# Audio-Visual Instance Segmentation

Ruohao Guo
Peking University
Yaru Chen
University of Surrey
Yanyu Qi
China Agricultural University
Wenzhen Yue
Peking University
Dantong Niu
University of California, Berkeley
Xianghua Ying*
Peking University

###### Abstract

In this paper, we propose a new multi-modal task, namely audio-visual instance
segmentation (AVIS), in which the goal is to identify, segment, and track
individual sounding object instances in audible videos, simultaneously. To our
knowledge, it is the first time that instance segmentation has been extended
into the audio-visual domain. To better facilitate this research, we construct
the first audio-visual instance segmentation benchmark (AVISeg). Specifically,
AVISeg consists of 1,258 videos with an average duration of 62.6 seconds from
YouTube and public audio-visual datasets, where 117 videos have been annotated
by using an interactive semi-automatic labeling tool based on the Segment
Anything Model (SAM). In addition, we present a simple baseline model for the
AVIS task. Our new model introduces an audio branch and a cross-modal fusion
module to Mask2Former to locate all sounding objects. Finally, we evaluate the
proposed method using two backbones on AVISeg. We believe that AVIS will
inspire the community towards a more comprehensive multi-modal understanding.

††* Corresponding author.

##  1 Introduction

Audio and vision are two highly relied upon modalities for humans to perceive,
navigate, and understand their environmental surroundings. Motivated by this
perceptual ability of humans, various audio-visual tasks have been studied,
such as audio-visual recognition [23, 13, 35], audio-visual event parsing [8,
20, 5], audio-visual question answering [28, 15, 33], and audio-visual
localization [24, 21, 25, 11]. However, owing to the lack of pixel-level
annotations, the above tasks are limited to video- or frame-level
classification problems eventually.

Figure 1: Comparison of different audio-visual segmentation tasks. (a) Audio-
Visual Saliency Detection (AVSD) only requires binary segmentation. (b) Audio-
Visual Semantic Segmentation (AVSS) associates one category with every pixel.
(c) Audio-Visual Instance Segmentation (AVIS) treats each sounding object of
the same class as an individual instance.

Recently, a novel pixel-level audio-visual segmentation (AVS) task has been
proposed in [36]. The goal of AVS task is to segment sounding objects from
videos, i.e., densely predict whether each pixel corresponds to the given
audio. It encompasses two sub-tasks: audio-visual saliency detection (AVSD)
and audio-visual semantic segmentation (AVSS). As illustrated in Fig. 1, AVSD
only requires binary segmentation, whereas AVSS entails generating semantic
maps that further predict the category information of the sounding objects. To
accomplish it, [36] construct the first pixel-level audio-visual segmentation
dataset, namely AVSBench, which provides ground truth labels for sounding
objects in audible videos. Then, they present a baseline model equipped with
well-designed temporal pixel-wise audio-visual interaction (TPAVI) modules for
cross-modal fusion and guiding audio-visual segmentation.

In this paper, we introduce a more difficult task: audio-visual instance
segmentation (AVIS), extending the instance segmentation problem to the audio-
visual domain. Different from AVSD and AVSS, the new problem aims at
simultaneously identifying, segmenting and tracking individual sounding object
instances in a given audible video. Fig. 1 shows a sample video with ground
truth annotations for this problem. This task provides opportunities for
practical applications which requires more fine-grained masks, including video
editing, virtual reality, robot navigation, etc. To our best knowledge, this
is the first work to address audio-visual instance segmentation problem.

To this end, we propose the first audio-visual instance segmentation
benchmark, termed AVISeg. AVISeg contains 1,258 videos with an average
duration of 62.6 seconds, where 117 videos have been annotated. The emphasized
properties of AVISeg are summarised as follows. (1) Long-term. Videos in
AVISeg last 62.6 seconds on average (5 seconds per video in AVSBench), which
is much closer to real applications. (2) Single and Multiple Sound Sources. In
a video, sound may originate from a single object or multiple objects
simultaneously. (3) Comprehensive Categories. AVISeg includes 26 object
classes (e.g., person, violin, dog, car, etc.) from 5 scenarios: music,
speaking, animal, machine and panorama. (4) Task Generality. Our new dataset
can also be served as a benchmark for AVSD and AVSS tasks. In addition, we
propose a simple baseline method for audio-visual instance segmentation. Based
upon Mask2Former [7, 6] which is a query-based method for image and video
instance segmentation, a new audio branch and a multi-modal fusion module are
added to the framework for segmenting and tracking sounding instances in
consecutive frames.

##  2 Related Work

###  2.1 Video Instance Segmentation

Video Instance Segmentation (VIS) task is first introduced in [31], including
identifying, segmenting, and tracking all individual objects within a video
simultaneously. Several studies have successfully attained the objective
through diverse approaches.

Based on Mask R-CNN [9], Mask Track R-CNN [31] designs a tracking head to
associate each instance within the video sequence. It also adopts an external
memory scheme to store instance-specific features across multiple consecutive
frames. Maskprop [2] extends the Hybrid Task Cascade Network [4] by leveraging
the previously predicted masks to crop the extracted features. It subsequently
propagates instance region features to adjacent frames, enabling the
incorporation of temporal dynamics into the modeling process. Based on the
one-stage FCOS framework [26], SipMask [3] introduces a spatial preservation
module that computes spatial coefficients for mask predictions. In STEm-Seg
[1], video clips are conceptualized as 3D spatiotemporal volumes. The
separation of object instances is achieved through clustering learned
embeddings. CrossVIS [32] presents a crossover learning framework that
harnesses contextual information across multiple video frames to enhance the
learning process. Most of the above methods depend on heuristic rules for
instance association, or involve iterative steps for mask generation and
optimization.

Transformer-based architectures later stand out as a notable innovation,
especially in terms of accuracy. VisTR [27] adopts a query-based paradigm to
jointly segment and track instances by extending DETR to the VIS domain.
Following this DETR-like framework, subsequent studies [6, 29] achieve
considerable performance improvement on benchmarks. In order to improve
computation efficiency, IFC [12] reduces the overhead of message passing
between frames by using memory tokens, and DVIS [34] designs a referring
tracker and temporal refiner to efficiently utilize the information of the
overall video.

###  2.2 Audio-Visual Segmentation

The Audio-Visual Segmentation (AVS) task aims to localize and predict the
pixel-wise segmentation mask of sounding objects. Three challenges are
inherent to this task: 1) the precise localization of audible frames; 2) the
accurate delineation of the contours of sounding objects; and 3) the
classification for the category semantics of different sounding objects. In
view of the difficulties above, the annotation of the first audio-visual
segmentation benchmark AVSBench [36] is carried out in two different manners:
1) binary masks of sounding objects representing audio-associated pixels of
sounding objects (for audio-visual saliency detection), and 2) semantic maps
indicating the object categories (for audio-visual semantic segmentation).

Figure 2: Example sequences of our audio-visual instance segmentation (AVISeg)
dataset. From top to bottom, these videos come from music, speaking, animal,
machine and panorama scenarios, respectively.

Zhou et al. [36] propose regularization loss and design a module for temporal
pixel-wise audio-visual interaction (TPAVI), a cross-modal attention based
fusion strategy where audio semantics serve as guidance for the visual
segmentation procedure. More recently, CATR [17] introduces a memory-efficient
and decoupled audio-video transformer that combines audio and video features
from their respective temporal and spatial dimensions. The audio-constrained
learnable query is proposed for incorporating audio information during
decoding, containing object-level information that can select which object is
being referred to segment. ECMVAE [19] factorizes the latent-space
representations for each modality with two components: a modality-shared
representation and a modality-specific representation. This factorization
process includes an orthogonality constraint imposed between the shared and
specific representations, ensuring the preservation of the unique attribute
embedded within the factorized latent code. A regularizing technique is also
designed to maximize mutual information and foster comprehensive exploration
of each modality.

##  3 Dataset

To construct AVISeg benchmark, we collect and choose 1,258 videos from YouTube
and publicly available audio-visual datasets (MUSIC-AVQA [16] and YT-ALL
[22]). AVISeg satisfies the four principles listed below to ensure the
diversity and complexity of audio-visual scenes.

(1) Long-term AVIS. Compared to the AVSBench[36] where the average length of
each video is only 5 seconds, we ensure videos in AVISeg last about 62.6
seconds. The duration is roughly 12.5 times longer than the short-term videos,
bringing it much closer to real applications.

(2) Single and Multiple Sound Sources. In a video, sound may originate from a
single object or multiple objects. For multi-source videos, some object
instances may emit sounds simultaneously, leading to audio signals becoming
entangled. Therefore, this multiple-object attribute requires the model to
accurately locate and distinguish the sound source among multiple potential
candidates.

Figure 3: The histogram of videos for 26 categories in our AVISeg dataset.
Objects are sorted by frequency of appearance. The entire category set roughly
covers various sounding objects that are commonly seen in real scenarios.

(3) Comprehensive Categories. AVISeg contains 26 common instance categories
(e.g., person, violin, piano, dog, cat, car, airplane, etc.) from 5 real-world
dynamic scenarios: music, speaking, animal, machine, and panorama.

(4) Task Generality. Our new dataset can also serve as a benchmark for various
pixel-level video understanding problems, such as AVSD and AVSS tasks.

Figure 4: The dependencies between categories and scenarios.

After reviewing over 20,000 potential candidates, we carefully selected the
most appropriate videos that satisfy the above criterion. Ultimately, we
select 1,258 videos in total to create a new benchmark whose average duration
is 62.6 seconds. We then adopt an interactive semi-automatic annotation tool
based on Segment Anything Model (SAM [14]) to label sounding object instances
belonging to the category set exhaustively in these videos. Fig. 2 shows some
sample videos with ground truth annotations.

Our new dataset AVISeg contains 1,258 videos over 26 classes. The detail and
distribution of videos can be shown in Fig. 3. Videos cover 5 real-world
scenarios with a cue of primary sound sources, i.e., music (e.g., singing,
instrument playing), speaking (e.g., conversation, monologue), animal (e.g.,
dog barking, cat meowing, lion roaring), machine (e.g., airplane flying, car
running, lawn mowing), and panorama (360° panoramic videos). The dependencies
between categories and scenarios are illustrated in Fig 4. We sample 78,665
frames with a frame rate of 1 FPS. In addition, 117 videos have been annotated
with high quality and they are divided into 84 training and 33 testing. There
are 6,973 frames, 281 objects, and 15,355 masks in total.

##  4 Method

Figure 5: The overview architecture of the proposed baseline model.

