then by the
trivial bound, this term is majorized by $X^{\frac{5}{6}}(\log X)^{k-1}$. Let
$S_{2}$ be a subset of $S$ with even cardinality $2t$:

$S_{2}=\\{i_{a_{1}},i_{a_{2}},\cdots,i_{a_{2t-1}},i_{a_{2t}}\\},\qquad
S_{2}^{c}=\\{i_{b_{1}},i_{b_{2}},\cdots,i_{b_{s}}\\}.$

There are $(2t)!/2^{t}$ ways to pair up two elements in $S_{2}$. For example,
we consider the following pairings.

$\displaystyle(i_{a_{1}},i_{a_{2}}),(i_{a_{3}},i_{a_{4}}),(i_{a_{5}},i_{a_{6}}),\cdots,(i_{a_{2t-1}},i_{a_{2t}}).$

This set of pairings corresponds the following sum

$\displaystyle\sum_{E\in\mathcal{E}(X)}\widehat{a}_{E}(p_{i_{a_{1}}})^{2}\widehat{a}_{E}(p_{i_{a_{3}}})^{2}\cdots\widehat{a}_{E}(p_{i_{a_{2t-1}}})^{2}\widehat{a}_{E}(q_{i_{b_{1}}}^{2})\widehat{a}_{E}(q_{i_{b_{2}}}^{2})\cdots\widehat{a}_{E}(q_{i_{b_{s}}}^{2})$

where $2t+s=k$. By the Frobenius trace formula, the above sum is

$\displaystyle|\mathcal{E}(X)|\cdot\left\\{\begin{array}[]{clc}1&\mbox{ if $s$
is even,}\\\ -1&\mbox{ if $s$ is odd}\end{array}\right.+O\left(p_{1}\cdots
p_{t}q_{1}\cdots
q_{s}X^{\frac{1}{2}}+\left(\frac{1}{p_{1}}+\cdots+\frac{1}{p_{t}}+\frac{1}{q_{1}}+\cdots\frac{1}{q_{s}}\right)X^{\frac{5}{6}}\right).$

The contribution from the error term $O(p_{1}\cdots p_{t}q_{1}\cdots
q_{s}X^{\frac{1}{2}})$ is dominated by

$(X^{\frac{2}{9n}}\log X)^{t}(X^{\frac{1}{9n}})^{s}X^{\frac{1}{2}}\ll
X^{\frac{1}{2}+\frac{2t+s}{9n}}(\log X)^{t}.$

The contribution from the error term
$O\left(\left(\frac{1}{p_{1}}+\cdots+\frac{1}{p_{t}}+\frac{1}{q_{1}}+\cdots\frac{1}{q_{s}}\right)X^{\frac{5}{6}}\right)$
is dominated by $X^{\frac{5}{6}}(\log X)^{k-1}$. The main term of the sum,
after being divided by $|\mathcal{E}(X)|\left(\frac{\log X}{-2}\right)^{k}$,
gives rise to

$\displaystyle\prod_{i=1}^{t}\left(\left(\frac{-2}{\log
X}\right)^{2}\sum_{p}\frac{\log^{2}p}{p}\widehat{\phi}_{n}\left(\frac{\log
p}{\log X}\right)^{2}\right)\times\prod_{j=1}^{s}\left(\frac{2}{\log
X}\sum_{q}\frac{\log q}{q}\widehat{\phi}_{n}\left(\frac{2\log q}{\log
X}\right)\right),$

which equals, by the prime number theorem,

$\displaystyle\left(2^{t}\prod_{i=1}^{t}\int_{\mathbb{R}}|u|\widehat{\phi}_{n}(u)^{2}du\right)\left(\left(\frac{1}{2}\right)^{s}\prod_{j=1}^{s}\int_{\mathbb{R}}\widehat{\phi}_{n}(u)du\right).$

Since there are $(2t)!/2^{t}$ ways to pair up two elements in $S_{2}$, the
claim follows. ∎