###  4.1 Problem Definition

The goal of the audio-visual instance segmentation task is to localize,
segment and track the sounding instances with the video and audio sequences as
input. Pre-define the category label set as $\mathcal{C}=\\{1,...,N\\}$, where
$N$ is the number of categories. Given the $T$ frames of video sequence with
the corresponding audio signals, we assume there are $M$ instances in total
belonging to $N$ categories that have been sounding in the sequences. For each
object $i$, $c^{i}\in\mathcal{C}$ indicates its category, and
$\mathbf{m}_{p...q}^{i}$ denotes its binary mask through the video, with
$\\{p...q\\}$ indicates the time that the object emits sound. Audio-visual
instance segmentation algorithm requires instance hypotheses. For each
hypothesis $j$, we predict its binary mask
$\tilde{\mathbf{m}}_{\tilde{p}...\tilde{q}}^{j}$, where
$\\{\tilde{p}...\tilde{q}\\}\in T$, category label
$\tilde{c}^{j}\in\mathcal{C}$ and confidence score $\tilde{s}^{j}\in[0,1]$.

During training, we aim to minimize the ground truth and the hypotheses, which
requires the algorithm to first detect and locate all sounding instances based
on the audio-visual information, and then track these sounding objects
precisely. The difference between this task and visual instance segmentation
[31] is that we focus more on the sounding object. In other words, the
$\\{\tilde{p}...\tilde{q}\\}$ needs to be predicted based on both of the
visual dynamics and the corresponding audio information. Compared to audio-
visual semantic segmentation, our new task requires more refined masks, i.e.,
identifying individual sounding instances.

###  4.2 Baseline Model

##### Encoder.

Following [36, 37], we apply an image encoder and audio encoder independently
to encode the corresponding signals. For the visual part, we denote the $T$
input frames as $x_{\text{visaul}}\in\mathbb{R}^{T\times 3\times H\times W}$.
A visual backbone (ResNet-50 or Swin-Base) is used to extract and produce
hierarchical features indicated as $\mathbf{F}_{i}\in\mathbb{R}^{T\times
h_{i}\times w_{i}\times C_{i}}$, where $(h_{i},w_{i})=(H,W)/2^{i}$,
$i=2,3,4,5$. For the audio part, following [10], we first resample the audio
signal to 16kHz mono audio
$x_{\text{audio}}\in\mathbb{R}^{N_{\text{samples}}\times 96\times 64}$,
$N_{\text{samples}}$ is related to the length of an audio clip. Then Fourier
transform is used to get a mel spectrum. Finally, we feed it to an audio
backbone (VGGish) to obtain the audio feature denoted as
$\mathbf{F}_{\text{audio}}\in\mathbb{R}^{T\times d}$, where $d$ is the feature
dimension set to 128.

##### Multi-modal Fusion.

With the visual feature $\mathbf{F}_{i}$ and audio feature
$\mathbf{F}_{\text{audio}}$ obtained above, we adopt the temporal pixel-wise
audio-visual interaction module (TPAVI) [37] to fuse visual features and the
corresponding audio features, so as to highlight multiple dynamic sounding
objects. Specifically, $\mathbf{F}_{\text{audio}}$ is further projected to a
feature space and reshaped to the exact same size as $\mathbf{F}_{i}$. Then
the interaction of the pixel-level visual feature and audio feature is modeled
in the TPAVI module to get the final fused feature $\mathbf{Z}_{i}(i=3,4,5)$
for subsequent decoding.

##### Decoder.

We first feed the output feature from the TPAVI module to a pixel decoder to
transform the low-resolution features to high-resolution per-pixel embeddings
$\hat{\mathbf{Z}}$. Then, like [7], we adopt a query-based Transformer decoder
to further learn the semantic-level features of the sounding object.
Specifically, we generate the audio query
$\mathbf{Q}_{\text{query}}\in\mathbb{R}^{N_{\text{query}}\times d}$ by
aggregating audio features $\mathbf{F}_{\text{audio}}$ on T dimension and
repeating the processed features $N_{\text{query}}$ times. We introduce a set
of learnable instance queries $\mathbf{Q}_{\text{ins}}\in\mathbb{R}^{T\times
N_{\text{query}}\times d}$, where each query represents a potential instance.
$\mathbf{Q}_{\text{query}}$ and $\mathbf{Q}_{\text{ins}}$ are added element-
wise and fed to the query-based transformer decoder. During the decoding
process, the aggregated queries continuously capture the audio-visual
information from the encoder’s outputs. This enables the model to process
semantic-level information of sounding objects with higher accuracy.

After obtaining the output query $\mathbf{Q}_{\text{output}}$ and high-
resolution feature $\hat{\mathbf{Z}}$ from pixel decoder, we predict the mask
$\mathcal{M}\in\mathbb{R}^{T\times N_{\text{query}}\times h\times w}$ by
multiplying the outputs from the two residual structures before a simple
linear layer. This can be formulated as Eq. 1, where $MLP$ is the multi-layer
projection process.

$\mathcal{M}=Linear(\hat{\mathbf{Z}}+MLP(\hat{\mathbf{Z}}\cdot\mathbf{Q}_{\text{output}}))$
(1)

##  5 Experiment

###  5.1 Experiment Setup

####  5.1.1 Evaluation Metrics

Following the metrics proposed in VIS [31], we employ average precision (AP)
and average recall (AR) for evaluation. AP is defined as the measurement of
the area under the precision-recall curve, which is subsequently averaged
across various intersection-over-union (IoU) thresholds. We adopt 10 IoU
thresholds ranging from 50% to 95% at 5% intervals. AR is defined as the
highest recall value given a specific number of segmented instances per video.
Both of these metrics undergo initial assessment on a per-category basis,
followed by averaging across the entire category set. The computation of IoU
between a ground truth instance $\mathbf{m}_{p...q}^{i}$ and a hypothetical
instance $\tilde{\mathbf{m}}_{\tilde{p}...\tilde{q}}^{j}$ is as follows.
First, extend $p$ and $\tilde{p}$ to 1, $q$ and $\tilde{q}$ to $T$ by padding
empty masks. Then,

$\operatorname{IoU}(i,j)=\frac{\Sigma_{t=1}^{T}\left|\mathbf{m}_{t}^{i}\cap\tilde{\mathbf{m}}_{t}^{j}\right|}{\Sigma_{t=1}^{T}\left|\mathbf{m}_{t}^{i}\cup\tilde{\mathbf{m}}_{t}^{j}\right|}$
(2)

This IoU metric quantifies the spatiotemporal consistency between the
predicted and ground truth segmentations. In cases where the algorithm
successfully identifies object masks but encounters challenges in maintaining
object tracking across consecutive frames, the resulting IoU score will be
comparatively low.

####  5.1.2 Implementation

In the training phase, we use Detectron2 [30] and follow the Mask2Former [6]
for VIS settings. To be specific, we adopt the AdamW optimizer [18] and the
step learning rate schedule. The initial learning rate is 0.0001 with a weight
decay of 0.05. A learning rate multiplier of 0.1 is implemented on the
backbone network, and we employ a learning rate decay strategy. Specifically,
the learning rate is decreased by a factor of 10 at two-thirds of the total
number of training steps. We train our models for 6k iterations, utilizing a
batch size of 12 for the AVISeg dataset. Each video clip consists of a
sequence of T = 2 frames, improving the overall training efficiency. All
experiments are conducted on 6 NVIDIA Quadro RTX 6000 GPUs.

During inference, our model exhibits the capability to handle video sequences
of different lengths. We feed the whole video sequence as input to the model.
The mask predictions are obtained without any post-processing. We keep
predictions with a confidence threshold greater than 0.5.

###  5.2 Results

As shown in Table 1, we report the quantitative results of our baseline model
with two image encoders on the AVISeg dataset. The baseline can achieve 31.5%
AP and 35.6% AP when using ResNet-50 and Swin-Base as image encoders,
respectively. The results show that the performance could be further improved
by using a better backbone.

Table 1: Quantitative results of our baseline model on AVISeg. Backbone | AP | AP50 | AP75 | AR1 | AR10
---|---|---|---|---|---
ResNet-50 | 31.5 | 54.2 | 26.2 | 30.5 | 38.3
Swin-Base | 35.6 | 60.7 | 28.6 | 31.1 | 38.8

We also present the visualization results on AVISeg dataset in Fig. 6. The
first column displays the raw frames, and the second column displays the
ground truth. The last column displays the predicted masks by our baseline
model with the Swin-Base backbone. It has a strong ability in target
localization and semantic understanding, which can accurately segment and
classify the sounding object instances.

Figure 6: Sample results of our baseline on AVISeg dataset.

##  6 Conclusion

In this work, we propose a novel multi-modal task named audio-visual instance
segmentation and a benchmark dataset named AVISeg. The new task is a
combination of sound source localization, instance segmentation, and multi-
object tracking, which poses specific challenges given complex audible videos
and real-world applications. Moreover, we present a simple baseline model
combining video instance segmentation and audio-visual fusion, which aims to
provide some early explorations towards this task. We believe AVIS will
innovate the research community on new research ideas and directions for real-
world understanding. We also hope that the AVISeg dataset can serve as a
platform to promote a comprehensive study of this task.

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# Generalized Polya-Szegö inequality

Hichem Hajaiej  Ipeit (Institut préparatoire aux études d’ingénieur de Tunis)
2, Rue Jawaher Lel Nahru - 1089 Montfleury - tunis Tunisie

###### Abstract.

We generalize Polya-Szegö inequality to integrands depending on $u$ and its
gradient. Under minimal additional assumptions, we establish equality cases in
this generalized inequality.

###### Key words and phrases:

Generalized Polya-Szegö inequalities, identity results, radial symmetry, non-
compact minimization problems

###### 2000 Mathematics Subject Classification:

46E35, 26B25, 26B99, 47B38

Ipeit (Institut préparatoire aux études d’ingénieur de Tunis) 2, Rue Jawaher
Lel Nahru - 1089 Montfleury - Tunis, Tunisie. E-mail<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 Preliminary stuff
  3. 3 Generalized Polya-Szegö inequality

##  1\. Introduction

The Polya-Szegö inequality asserts that the $L^{2}$ norm of the gradient of a
positive function $u$ in $W^{1,p}({\mathbb{R}}^{N})$ cannot increase under
Schwarz symmetrization,

(1.1) $\int_{{\mathbb{R}}^{N}}|\nabla
u^{*}|^{2}dx\leq\int_{{\mathbb{R}}^{N}}|\nabla u|^{2}dx.$

The Schwarz rearrangement of $u$ is denoted here by $u^{*}$. Inequality (1.1)
has numerous applications in physics. It was first used in 1945 by G. Polya
and G. Szegö to prove that the capacity of a condenser diminishes or remains
unchanged by applying the process of Schwarz symmetrization (see [30]).
Inequality (1.1) was also the key ingredients to show that, among all bounded
bodies with fixed measure, balls have the minimal capacity (see [26, Theorem
11.17]). Finally (1.1) has also played a crucial role in the solution of the
famous Choquard’s conjecture (see [25]). It is heavily connected to the
isoperimetric inequality and to Riesz-type rearrangement inequalities.
Moreover, it turned out that (1.1) is extremely helpful in establishing the
existence of ground states solutions of the nonlinear Schrödinger equation

(1.2) $\begin{cases}{\rm i}\partial_{t}\Phi+\Delta\Phi+f(|x|,\Phi)=0&\text{in
${\mathbb{R}}^{N}\times(0,\infty)$},\\\ \Phi(x,0)=\Phi_{0}(x)&\text{in
${\mathbb{R}}^{N}$}.\end{cases}$

A ground state solution of equation (1.2) is a positive solution to the
following associated variational problem

(1.3) $\inf\left\\{\frac{1}{2}\int_{{\mathbb{R}}^{N}}|\nabla
u|^{2}dx-\int_{{\mathbb{R}}^{N}}F(|x|,u)dx:\,u\in
H^{1}({\mathbb{R}}^{N}),\,\,\|u\|_{L^{2}}=1\right\\},$

where $F(|x|,s)$ is the primitive of $f(|x|,\cdot)$ with $F(|x|,0)=0$.
Inequality (1.1) together with the generalized Hardy-Littlewood inequality
were crucial to prove that (1.3) admits a radial and radially decreasing
solution. Furthermore, under appropriate regularity assumptions on the
nonlinearity $F$, there exists a Lagrange multiplier $\lambda$ such that any
minimizer of (1.3) is a solution of the following semi-linear elliptic PDE

$-\Delta u+f(|x|,u)+\lambda u=0,\quad\text{in ${\mathbb{R}}^{N}$}.$

We refer the reader to [20] for a detailed analysis. The same approach applies
to the more general quasi-linear PDE

$-\Delta_{p}u+f(|x|,u)+\lambda u=0,\quad\text{in ${\mathbb{R}}^{N}$}.$

where $\Delta_{p}u$ means ${\rm div}(|\nabla u|^{p-2}\nabla u)$, and we can
derive similar properties of ground state solutions since (1.1) extends to
gradients that are in $L^{p}({\mathbb{R}}^{N})$ in place of
$L^{2}({\mathbb{R}}^{N})$, namely

(1.4) $\int_{{\mathbb{R}}^{N}}|\nabla
u^{*}|^{p}dx\leq\int_{{\mathbb{R}}^{N}}|\nabla u|^{p}dx.$

Due to the multitude of applications in physics, rearrangement inequalities
like (1.1) and (1.4) have attracted a huge number of mathematicians from the
middle of the last century. Different approaches were built up to establish
these inequalities such as heat-kernel methods, slicing and cut-off techniques
and two-point rearrangement.

A generalization of inequality (1.4) to suitable convex integrands
$A:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$,

(1.5) $\int_{{\mathbb{R}}^{N}}A(|\nabla
u^{*}|)dx\leq\int_{{\mathbb{R}}^{N}}A(|\nabla u|)dx,$

was first established by Almgren and Lieb (see [1]). Inequality (1.5) is
important in studying the continuity and discontinuity of Schwarz
symmetrization in Sobolev spaces (see e.g. [1, 11]). It also permits us to
study symmetry properties of variational problems involving integrals of type
$\int_{{\mathbb{R}}^{N}}A(|\nabla u|)dx$. Extensions of Polya-Szegö inequality
to more general operators of the form

$j(s,\xi)=b(s)A(|\xi|),\quad s\in{\mathbb{R}},\,\xi\in{\mathbb{R}}^{N},$

on bounded domains have been investigated by Kawohl, Mossino and Bandle. More
precisely, they proved that

(1.6) $\int_{\Omega^{*}}b(u^{*})A(|\nabla
u^{*}|)dx\leq\int_{\Omega}b(u)A(|\nabla u|)dx,$

where $\Omega^{*}$ denotes the ball in ${\mathbb{R}}^{N}$ centered at the
origin having the Lebesgue measure of $\Omega$, under suitably convexity,
monotonicity and growth assumptions (see e.g. [3, 24, 29]). Numerous
applications of (1.6) have been discussed in the above references. In [35],
Tahraoui claimed that a general integrand $j(s,\xi)$ with appropriate
properties can be written in the form

$\sum_{i=1}^{\infty}b_{i}(s)A_{i}(|\xi|)+R_{1}(s)+R_{2}(\xi),\quad
s\in{\mathbb{R}},\,\xi\in{\mathbb{R}}^{N},$

where $b_{i}$ and $A_{i}$ are such that inequality (1.6) holds. However, there
are some mistakes in [35] and we do not believe that this density type result
holds true. Until quite recently there were no results dealing with the
generalized Polya-Szegö inequality, namely

(1.7) $\int_{\Omega^{*}}j(u^{*},|\nabla u^{*}|)dx\leq\int_{\Omega}j(u,|\nabla
u|)dx.$

While writing down this paper we have learned about a very recent survey by F.
Brock [6] who was able to prove (1.7) under continuity, monotonicity,
convexity and growth conditions.

Following a completely different approach, we prove (1.7) without requiring
any growth conditions on $j$. As it can be easily seen it is important to drop
these conditions to the able to cover some relevant applications. Our approach
is based upon a suitable approximation of the Schwarz symmetrized $u^{*}$ of a
function $u$. More precisely, if $(H_{n})_{n\geq 1}$ is a dense sequence in
the set of closed half spaces $H$ containing $0$ and $u\in
L^{p}_{+}({\mathbb{R}}^{N})$, there exists a sequence $(u_{n})$ consisting of
iterated polarizations of the $H_{n}$s which converges to $u^{*}$ in
$L^{p}({\mathbb{R}}^{N})$ (see [17, 38]). On the other hand, a straightforward
computation shows that

$\|\nabla u\|_{L^{p}({\mathbb{R}}^{N})}=\|\nabla
u_{0}\|_{L^{p}({\mathbb{R}}^{N})}=\cdots=\|\nabla
u_{n}\|_{L^{p}({\mathbb{R}}^{N})},\quad\text{for all $n\in{\mathbb{N}}$}.$

By combining these properties with the weak lower semicontinuity of the
functional $J(u)=\int j(u,|\nabla u|)dx$ enable us to conclude (see Theorem
3.1). Note that (1.5) was proved using coarea formula; however this approach
does not apply to integrands depending both on $u$ and its gradient since one
has to apply simultaneously the coarea formula to $|\nabla u|$ and to
decompose $u$ with the Layer-Cake principle.

Notice that Brock’s method is based on an intermediate maximization problem
and cannot yield to the establishment of equality cases. Our approximation
approach was also fruitful in determining the relationship between $u$ and
$u^{*}$ such that

(1.8) $\int_{{\mathbb{R}}^{N}}j(u^{*},|\nabla
u^{*}|)dx=\int_{{\mathbb{R}}^{N}}j(u,|\nabla u|)dx.$

Indeed, under very general conditions on $j$, we prove that (1.8) is
equivalent to

$\int_{{\mathbb{R}}^{N}}|\nabla u^{*}|^{p}dx=\int_{{\mathbb{R}}^{N}}|\nabla
u|^{p}dx.$

For $j(\xi)=|\xi|^{p}$, identity cases were completely studied in the
breakthrough paper of Brothers and Ziemer [10].

The paper is organized as follows. Section 2 is dedicated to some preliminary
stuff, especially the ones concerning the invariance of a class of functionals
under polarization. These observations are crucial, in Section 3, to establish
in a simple way the generalized Polya-Szegö inequality.

Notations.

  1. (1)

For $N\in{\mathbb{N}}$, $N\geq 1$, we denote by $|\cdot|$ the euclidean norm
in ${\mathbb{R}}^{N}$.

  2. (2)

${\mathbb{R}}_{+}$ (resp. ${\mathbb{R}}_{-}$) is the set of positive (resp.
negative) real values.

  3. (3)

$\mu$ denotes the Lebesgue measure in ${\mathbb{R}}^{N}$.

  4. (4)

$M({\mathbb{R}}^{N})$ is the set of measurable functions in
${\mathbb{R}}^{N}$.

  5. (5)

For $p>1$ we denote by $L^{p}({\mathbb{R}}^{N})$ the space of $f$ in
$M({\mathbb{R}}^{N})$ with $\int_{{\mathbb{R}}^{N}}|f|^{p}dx<\infty$.

  6. (6)

The norm $(\int_{{\mathbb{R}}^{N}}|f|^{p}dx)^{1/p}$ in
$L^{p}({\mathbb{R}}^{N})$ is denoted by $\|\cdot\|_{p}$.

  7. (7)

For $p>1$ we denote by $W^{1,p}({\mathbb{R}}^{N})$ the Sobolev space of
functions $f$ in $L^{p}({\mathbb{R}}^{N})$ having generalized partial
derivatives $D_{i}f$ in $L^{p}({\mathbb{R}}^{N})$, for $i=1,\dots,N$.

  8. (8)

$D^{1,p}({\mathbb{R}}^{N})$ is the space of measurable functions whose
gradient is in $L^{p}({\mathbb{R}}^{N})$.

  9. (9)

$L^{p}_{+}({\mathbb{R}}^{N})$ is the cone of positive functions of
$L^{p}({\mathbb{R}}^{N})$.

  10. (10)

$W^{1,p}_{+}({\mathbb{R}}^{N})$ is the cone of positive functions of
$W^{1,p}({\mathbb{R}}^{N})$.

  11. (11)

For $R>0$, $B(0,R)$ is the ball in ${\mathbb{R}}^{N}$ centered at zero with
radius $R$.

##  2\. Preliminary stuff

In the following $H$ will design a closed half-space of ${\mathbb{R}}^{N}$
containing the origin, $0_{{\mathbb{R}}^{N}}\in H$. We denote by
${\mathcal{H}}$ the set of closed half-spaces of ${\mathbb{R}}^{N}$ containing
the origin. We shall equip ${\mathcal{H}}$ with a topology ensuring that
$H_{n}\to H$ as $n\to\infty$ if there is a sequence of isometries
$i_{n}:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ such that $H_{n}=i_{n}(H)$ and
$i_{n}$ converges to the identity as $n\to\infty$. We first recall some basic
notions. For more details, we refer the reader to [12].

######  Definition 2.1.

A reflection $\sigma:{\mathbb{R}}^{N}\to{\mathbb{R}}^{N}$ with respect to $H$
is an isometry such that the following properties hold

  1. (1)

$\sigma\circ\sigma(x)=x$, for all $x\in{\mathbb{R}}^{N}$;

  2. (2)

the fixed point set of $\sigma$ separates ${\mathbb{R}}^{N}$ in $H$ and
${\mathbb{R}}^{N}\setminus H$ (interchanged by $\sigma$);

  3. (3)

$|x-y|<|x-\sigma(y)|$, for all $x,y\in H$.

Given $x\in{\mathbb{R}}^{N}$, the reflected point $\sigma_{H}(x)$ will also be
denoted by $x^{H}$.