By Propositions 4.1 and 4.2, and (4.1) we have the following inequality

$\displaystyle
D_{n}^{*}(\mathcal{E},\Phi)\leq\phi_{n}(0)^{n}\sum_{S}\left(\frac{1}{\sigma_{n}}\right)^{|S^{c}|}\sum_{\begin{subarray}{c}S_{2}\subset
S\\\
|S_{2}|\text{even}\end{subarray}}\left(\frac{1}{2}\right)^{|S_{2}^{c}|}\left|S_{2}\right|!\left(\frac{1}{6}\right)^{|S_{2}|/2}+O\left(\frac{1}{\log
X}\right),$

and, by $(\ref{moment-ineq})$, Theorem 1.3 follows.

###  4.1. Proof of Theorem 1.1

We choose the test function $\phi_{2n}(x)$. Then
$\widehat{\phi}_{2n}(0)=\frac{1}{4}\sigma_{2n}$, and
$\phi_{2n}(0)=\frac{1}{4}\sigma_{2n}^{2}$.

By Weil’s explicit formula, we have

$r_{E}\phi_{2n}(0)\leq\widehat{\phi}_{2n}(0)-\frac{2}{\log
X}\sum_{m_{i}}\frac{\widehat{a}_{E}(m_{i})\Lambda(m_{i})}{\sqrt{m_{i}}}\widehat{\phi}_{2n}\left(\frac{\log
m_{i}}{\log X}\right)+O\left(\frac{1}{\log X}\right),$

hence

$\displaystyle
r_{E}\leq\frac{1}{\sigma_{2n}}+\frac{4}{\sigma_{2n}^{2}}\left(-\frac{2}{\log
X}\sum_{m_{i}}\frac{\widehat{a}_{E}(m_{i})\Lambda(m_{i})}{\sqrt{m_{i}}}\widehat{\phi}_{2n}\left(\frac{\log
m_{i}}{\log X}\right)\right)+O\left(\frac{1}{\sigma_{2n}^{2}\log X}\right).$

Now assume that $r_{E}\geq\frac{1+C}{\sigma_{2n}}$ with some positive constant
$C$. Then, for sufficiently large $X$,

$\displaystyle-\frac{2}{\log
X}\sum_{m_{i}}\frac{\widehat{a}_{E}(m_{i})\Lambda(m_{i})}{\sqrt{m_{i}}}\widehat{\phi}_{2n}\left(\frac{\log
m_{i}}{\log X}\right)\geq\frac{C\sigma_{2n}}{4}.$

Therefore,

$\displaystyle\left|\\{E\in\mathcal{E}(X)|r_{E}\geq\frac{1+C}{\sigma_{2n}}\\}\right|$
$\displaystyle\left(\frac{C\sigma_{2n}}{4}\right)^{2n}\leq\sum_{E\in\mathcal{E}(X)}\left(-\frac{2}{\log
X}\sum_{m_{i}}\frac{\widehat{a}_{E}(m_{i})\Lambda(m_{i})}{\sqrt{m_{i}}}\widehat{\phi}_{2n}\left(\frac{\log
m_{i}}{\log X}\right)\right)^{2n}$
$\displaystyle\leq\left(\frac{\sigma_{2n}^{2}}{4}\right)^{2n}\sum_{S_{2}\subset\\{1,2,3,\dots,2n\\}}\left(\frac{1}{2}\right)^{|S_{2}^{c}|}|S_{2}|!\left(\frac{1}{6}\right)^{|S_{2}|/2}|\mathcal{E}(X)|+O\left(\frac{X^{\frac{5}{6}}}{\log
X}\right),$

where the second inequality is justified by Propositions 4.1, 4.2, and finally
we obtain

$\displaystyle P\left(r_{E}\geq(1+C)\cdot
9n\right)\leq\frac{\sum_{k=0}^{n}{{2n}\choose{2k}}\left(\frac{1}{2}\right)^{2n-2k}(2k)!\left(\frac{1}{6}\right)^{k}}{(C\cdot
9n)^{2n}}.$

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