######  Definition 2.2.

Let $H$ be a given half-space in ${\mathbb{R}}^{N}$. The two-point
rearrangement (or polarization) of a nonnegative real valued function
$u:{\mathbb{R}}^{N}\to{\mathbb{R}}_{+}$ with respect to a given reflection
$\sigma_{H}$ (with respect to $H$) is defined as

$u^{H}(x):=\begin{cases}\max\\{u(x),u(\sigma_{H}(x))\\},&\text{for $x\in
H$},\\\ \min\\{u(x),u(\sigma_{H}(x))\\},&\text{for
$x\in{\mathbb{R}}^{N}\setminus H$}.\end{cases}$

######  Definition 2.3.

We say that a nonnegative measurable function $u$ is symmetrizable if
$\mu(\\{x\in{\mathbb{R}}^{N}:u(x)>t\\})<\infty$ for all $t>0$. The space of
symmetrizable functions is denoted by $F_{N}$ and, of course,
$L^{p}_{+}({\mathbb{R}}^{N})\subset F_{N}$. Also, two functions $u,v$ are said
to be equimeasurable (and we shall write $u\sim v$) when

$\mu(\\{x\in{\mathbb{R}}^{N}:u(x)>t\\})=\mu(\\{x\in{\mathbb{R}}^{N}:v(x)>t\\}),$

for all $t>0$.

######  Definition 2.4.

For a given $u$ in $F_{N}$, the Schwarz symmetrization $u^{*}$ of $u$ is the
unique function with the following properties (see e.g. [19])

  1. (1)

$u$ and $u^{*}$ are equimeasurable;

  2. (2)

$u^{*}(x)=h(|x|)$, where $h:(0,\infty)\to{\mathbb{R}}_{+}$ is a continuous and
decreasing function.

In particular, $u$, $u^{H}$ and $u^{*}$ are all equimeasurable functions (see
e.g. [2]).

######  Lemma 2.5.

Let $u\in W^{1,p}_{+}({\mathbb{R}}^{N})$ and let $H$ be a given half-space.
Then $u^{H}\in W^{1,p}_{+}({\mathbb{R}}^{N})$ and, setting

$v(x):=u(x^{H}),\quad w(x):=u^{H}(x^{H}),\qquad x\in{\mathbb{R}}^{N},$

the following facts hold:

  1. (1)

We have

$\displaystyle\nabla u^{H}(x)$ $\displaystyle=\begin{cases}\nabla
u(x)&\text{for $x\in\\{u>v\\}\cap H$},\\\ \nabla v(x)&\text{for $x\in\\{u\leq
v\\}\cap H$},\\\ \end{cases}$ $\displaystyle\nabla w(x)$
$\displaystyle=\begin{cases}\nabla v(x)&\text{for $x\in\\{u>v\\}\cap H$},\\\
\nabla u(x)&\text{for $x\in\\{u\leq v\\}\cap H$}.\\\ \end{cases}$

  2. (2)

For all $i=1,\dots,N$ and $p\in(1,\infty)$, we have

(2.1)
$\|D_{i}u^{H}\|_{L^{p}({\mathbb{R}}^{N})}=\|D_{i}u\|_{L^{p}({\mathbb{R}}^{N})}.$

  3. (3)

Let $j:[0,\infty)\times[0,\infty)\to{\mathbb{R}}$ be a Borel measurable
function. Then

(2.2) $\int_{{\mathbb{R}}^{N}}j(u,|\nabla
u|)dx=\int_{{\mathbb{R}}^{N}}j(u^{H},|\nabla u^{H}|)dx,$

provided that $0\in H$ and that both integrals are finite.

###### Proof.

Observing that, for all $x\in H$, we have

$u^{H}(x)=v(x)+(u(x)-v(x))^{+},\qquad w(x)=u(x)-(u(x)-v(x))^{+},$

in light of [26, Corollary 6.18] it follows that $v,w$ belong to
$W^{1,p}_{+}({\mathbb{R}}^{N})$. Assertion (1) follows by a simple direct
computation. Assertion (2) follows as a consequence of assertion (1).
Concerning (3), writing $\sigma_{H}$ as $\sigma_{H}(x)=x_{0}+Rx$, where $R$ is
an orthogonal linear transformation, taking into account that $|{\rm
det}\,R|=1$ and

$|\nabla v(x)|=|\nabla(u(\sigma_{H}(x)))|=|R(\nabla
u(\sigma_{H}(x)))|=|(\nabla u)(\sigma_{H}(x))|,$

we have

$\displaystyle\int_{{\mathbb{R}}^{N}}j(u,|\nabla u|)dx$
$\displaystyle=\int_{H}j(u,|\nabla u|)dx+\int_{{\mathbb{R}}^{N}\setminus
H}j(u,|\nabla u|)dx$ $\displaystyle=\int_{H}j(u,|\nabla
u|)dx+\int_{H}j(u(\sigma_{H}(x)),|(\nabla u)(\sigma_{H}(x))|)dx$
$\displaystyle=\int_{H}j(u,|\nabla u|)dx+\int_{H}j(v,|\nabla v|)dx.$

In a similar fashion, we have

$\displaystyle\int_{{\mathbb{R}}^{N}}j(u^{H},|\nabla u^{H}|)dx$
$\displaystyle=\int_{H}j(u^{H},|\nabla
u^{H}|)dx+\int_{H}j(u^{H}(\sigma_{H}(x)),|(\nabla u^{H})(\sigma_{H}(x))|)dx$
$\displaystyle=\int_{H}j(u^{H},|\nabla u^{H}|)dx+\int_{H}j(w,|\nabla w|)dx$
$\displaystyle=\int_{\\{u>v\\}\cap H}j(u,|\nabla u|)dx+\int_{\\{u>v\\}\cap
H}j(v,|\nabla v|)dx$ $\displaystyle+\int_{\\{u\leq v\\}\cap H}j(v,|\nabla
v|)dx+\int_{\\{u\leq v\\}\cap H}j(u,|\nabla u|)dx$
$\displaystyle=\int_{H}j(u,|\nabla u|)dx+\int_{H}j(v,|\nabla v|)dx,$

which concludes the proof ∎

##  3\. Generalized Polya-Szegö inequality

The first main result of the paper is the following

######  Theorem 3.1.

Let $\varrho:[0,\infty)\times{\mathbb{R}}^{N}\to{\mathbb{R}}$ be a Borel
measurable function. For any function $u\in W^{1,p}_{+}({\mathbb{R}}^{N})$,
let us set

$J(u)=\int_{{\mathbb{R}}^{N}}\varrho(u,\nabla u)dx.$

Moreover, let $(H_{n})_{n\geq 1}$ be a dense sequence in the set of closed
half spaces containing $0_{{\mathbb{R}}^{N}}$. For $u\in
W^{1,p}_{+}({\mathbb{R}}^{N})$, define a sequence $(u_{n})$ by setting

$\begin{cases}u_{0}=u&\\\ u_{n+1}=u_{n}^{H_{1}\ldots H_{n+1}}.&\end{cases}$

Assume that the following conditions hold:

  1. (1)

$-\infty<J(u)<+\infty;$

  2. (2)

(3.1) $\liminf_{n}J(u_{n})\leq J(u);$

  3. (3)

if $(u_{n})$ converges weakly to some $v$ in $W^{1,p}_{+}({\mathbb{R}}^{N})$,
then

$J(v)\leq\liminf_{n}J(u_{n}).$

Then

$J(u^{*})\leq J(u).$

###### Proof.

By the (explicit) approximation results contained in [17, 38], we know that
$u_{n}\to u^{*}$ in $L^{p}({\mathbb{R}}^{N})$ as $n\to\infty$. Moreover, by
Lemma 2.5 applied with $j(s,|\xi|)=|\xi|^{p}$, we have

(3.2) $\|\nabla u\|_{L^{p}({\mathbb{R}}^{N})}=\|\nabla
u_{0}\|_{L^{p}({\mathbb{R}}^{N})}=\cdots=\|\nabla
u_{n}\|_{L^{p}({\mathbb{R}}^{N})},\quad\text{for all $n\in{\mathbb{N}}$}.$

In particular, up to a subsequence, $(u_{n})$ is weakly convergent to some
function $v$ in $W^{1,p}({\mathbb{R}}^{N})$. By uniqueness of the weak limit
in $L^{p}({\mathbb{R}}^{N})$ one can easily check that $v=u^{*}$, namely
$u_{n}\rightharpoonup u^{*}$ in $W^{1,p}({\mathbb{R}}^{N})$. Hence, using
assumption (3) and (3.1), we have

(3.3) $J(u^{*})\leq\liminf_{n}J(u_{n})\leq J(u),$

concluding the proof. ∎

######  Remark 3.2.

A quite large class of functionals $J$ which satisfy assumption (3.1) of the
previous Theorem is provided by Lemma 2.5.

######  Corollary 3.3.

Let $j:[0,\infty)\times[0,\infty)\to{\mathbb{R}}$ be a function satisfying the
following assumptions:

  1. (1)

$j(\cdot,t)$ is continuous for all $t\in[0,\infty)$;

  2. (2)

$j(s,\cdot)$ is convex for all $s\in[0,\infty)$ and continuous at zero;

  3. (3)

$j(s,\cdot)$ is nondecreasing for all $s\in[0,\infty)$.

Then, for all function $u\in W^{1,p}_{+}({\mathbb{R}}^{N})$ such that

$\int_{{\mathbb{R}}^{N}}j(u,|\nabla u|)dx<\infty,$

we have

$\int_{{\mathbb{R}}^{N}}j(u^{*},|\nabla
u^{*}|)dx\leq\int_{{\mathbb{R}}^{N}}j(u,|\nabla u|)dx.$

###### Proof.

The assumptions on $j$ imply that $\\{\xi\mapsto j(s,|\xi|)\\}$ is convex so
that the weak lower semicontinuity assumption of Theorem 3.1 holds (we refer
the reader e.g. to the papers [21, 22] by A. Ioffe). Also, assumption (3.1) of
Theorem 3.1 is provided by means of Lemma 2.5. ∎

######  Remark 3.4.

In [6, Theorem 4.3], F. Brock proved Corollary 3.3 for Lipschitz functions
having compact support. In order to prove the most interesting cases in the
applications, the inequality has to hold for functions $u$ in
$W^{1,p}_{+}({\mathbb{R}}^{N})$. This forces him to assume some growth
conditions of the Lagrangian $j$, for instance to assume that there exists a
positive constant $K$ and $q\in[p,p^{*}]$ such that

$|j(s,|\xi|)|\leq K(s^{q}+|\xi|^{p}),\quad\text{for all $s\in{\mathbb{R}}_{+}$
and $\xi\in{\mathbb{R}}^{N}$}.$

By our approach, instead, can include integrands such as

$j(s,|\xi|)=\frac{1}{2}(1+s^{2\alpha})|\xi|^{p},\quad\text{for all
$s\in{\mathbb{R}}_{+}$ and $\xi\in{\mathbb{R}}^{N}$},$

for some $\alpha>0$, which have meaningful physical applications (for instance
quasi-linear Schrödinger equations, see [27] and references therein). We also
stress that the approach of [6] cannot yield the establishment of equality
cases (see Theorem 3.6).

######  Corollary 3.5.

Let $m\geq 1$ and $p_{1},\dots,p_{m}\in(1,\infty)$. Then

$\sum_{i=1}^{m}\int_{{\mathbb{R}}^{N}}|D_{i}u^{*}|^{p_{i}}dx\leq\sum_{i=1}^{m}\int_{{\mathbb{R}}^{N}}|D_{i}u|^{p_{i}}dx,$

for all $u\in\bigcap_{i=1}^{m}W^{1,p_{i}}_{+}({\mathbb{R}}^{N})$.

###### Proof.

The assertion follows by a simple combination of Theorem 3.1 with inequality
(2.1) of Lemma 2.5. ∎

######  Theorem 3.6.

In addition to the assumptions of Theorem 3.1, assume that

(3.4) $\text{$J(u_{n})\to J(u^{*})$ as $n\to\infty$ implies that $u_{n}\to
u^{*}$ in $D^{1,p}({\mathbb{R}}^{N})$ as $n\to\infty$}.$

Then

$J(u)=J(u^{*})\,\,\Longrightarrow\,\,\|\nabla
u\|_{L^{p}({\mathbb{R}}^{N})}=\|\nabla u^{*}\|_{L^{p}({\mathbb{R}}^{N})}.$

###### Proof.

Assume that $J(u)=J(u^{*})$. Then, by assumption (3.1), we obtain

$J(u^{*})=\lim_{n}J(u_{n})=J(u).$

In turn, by assumption, $u_{n}\to u^{*}$ in $D^{1,p}({\mathbb{R}}^{N})$ as
$n\to\infty$. Then, taking the limit inside equalities (3.2), we conclude the
assertion. ∎

######  Remark 3.7.

Assume that $\\{\xi\mapsto j(s,|\xi|)\\}$ is strictly convex for any
$s\in{\mathbb{R}}_{+}$ and there exists $\nu^{\prime}>0$ such that
$j(s,|\xi|)\geq\nu^{\prime}|\xi|^{p}$ for all $s\in{\mathbb{R}}_{+}$ and
$\xi\in{\mathbb{R}}^{N}$. Then assumption (3.4) is fulfilled for
$J(u)=\int_{{\mathbb{R}}^{N}}j(u,|\nabla u|)dx$. We refer to [39, Section 3].

######  Remark 3.8.

Equality cases of the type $\|\nabla u\|_{L^{p}({\mathbb{R}}^{N})}=\|\nabla
u^{*}\|_{L^{p}({\mathbb{R}}^{N})}$ have been completely characterized in the
breakthrough paper by Brothers and Ziemer [10].

Let us now set

$\displaystyle{M={\rm esssup}_{{\mathbb{R}}^{N}}u={\rm
esssup}_{{\mathbb{R}}^{N}}u^{*}},\qquad C^{*}=\\{x\in{\mathbb{R}}^{N}:\nabla
u^{*}(x)=0\\}.$

######  Corollary 3.9.

Assume that $\\{\xi\mapsto j(s,|\xi|)\\}$ is strictly convex and there exists
a positive constant $\nu^{\prime}$ such that

$j(s,|\xi|)\geq\nu^{\prime}|\xi|^{p},\quad\text{for all $s\in{\mathbb{R}}$ and
$\xi\in{\mathbb{R}}^{N}$}.$

Moreover, assume that

$\int_{{\mathbb{R}}^{N}}j(u,|\nabla
u|)dx=\int_{{\mathbb{R}}^{N}}j(u^{*},|\nabla
u^{*}|)dx,\quad\mu(C^{*}\cap(u^{*})^{-1}(0,M))=0.$

Then there exists $x_{0}\in{\mathbb{R}}^{N}$ such that

$u(x)=u^{*}(x-x_{0}),\quad\text{for all $x\in{\mathbb{R}}^{N}$},$

namely $u$ is radially symmetric after a translation in ${\mathbb{R}}^{N}$.

###### Proof.

It is sufficient to combine Theorem 3.6 with [10, Theorem 1.1]. ∎

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# Rational points on quadratic elliptic surfaces

Mohammad Sadek  Faculty of Engineering and Natural Sciences, Sabancı
University, Tuzla, İstanbul, 34956 Turkey<EMAIL_ADDRESS>
###### Abstract.

We consider elliptic surfaces whose coefficients are degree $2$ polynomials in
a variable $T$. It was recently shown, [8], that for infinitely many rational
values of $T$ the resulting elliptic curves have rank at least $1$. In this
article, we prove that the Mordell-Weil rank of each such elliptic surface is
at most $6$ over ${\mathbb{Q}}$. In fact, we show that the Mordell-Weil rank
of these elliptic surfaces is controlled by the number of zeros of a certain
polynomial over ${\mathbb{Q}}$.

††footnotetext:  Keywords: elliptic surfaces, elliptic curves, Nagao’s
conjecture, Mordell-Weil rank
The datasets generated during and/or analysed during the current study are
available from the corresponding author on reasonable request.
2010 Mathematics Subject Classification: 11G30, 14H25

##  1\. Introduction

Let ${\mathcal{E}}$ be a nonconstant elliptic surface defined by the following
equation over ${\mathbb{Q}}$

(1) $\displaystyle y^{2}=a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T),$

where $a_{i}(T)\in{\mathbb{Q}}[T]$. The elliptic surface ${\mathcal{E}}$ can
be thought of as a family of elliptic curves parametrized by
$T\in{\mathbb{Q}}$. The Lang-Néron theorem asserts that the abelian group
${\mathcal{E}}({\mathbb{Q}}(T))$ of rational points is finitely generated. In
addition, for all but finitely many rational values of $T=t$, specialization
yields an elliptic curve ${\mathcal{E}}_{t}$. The Mordell-Weil theorem states
that the abelian group ${\mathcal{E}}_{t}({\mathbb{Q}})$ is also finitely
generated. Thanks to Silverman’s specialization theorem, one knows that for
all but finitely many values $t\in{\mathbb{Q}}$, the Mordell-Weil rank of
${\mathcal{E}}_{t}({\mathbb{Q}})$ is at least the Mordell-Weil rank of
${\mathcal{E}}({\mathbb{Q}}(T))$.

The construction of elliptic surfaces over ${\mathbb{Q}}$ with large Mordell-
Weil rank has been an interesting target for many researchers. One of the main
reasons is that among specializations of these elliptic surfaces one may find
elliptic curves with large Mordell-Weil rank over ${\mathbb{Q}}$. In [10, 11],
Mestre gave examples of elliptic surfaces of Mordell-Weil rank $\geq 11,12$
over ${\mathbb{Q}}(T)$. Nagao used ideas of Mestre to search for elliptic
surfaces of larger Mordell-Weil rank over ${\mathbb{Q}}(T)$, see [13]. The
arithmetic of these elliptic surfaces was employed to find elliptic curves of
Mordell-Weil ranks $\geq 19,20,21,22$ in [3], [12], [14], and [5],
respectively.

The following question is frequently posed: can we find rational values for
$t$ such that the specialized elliptic curves ${\mathcal{E}}_{t}$ satisfy that
$\operatorname{rank}({\mathcal{E}}_{t}({\mathbb{Q}}))\geq\operatorname{rank}({\mathcal{E}}({\mathbb{Q}}(T)))+1$?
If the answer is positive, then are there infinitely many such rational values
for $t$? If the answer again is in the affirmative, then is the density
(appropriately defined) of such rational values for $t$ positive?

Elliptic surfaces defined by equation (1) over ${\mathbb{Q}}$ where the
polynomials $a_{i}(t)$ are of degree at most $1$ were studied in [9]. It was
proven that the Mordell-Weil rank of such an elliptic surface is at most $3$,
see [9, Lemma 3.1]. Explicit families of such elliptic surfaces with Mordell-
Weil rank $0,1,2,$ or $3$ can be found in [4, 9].

In this article, we consider the elliptic surface ${\mathcal{E}}$ defined by
$y^{2}=a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)$, where
$a_{i}(T)\in{\mathbb{Q}}[T]$ are polynomials of degree at most $2$. We assume
moreover it is a nontrivial elliptic surface, i.e., at least two of the curves
${\mathcal{E}}_{t}$ are smooth, elliptic and not isomorphic to each other over
${\mathbb{Q}}$. This quadratic family of elliptic curves was studied in [8].
The authors prove that when the whole family is viewed as a single algebraic
surface in $\mathbb{A}^{3}_{xyt}$, then the resulting surface is unirational
over ${\mathbb{Q}}$. As a consequence, it is proved that there are infinitely
many rational values $T=t$ such that
$\operatorname{rank}({\mathcal{E}}_{t}({\mathbb{Q}}))>0$. In fact, it is
conjectured that such rational values constitute a positive proportion of all
rational numbers. A key ingredient to reach the aforementioned results is the
observation

$a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)=A(x)T^{2}+B(x)T+C(x)$

where $A(x),B(x),C(x)\in{\mathbb{Q}}[x]$ are of degree at most $3$. In
particular, ${\mathcal{E}}$ is birational to a conic bundle with at most $7$
singular fibers.

Nagao conjectured that the Mordell-Weil rank of an elliptic surface can be
recovered from local information of the elliptic surface modulo primes, see
[15]. This conjecture was proved unconditionally for rational elliptic
surfaces, and conditionally on Sato-Tate conjecture for other elliptic
surfaces by Rosen and Silverman, see [16]. We show that the quadratic elliptic
surface

${\mathcal{E}}:y^{2}=a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)=A(x)T^{2}+B(x)T+C(x),$

are rational elliptic surfaces under some mild conditions.

Making use of the fact that the aformentioned quadratic elliptic surfaces are
rational enables us to evaluate their Mordell-Weil rank. In fact, we show that
the Mordell-Weil rank depends on the factorization behavior of the polynomial
$D(x):=B(x)^{2}-4A(x)C(x)$ in ${\mathbb{Q}}[x]$ where the latter polynomial
$D(x)$ is of degree at most $6$. In particular, we show that the largest
possible value for the Mordell-Weil rank $r_{{\mathcal{E}}}$ of
${\mathcal{E}}$ is $6$. Moreover, this maximal rank occurs if and only if
$D(x)$ splits completely into distinct linear factors over ${\mathbb{Q}}$ and
the value of $A(x)$ at each of the simple zeros of $D(x)$ is a nonzero
rational square. In addition, we show the existence of quadratic elliptic
surfaces with Mordell-Weil rank $r$ where $0\leq r\leq 6$. The aforementioned
result asserts that for all but finitely many rational values of $t$, the
Mordell-Weil rank of the specialized elliptic curve ${\mathcal{E}}_{t}$ is at
least $r_{{\mathcal{E}}}$. This provides an improvement on the results of [8]
mentioned above.

It is worth mentioning that elliptic surfaces defined by equation (1) over
${\mathbb{Q}}$, where at least one of the polynomials $a_{i}(t)$ is of degree
at least $3$, are not in general rational elliptic surfaces. However, assuming
Sato-Tate conjecture for these surfaces, Nagao’s conjecture on their Mordell-
Weil ranks holds true, see [16]. Yet, since these elliptic surfaces are not
birational to conic bundles, the techniques used in this work can not be
applied to evaluate their Mordell-Weil ranks.

After the submission of this article, it came to the attention of the author
that the authors of [2] have been using similar techniques to find expressions
for the rank of some quadratic elliptic surfaces over ${\mathbb{Q}}(T)$
depending on the factorization and other properties of certain polynomials. In
fact, the authors treat elliptic surfaces over ${\mathbb{Q}}$ whose
Weierstrass equations can be written as

$y^{2}=A(x)T^{2}+B(x)T+C(x),$

where $A(x),B(x)\in{\mathbb{Q}}[x]$ are of degree at most $2$ and
$C(x)\in{\mathbb{Q}}[x]$ is monic of degree $3$, and one of $A(X)$ or $B(X)$
is not the zero polynomial. The authors then introduce conditions under which
these elliptic surfaces possess positive Mordell-Weil ranks. This is
approached by considering several subcases based on whether $A(x)$ is a
nonconstant polynomial, zero, a nonzero rational square, or a rational number
which is not a rational square.

### Acknowledgments

The author would like to thank the anonymous referees for several comments and
suggestions. This work is partially supported by BAGEP Award of the Science
Academy, Turkey.

##  2\. Elliptic surfaces

Let ${\mathcal{E}}\to{\mathbb{P}}^{1}$ be a (non-split) elliptic surface over
${\mathbb{Q}}$ described by a Weierstrass equation

${\mathcal{E}}:y^{2}=x^{3}+A(T)x+B(T),\,A,B\in{\mathbb{Z}}[T],\,\Delta(T)=4A(T)^{3}+27B(T)^{2}\neq
0.$

By evaluating the polynomials $A(T)$ and $B(T)$ at integer values for $T$, we
obtain elliptic curves over ${\mathbb{Q}}$.

Silverman’s Specialization Theorem implies that, for all but finitely many
values $t\in{\mathbb{Z}}$, the Mordell-Weil rank of the fiber
${\mathcal{E}}_{t}$ over ${\mathbb{Q}}$ is at least that of the curve
${\mathcal{E}}$ over ${\mathbb{Q}}(T)$.

Let $t\in{\mathbb{Z}}$ and let $p$ be a prime. If $p|\Delta(t)$, then we let
$a_{p}({\mathcal{E}}_{t})=0$. Otherwise, we let $a_{p}({\mathcal{E}}_{t})$ be
the $p$-th coefficient of the $L$-series of the fiber ${\mathcal{E}}_{t}$. We
set

$A_{p}({\mathcal{E}})=\frac{1}{p}\sum_{t=1}^{p}a_{p}({\mathcal{E}}_{t})$

to be the average of the $a_{p}$’s over the fibers.

Nagao [15] conjectured that

$\lim_{X\to\infty}S(X,{\mathcal{E}})=\operatorname{rank}{\mathcal{E}}({\mathbb{Q}}(T)),\textrm{
where }S(X,{\mathcal{E}})=\frac{1}{X}\sum_{p\leq X}-A_{p}({\mathcal{E}})\log
p.$

In fact, he proved the conjecture for several elliptic surfaces. In [16], it
was proved, among other results, that Nagao’s conjecture is true for rational
elliptic surfaces, namely, if ${\mathcal{E}}$ is birational to
${\mathbb{P}}^{2}$. In terms of a minimal Weierstrass equation
$y^{2}=x^{3}+A(T)x+B(T)$ for ${\mathcal{E}}$ over ${\mathbb{Q}}(T)$, an
elliptic surface is rational if and only if one of the following two
conditions holds:

  * (i)

$0<\max\\{3\deg A,2\deg B\\}<12$.

  * (ii)

$3\deg A=2\deg B=12$ and $\operatorname{ord}_{T=0}T^{12}\Delta(T^{-1})=0$,

see [16, Remark 1.3.1]

##  3\. Quadratic families of elliptic curves

Let ${\mathcal{E}}$ be an elliptic surface defined over ${\mathbb{Q}}$ by the
following equation

$y^{2}=a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)$

where $a_{i}\in{\mathbb{Z}}[T]$, $\max_{i}\deg a_{i}(T)=2$, $i=0,1,2,3$.

We only consider non-split elliptic surfaces. Therefore, at least two of the
curves ${\mathcal{E}}_{t}$ are smooth, elliptic and not isomorphic to each
other over ${\mathbb{Q}}$; $a_{3}(T)$ is not identically $0$; and not all the
$a_{i}(T)$ are constant multiples of the same square $(aT+b)^{2}$.

One uses the transformation $x\mapsto x/a_{3}(T)$ and $y\mapsto y/a_{3}(T)$ to
see that ${\mathcal{E}}$ is described by
$y^{2}=x^{3}+a_{2}(T)\,x^{2}+a_{1}(T)a_{3}(T)\,x+a_{0}(T)a_{3}(T)^{2}$. Then
one sees that the transformation $x\mapsto x-a_{2}(T)/3$ allows
${\mathcal{E}}$ to be described by

$y^{2}=x^{3}+(a_{1}(T)a_{3}(T)-a_{2}(T)^{2}/3)x+a_{0}(T)a_{3}(T)^{2}-a_{1}(T)a_{2}(T)a_{3}(T)/3+2a_{2}(T)^{3}/27.$

The discriminant $\Delta(T)$ of ${\mathcal{E}}$ is

$a_{3}(T)^{2}\left[-a_{1}(T)^{2}a_{2}(T)^{2}+4a_{1}(T)^{3}a_{3}(T)-18a_{0}(T)a_{1}(T)a_{2}(T)a_{3}(T)+a_{0}(T)\left(4a_{2}(T)^{3}+27a_{0}(T)a_{3}(T)^{2}\right)\right].$

######  Lemma 3.1.

Let ${\mathcal{E}}$ be an elliptic surface defined over ${\mathbb{Q}}$ by the
following equation

$y^{2}=a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)$

where $a_{i}(T)\in{\mathbb{Z}}[T]$ and $\deg a_{i}(T)\leq 2$.

If the fiber ${\mathcal{E}}_{\infty}$ is an elliptic curve, then
${\mathcal{E}}$ is a rational elliptic surface. In particular,
$\lim_{X\to\infty}S(X,{\mathcal{E}})=\operatorname{rank}({\mathcal{E}}({\mathbb{Q}}(T)))$.

Proof: We will show that the condition ${\mathcal{E}}_{\infty}$ is an elliptic
curve implies that ${\mathcal{E}}$ is an elliptic surface that satisfies
$\operatorname{ord}_{T=0}T^{12}\Delta(T^{-1})=0$, hence ${\mathcal{E}}$ is a
rational elliptic surface, see §2. The formula for $\Delta(T)$ implies that
$T^{12}\Delta(T^{-1})$ at $T=0$ is given by

$\Delta=a_{3}^{2}\left(-a_{1}^{2}a_{2}^{2}+4a_{1}^{3}a_{3}-18a_{0}a_{1}a_{2}a_{3}+a_{0}\left(4a_{2}^{3}+27a_{0}a_{3}^{2}\right)\right)$

where $a_{i}$ is the coefficient of $T^{2}$ in $a_{i}(T)$. But $\Delta$ is the
discriminant of the cubic curve

${\mathcal{E}}_{\infty}:y^{2}=x^{3}+(a_{1}a_{3}-a_{2}^{2}/3)x+a_{0}a_{3}^{2}-a_{1}a_{2}a_{3}/3+2a_{2}^{3}/27.$

If ${\mathcal{E}}_{\infty}$ is an elliptic curve, then $\Delta\neq 0$, hence
the result. $\Box$

If ${\mathcal{E}}_{\infty}$ is an elliptic curve, then $\Delta\neq 0$, and so
$a_{3}\neq 0$. It is easy to see that

$a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)=A(x)T^{2}+B(x)T+C(x)$

where $\deg A=3$ and $\deg B,\deg C\leq 3$. This holds as the leading
coefficient of $A(x)$ is $a_{3}$.

The elliptic surface ${\mathcal{E}}$ can be viewed as the family of conics

$y^{2}=A(x)T^{2}+B(x)T+C(x)$

over ${\mathbb{Q}}(x)$. The singular conics are exactly those for which $x$ is
a root of the discriminant $B(x)^{2}-4A(x)C(x)$. There are in general $6$, not
necessarily distinct, such values for $x$. One may see that, after a
birational transformation, the conic at infinity is isomorphic to the conic
described by $a_{3}^{\prime}(S,T)=0$ in ${\mathbb{P}}^{2}_{STW}$, where
$a_{3}^{\prime}(S,T):=S^{2}a_{3}(T/S)$ is the homogenization of $a_{3}(T)$. If
$a_{3}(T)$ is irreducible in ${\mathbb{Q}}[T]$, then the conic at infinity is
singular, hence there are $\leq 7$ singular conics in the family. If
$a_{3}(T)$ is reducible, then the conic at infinity is reducible and we can
contract either of its irreducible components. In the latter case, the conic
bundle has at most $6$ singular conics, see [8, §0]. Therefore, from now on we
assume that $a_{3}(T)$ is irreducible of degree $2$.

######  Definition 3.2.

Let ${\mathcal{E}}$ be a non-split elliptic surface defined over
${\mathbb{Q}}$ by the equation

$y^{2}=a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)$

where $a_{i}(T)\in{\mathbb{Z}}[T]$, $\deg a_{3}(T)=2$, $\deg a_{i}(T)\leq 2$,
$i=0,1,2$, and $a_{3}(T)$ is irreducible. Assume moreover that
${\mathcal{E}}_{\infty}$ is an elliptic curve. Then ${\mathcal{E}}$ will be
called a quadratic elliptic surface.

##  4\. Evaluation of some character sums

Let $p$ be a prime. Throughout the paper, ${\mathbb{F}_{p}}$ will be the
finite field with $p$ elements, and $\phi_{p}$ will be the unique quadratic
character modulo $p$.

One will need the following standard lemma to evaluate character sums of
quadratic polynomials, see for example [1, Lemma A.2].

######  Lemma 4.1.

Let $p>2$ be a prime. Let $a,b,c\in{\mathbb{F}_{p}}$ be such that either $a$
or $b$ are nonzero. One then has

$\displaystyle\sum_{t=0}^{p-1}\phi_{p}(at^{2}+bt+c)=\begin{cases}(p-1)\phi_{p}(a)&\textrm{
if }p|(b^{2}-4ac)\\\ -\phi_{p}(a)&\textrm{ otherwise}\end{cases}$

######  Lemma 4.2.

Let $f(x)\in{\mathbb{F}_{p}}[x]$ be a cubic polynomial. One has

$\displaystyle\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))=\begin{cases}-a_{p}(E)&\textrm{
if $f(x)$ has no repeated root}\\\
-\phi_{p}(-f_{1}^{-1}f_{2}f_{3}+f_{4})&\textrm{ if
}f(x)=(f_{1}x+f_{2})^{2}(f_{3}x+f_{4}),\;f_{1}f_{3}\in\mathbb{F}_{p}^{\times}\\\
0&\textrm{ if
}f(x)=(f_{1}x+f_{2})^{3},\;f_{1}\in\mathbb{F}_{p}^{\times}\end{cases}$

where $a_{p}(E)$ is the trace of Frobenius of the elliptic curve $E$ defined
by $y^{2}=f(x)$ over ${\mathbb{F}_{p}}$.

Proof: If $f(x)$ has no repeated root, then one has
$a_{p}(E)=-\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))$.

One now assumes that $f(x)$ has a repeated root. If
$f(x)=(f_{1}x+f_{2})^{2}(f_{3}x+f_{4})$, with
$f_{1}f_{3}\in\mathbb{F}_{p}^{\times}$, then

$\displaystyle\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))$ $\displaystyle=$
$\displaystyle\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}^{2}(f_{1}x+f_{2})\phi_{p}(f_{3}x+f_{4})=\sum_{x\neq-
f_{1}^{-1}f_{2}}\phi_{p}(f_{3}x+f_{4})$ $\displaystyle=$
$\displaystyle\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f_{3}x+f_{4})-\phi_{p}(-f_{1}^{-1}f_{2}f_{3}+f_{4})=-\phi_{p}(-f_{1}^{-1}f_{2}f_{3}+f_{4}).$

As for $f(x)=(f_{1}x+f_{2})^{3}$, $f_{1}\in\mathbb{F}_{p}^{\times}$, one has

$\displaystyle\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))$ $\displaystyle=$
$\displaystyle\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}^{3}(f_{1}x+f_{2})=\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f_{1}x+f_{2})=0.$

$\Box$

For $N\geq 2$, we set $\pi(N)$ to be the number of primes less than or equal
to $N$. The following is Lemma 2.1 in [15].

######  Lemma 4.3.

Let $\\{c_{p}\\}_{p}$ be a bounded sequence of nonnegative numbers indexed by
primes $p$. If one of the sequences
$\left\\{\displaystyle\frac{1}{\pi(N)}\sum_{\begin{subarray}{c}p\leq N\\\
p:\textrm{ prime}\end{subarray}}c_{p}\right\\}$ and
$\left\\{\displaystyle\frac{1}{N}\sum_{\begin{subarray}{c}p\leq N\\\
p:\textrm{ prime}\end{subarray}}c_{p}\log p\right\\}$ converges, then both
converge to the same limit.

######  Lemma 4.4.

Let $f(x)\in{\mathbb{Z}}[x]$ be a cubic polynomial. One has

$\lim_{N\to\infty}\frac{1}{N}\sum_{\begin{subarray}{c}p\leq N\\\ p:\textrm{
prime}\end{subarray}}\frac{\log
p}{p}\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))=0.$

Proof: We start with assuming that $f(x)$ has a repeated root over
${\mathbb{Q}}$. If $f(x)$ has a root of multiplicity three, then this means
that the reduction of $f$ modulo every prime $p$ is of multiplicity three.
Lemma 4.2 implies that
$\displaystyle\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))=0$, hence the result.

If $f(x)$ has a root of multiplicity two, then this means that for all but
finitely many primes the reduction of $f(x)$ has a double root modulo $p$. In
view of Lemma 4.2, one has

$\left|\sum_{\begin{subarray}{c}p\leq N\\\ p:\textrm{
prime}\end{subarray}}\frac{\log
p}{p}\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))\right|\leq\sum_{\begin{subarray}{c}p\leq
N\\\ p:\textrm{ prime}\end{subarray}}\frac{\log
p}{p}\left|\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))\right|\leq\sum_{\begin{subarray}{c}p\leq
N\\\ p:\textrm{ prime}\end{subarray}}\frac{\log p}{p}<2+\log N,$

where the last inequality is Mertens’ first theorem. Thus the result follows
in this case.

Now we assume $f(x)$ does not have a multiple root over ${\mathbb{Q}}$. This
means that for all but finitely many primes, the reduction of $f(x)$ has no
multiple roots. In particular, the equation $E:y^{2}=f(x)$ represents an
elliptic curve for all but finitely many primes. Setting $\Delta(E)$ to be the
discriminant of $E$, one obtains

$\displaystyle\left|\sum_{\begin{subarray}{c}p\leq N\\\ p:\textrm{
prime}\end{subarray}}\frac{\log
p}{p}\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))\right|$ $\displaystyle=$
$\displaystyle\left|-\sum_{\begin{subarray}{c}p\leq N\\\
p\nmid\Delta(E)\end{subarray}}\frac{\log
p}{p}a_{p}(E)+\sum_{\begin{subarray}{c}p\leq N\\\
p\mid\Delta(E)\end{subarray}}\frac{\log
p}{p}\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))\right|$ $\displaystyle\leq$
$\displaystyle\sum_{\begin{subarray}{c}p\leq N\\\
p\nmid\Delta(E)\end{subarray}}\frac{\log
p}{p}|a_{p}(E)|+\sum_{\begin{subarray}{c}p\leq N\\\
p\mid\Delta(E)\end{subarray}}\frac{\log
p}{p}\left|\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))\right|$
$\displaystyle\leq$ $\displaystyle 2\sum_{\begin{subarray}{c}p\leq N\\\
p\nmid\Delta(E)\end{subarray}}\frac{\log
p}{\sqrt{p}}+\sum_{\begin{subarray}{c}p\leq N\\\
p\mid\Delta(E)\end{subarray}}\frac{\log
p}{p}\left|\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(f(x))\right|,$

where the last inequality follows from Hasse’s bound, and the second term in
the third line is a constant. Now Lemma 4.3 shows that

$\lim_{N\to\infty}\frac{1}{N}\sum_{\begin{subarray}{c}p\leq N\\\ p:\textrm{
prime}\end{subarray}}\frac{\log
p}{\sqrt{p}}=\lim_{N\to\infty}\frac{1}{\pi(N)}\sum_{\begin{subarray}{c}p\leq
N\\\ p:\textrm{ prime}\end{subarray}}\frac{1}{\sqrt{p}}.$

One notices that

$\sum_{\begin{subarray}{c}p\leq N\\\ p:\textrm{
prime}\end{subarray}}\frac{1}{\sqrt{p}}\leq\sum_{2\leq i\leq
N}\frac{1}{\sqrt{i}}\leq\int_{1}^{N}\frac{1}{\sqrt{x}}\,dx=2\sqrt{N}-1.$

Finally, the Prime Number Theorem yields that

$\lim_{N\to\infty}\frac{1}{N}\sum_{\begin{subarray}{c}p\leq N\\\ p:\textrm{
prime}\end{subarray}}\frac{\log p}{\sqrt{p}}=0,$

hence follows the result. $\Box$

##  5\. The Mordell-Weil rank

Let ${\mathcal{E}}$ be a quadratic elliptic surface defined by
$y^{2}=a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)$. One recalls that

$a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)=A(x)T^{2}+B(x)T+C(x)$

where $A,B,C\in{\mathbb{Z}}[x]$, $\deg A=3$, and $\deg B,\deg C\leq 3$. The
following theorem allows one to evaluate $A_{p}({\mathcal{E}})$.

######  Theorem 5.1.

Let ${\mathcal{E}}$ be a quadratic elliptic surface defined over
${\mathbb{Q}}$ by the equation

$y^{2}=a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)=A(x)T^{2}+B(x)T+C(x)$

where $A,B,C\in{\mathbb{Z}}[x]$, $\deg A=3$, and $\deg B,\deg C\leq 3$. Then

$\displaystyle-
pA_{p}({\mathcal{E}})=p\sum_{\begin{subarray}{c}x\in{\mathbb{F}_{p}}\\\
B(x)^{2}-4A(x)C(x)\equiv 0\textrm{ mod
}p\end{subarray}}\phi_{p}(A(x))-\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(A(x))+R_{p},$

where $|R_{p}|\leq 12.$

Proof: One starts with evaluating $A_{p}({\mathcal{E}})$ recalling that if
$p|\Delta(t)$, then $a_{p}({\mathcal{E}}_{t})=0$:

$\displaystyle-pA_{p}({\mathcal{E}})$ $\displaystyle=$
$\displaystyle-\sum_{t\in{\mathbb{F}_{p}}}a_{p}({\mathcal{E}}_{t})$
$\displaystyle=$
$\displaystyle\sum_{\begin{subarray}{c}t\in{\mathbb{F}_{p}}\\\
p\nmid\Delta(t)\end{subarray}}\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(a_{3}(t)x^{3}+a_{2}(t)x^{2}+a_{1}(t)x+a_{0}(t))$
$\displaystyle=$
$\displaystyle\sum_{x\in{\mathbb{F}_{p}}}\sum_{\begin{subarray}{c}t\in{\mathbb{F}_{p}}\\\
p\nmid\Delta(t)\end{subarray}}\phi_{p}(A(x)t^{2}+B(x)t+C(x))$ $\displaystyle=$
$\displaystyle\left(\sum_{\begin{subarray}{c}x\in{\mathbb{F}_{p}}\\\
B(x)^{2}-4A(x)C(x)\equiv 0\textrm{ mod
}p\end{subarray}}+\sum_{\begin{subarray}{c}x\in{\mathbb{F}_{p}}\\\
B(x)^{2}-4A(x)C(x)\not\equiv 0\textrm{ mod
}p\end{subarray}}\right)\sum_{\begin{subarray}{c}t\in{\mathbb{F}_{p}}\\\
p\nmid\Delta(t)\end{subarray}}\phi_{p}(A(x)t^{2}+B(x)t+C(x)).$

Now one may use Lemma 4.1 to see that for a fixed $x\in{\mathbb{F}_{p}}$, one
has

$\displaystyle\sum_{t\in{\mathbb{F}_{p}}}\phi_{p}(A(x)t^{2}+B(x)t+C(x))=\begin{cases}(p-1)\phi_{p}(A(x))&\textrm{
if }B(x)^{2}-4A(x)C(x)\equiv 0\textrm{ mod }p\\\ -\phi_{p}(A(x))&\textrm{ if
}B(x)^{2}-4A(x)C(x)\not\equiv 0\textrm{ mod }p.\end{cases}$

This yields that

$\displaystyle-pA_{p}({\mathcal{E}})$ $\displaystyle=$
$\displaystyle(p-1)\sum_{\begin{subarray}{c}x\in{\mathbb{F}_{p}}\\\
B(x)^{2}-4A(x)C(x)\equiv 0\textrm{ mod
}p\end{subarray}}\phi_{p}(A(x))-\sum_{\begin{subarray}{c}x\in{\mathbb{F}_{p}}\\\
B(x)^{2}-4A(x)C(x)\not\equiv 0\textrm{ mod }p\end{subarray}}\phi_{p}(A(x))$
$\displaystyle-$
$\displaystyle\sum_{x\in{\mathbb{F}_{p}}}\sum_{\begin{subarray}{c}t\in{\mathbb{F}_{p}}\\\
p|\Delta(t)\end{subarray}}\phi_{p}(A(x)t^{2}+B(x)t+C(x))$

Using Lemma 4.2, one obtains that

$\displaystyle-pA_{p}({\mathcal{E}})$ $\displaystyle=$ $\displaystyle
p\sum_{\begin{subarray}{c}x\in{\mathbb{F}_{p}}\\\ B(x)^{2}-4A(x)C(x)\equiv
0\textrm{ mod
}p\end{subarray}}\phi_{p}(A(x))-\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(A(x))-\sum_{x\in{\mathbb{F}_{p}}}\sum_{t\in{\mathbb{F}_{p}}}^{\prime}\phi_{p}(A(x)t^{2}+B(x)t+C(x))$

where $\sum^{\prime}$ means that the sum runs over $t\in{\mathbb{F}_{p}}$ such
that $p$ is a prime of multiplicative reduction of ${\mathcal{E}}_{t}$. Again,
according to Lemma 4.2 and the fact that the discriminant of ${\mathcal{E}}$
is a polynomial of degree $12$ in $T$, one knows that

$\left|\sum_{x\in{\mathbb{F}_{p}}}\sum_{t\in{\mathbb{F}_{p}}}^{\prime}\phi_{p}(A(x)t^{2}+B(x)t+C(x))\right|=\left|\sum_{t\in{\mathbb{F}_{p}}}^{\prime}\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(a_{3}(t)x^{3}+a_{2}(t)x^{2}+a_{1}(t)x+a_{0}(t))\right|\leq
12,$

hence the conclusion. $\Box$

######  Remark 5.2.

Let ${\mathcal{E}}$ be a quadratic elliptic surface defined over
${\mathbb{Q}}$ by the following equation

$y^{2}=a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)=A(x)T^{2}+B(x)T+C(x)$

where $A,B,C\in{\mathbb{Z}}[x]$, $\deg A=3$, and $\deg B,\deg C\leq 3$. Then
one has

$|A_{p}({\mathcal{E}})|\leq\begin{cases}6+\frac{2}{\sqrt{p}}+\frac{12}{p}&\textrm{if
$A(x)$ has no repeated root}\\\ 6+\frac{13}{p}&\textrm{if $A(x)$ has a root of
multiplicity $2$}\\\ 6+\frac{12}{p}&\textrm{if $A(x)$ has a root of
multiplicity $3$}\end{cases}$

This follows directly from Lemma 4.2 and Theorem 5.1 together with Hasse’s
bound when $f(x)$ has no repeated root.

Now the bounds of the Mordell-Weil rank of a quadratic elliptic surface
follow.

######  Theorem 5.3.

Let ${\mathcal{E}}$ be a quadratic elliptic surface defined over
${\mathbb{Q}}$ by the following equation

$y^{2}=a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)=A(x)T^{2}+B(x)T+C(x)$

where $A,B,C\in{\mathbb{Z}}[x]$, $\deg A=3$, and $\deg B,\deg C\leq 3$. We set

$S_{1}=\\{x\in{\mathbb{Q}}:B(x)^{2}-4A(x)C(x)=0\textrm{ and $A(x)$ is a
nonzero rational square}\\}.$

The Mordell-Weil rank of ${\mathcal{E}}({\mathbb{Q}}(T))$ is given by

$\begin{cases}\operatorname{rank}({\mathcal{E}}({\mathbb{Q}}(T)))=|S_{1}|&\textrm{
if }B(x)^{2}-4A(x)C(x)\textrm{ splits completely over }{\mathbb{Q}}\\\
|S_{1}|\leq\operatorname{rank}({\mathcal{E}}({\mathbb{Q}}(T)))\leq|S_{1}|+\delta&\textrm{
otherwise}\end{cases}$

where $\delta$ is the number of irreducible factors of $B(x)^{2}-4A(x)C(x)$
over ${\mathbb{Q}}$ with degree at least $2$.

In particular,
$|S_{1}|\leq\operatorname{rank}({\mathcal{E}}({\mathbb{Q}}(T)))\leq|S_{1}|+3.$

Proof: Since ${\mathcal{E}}$ is a rational elliptic surface, it follows that
$\displaystyle\operatorname{rank}({\mathcal{E}}({\mathbb{Q}}(T)))=\lim_{X\to\infty}\frac{1}{X}\sum_{\begin{subarray}{c}p\leq
X\\\ p:\textrm{ prime}\end{subarray}}-A_{p}({\mathcal{E}})\log p$. As
$\displaystyle\lim_{X\to\infty}\frac{1}{X}\sum_{\begin{subarray}{c}p\leq X\\\
p:\textrm{
prime}\end{subarray}}\sum_{x\in{\mathbb{F}_{p}}}\phi_{p}(A(x))\frac{\log
p}{p}=0$ by Lemma 4.4, Theorem 5.1 implies that

$\operatorname{rank}({\mathcal{E}}({\mathbb{Q}}(T)))=\lim_{X\to\infty}\frac{1}{X}\sum_{\begin{subarray}{c}p\leq
X\\\ p:\textrm{
prime}\end{subarray}}\sum_{\begin{subarray}{c}x\in{\mathbb{F}_{p}}\\\
B(x)^{2}-4A(x)C(x)\equiv 0\textrm{ mod }p\end{subarray}}\phi_{p}(A(x))\log p.$

Now we set

$S_{2}=\\{x\in{\mathbb{Q}}:B(x)^{2}-4A(x)C(x)=0\textrm{ and $A(x)$ is not a
nonzero rational square}\\}.$

Let $B(x)^{2}-4A(x)C(x)=h_{1}(x)h_{2}(x)g(x)$ where
$h_{i}(x)\in{\mathbb{Z}}[x]$ is such that $h_{i}(x)=0$ if and only if $x\in
S_{i}$. It follows that $g(x)$ is a product of at most three irreducible
polynomials each of degree $\geq 2$. One has

$\sum_{\begin{subarray}{c}x\in{\mathbb{F}_{p}}\\\ B(x)^{2}-4A(x)C(x)\equiv
0\textrm{ mod }p\end{subarray}}\phi_{p}(A(x))=\sum_{x\in
S_{1}}\phi_{p}(A(x))+\sum_{x\in
S_{2}}\phi_{p}(A(x))+\sum_{\begin{subarray}{c}x\in{\mathbb{F}_{p}}\\\
g(x)\equiv 0\textrm{ mod }p\end{subarray}}\phi_{p}(A(x)).$

One knows that for any $x\in S_{1}$, $\phi_{p}(A(x))=1$. Moreover, for any
$x\in S_{2}$, either $A(x)=0$, hence it does not contribute to the second sum,
or $A(x)$ is not a square in ${\mathbb{Q}}$. In the latter case, one needs to
evaluate the following limit for $x\in S_{2}$

(2) $\lim_{X\to\infty}\frac{1}{X}\sum_{\begin{subarray}{c}p\leq X\\\
p:\textrm{ prime}\end{subarray}}\phi_{p}(A(x))\log
p\leq\lim_{X\to\infty}\frac{\log X}{X}\sum_{\begin{subarray}{c}p\leq X\\\
p:\textrm{ prime}\end{subarray}}\phi_{p}(A(x)).$

In accordance to (5.79) of [7], one has

$\sum_{\begin{subarray}{c}p\leq X\\\ p:\textrm{
prime}\end{subarray}}\phi_{p}(A(x))=O(X(\log X)^{-\alpha})$

for any $\alpha>0$ where the implied constant depends only on $\alpha$. In
particular, by choosing $\alpha>1$, one obtains that the limit in (2) is zero.
Now, the latter argument together with the Prime Number Theorem and Lemma 4.3
yield that

$\operatorname{rank}({\mathcal{E}}({\mathbb{Q}}(T)))=|S_{1}|+\lim_{X\to\infty}\frac{1}{X}\sum_{\begin{subarray}{c}p\leq
X\\\ p:\textrm{
prime}\end{subarray}}\sum_{\begin{subarray}{c}x\in{\mathbb{F}_{p}}\\\
g(x)\equiv 0\textrm{ mod }p\end{subarray}}\phi_{p}(A(x))\log p.$

Now, if $m(x)$ is an irreducible factor of $g(x)$, $\deg m\geq 2$, the
following inequality holds

$\sum_{\begin{subarray}{c}x\in{\mathbb{F}_{p}}\\\ m(x)\equiv 0\textrm{ mod
}p\end{subarray}}\phi_{p}(A(x))\leq r_{p}(m(x))$

where $r_{p}(m(x))$ is the number of zeros of $m(x)$ mod $p$.

Since $m(x)$ is irreducible, then by Chebotarev’s density theorem, the density
of primes $p$ such that $m$ has $k$ roots mod $p$ is equal to the proportion
of elements in the Galois group of $m(x)$ which fix $k$ roots when acting on
the roots. As the Galois group acts transitively on the roots, the average
number of fixed points is given by

$\lim_{X\to\infty}\frac{1}{X}\sum_{\begin{subarray}{c}p\leq X\\\ p:\textrm{
prime}\end{subarray}}r_{p}(m(x))\log
p=\lim_{X\to\infty}\frac{1}{\pi(X)}\sum_{\begin{subarray}{c}p\leq X\\\
p:\textrm{ prime}\end{subarray}}r_{p}(m(x))=1$

where the first equality is due to Lemma 4.3, whence the result follows.
$\Box$

######  Remark 5.4.

Let ${\mathcal{E}}$ be a quadratic elliptic surface defined over
${\mathbb{Q}}$ by the equation

$y^{2}=a_{3}(T)x^{3}+a_{2}(T)x^{2}+a_{1}(T)x+a_{0}(T)=A(x)T^{2}+B(x)T+C(x)$

where $A,B,C\in{\mathbb{Z}}[x]$, $\deg A=3$, and $\deg B,\deg C\leq 3$. One
has

$0\leq\operatorname{rank}({\mathcal{E}}({\mathbb{Q}}(T)))\leq 6.$

If one wants to construct a quadratic elliptic surface with Mordell-Weil rank
$r$, namely $1\leq r\leq 6$, then one considers the polynomial

$D(x)=4(x-v_{1})(x-v_{2})(x-v_{3})(x-v_{4})(x-v_{5})(x-v_{6}),\quad
v_{i}\in{\mathbb{Z}},\;v_{i}\neq v_{j}\textrm{ if }i\neq j.$

One then chooses $A(x),B(x),C(x)\in{\mathbb{Z}}[x]$, with $\deg A(x)=3$ and
$B(x),C(x)$ of degree at most $3$ such that $D(x)-B(x)^{2}=-4A(x)C(x)$,
$B(v_{i})\neq 0$, $i=1,\cdots,r$; and $A(v_{i})$ is a nonzero rational sqaure
for every $i=1,\ldots,r$. Now if we consider the elliptic surface

${\mathcal{E}}:y^{2}=A(x)T^{2}+B(x)T+C(x)$

over ${\mathbb{Q}}$, then according to Theorem 5.3, one has
$\operatorname{rank}({\mathcal{E}}({\mathbb{Q}}(T)))=r$. In [1], examples of
such elliptic surfaces with maximal rank $6$ were given. In current work, we
plan to present how the polynomials $A(x),B(x),$ and $C(x)$ may be chosen to
produce an elliptic surface over ${\mathbb{Q}}$ with Mordell-Weil rank $r$,
$1\leq r\leq 6$.

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