# GridPull: Towards Scalability in Learning Implicit Representations from 3D
Point Clouds

Chao Chen1 Yu-Shen Liu1 Zhizhong Han2
1School of Software, Tsinghua University, Beijing, China
2Department of Computer Science, Wayne State University, Detroit, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>The
corresponding author is Yu-Shen Liu. This work was supported by National Key
R&D Program of China (2022YFC3800600), the National Natural Science Foundation
of China (62272263, 62072268), and in part by Tsinghua-Kuaishou Institute of
Future Media Data.

###### Abstract

Learning implicit representations has been a widely used solution for surface
reconstruction from 3D point clouds. The latest methods infer a distance or
occupancy field by overfitting a neural network on a single point cloud.
However, these methods suffer from a slow inference due to the slow
convergence of neural networks and the extensive calculation of distances to
surface points, which limits them to small scale points. To resolve the
scalability issue in surface reconstruction, we propose GridPull to improve
the efficiency of learning implicit representations from large scale point
clouds. Our novelty lies in the fast inference of a discrete distance field
defined on grids without using any neural components. To remedy the lack of
continuousness brought by neural networks, we introduce a loss function to
encourage continuous distances and consistent gradients in the field during
pulling queries onto the surface in grids near to the surface. We use uniform
grids for a fast grid search to localize sampled queries, and organize surface
points in a tree structure to speed up the calculation of distances to the
surface. We do not rely on learning priors or normal supervision during
optimization, and achieve superiority over the latest methods in terms of
complexity and accuracy. We evaluate our method on shape and scene benchmarks,
and report numerical and visual comparisons with the latest methods to justify
our effectiveness and superiority. The code is available at
https://github.com/chenchao15/GridPull.

##  1 Introduction

It is vital to reconstruct surfaces from 3D point clouds for downstream
applications. A widely used strategy is to learn implicit representations [59,
62, 26, 10, 91, 77, 54, 68, 88] from 3D point clouds in a data-driven manner.
With the learned implicit representations, we can reconstruct surfaces in
meshes by running the marching cubes algorithm [48]. By learning priors from
large scale datasets during training, previous methods [57, 21, 31, 46, 78,
73, 17] generalize the learned priors to either global implicit functions [89,
80, 8, 32, 6, 50] or local ones for unseen point clouds. However, the
generalization ability limits their performances on large structure or
geometry variations of training samples.

More recent methods [23, 12, 2, 100, 3, 95, 4, 49, 14, 65, 35] achieved better
generalization by directly overfitting neural networks on single unseen point
clouds. They rely on extensive calculations of distances between queries and
surface points to probe the space, which supervises neural networks to
converge to a distance or occupancy field [49, 23, 2, 100, 3, 11, 52, 83, 40].
However, this inference procedure is time consuming due to the extensive
distance calculations and the slow convergence of neural networks. This
demerit makes these methods hard to scale up to large scale point clouds for
surface reconstruction.

To address the scalability challenge, we propose GridPull to speed up the
learning of implicit function from large scale point clouds. GridPull does not
require learned priors or point normal, and directly infers a distance field
from a point cloud without using any neural components. We infer the distance
field on grids near the surface, which reduces the number of grids we need to
infer. Moreover, we organize surface points in a tree structure to speed up
the nearest neighbor search for the calculation of distances to the surface.
Specifically, we infer the discrete distance field by pulling queries onto the
surface in grids of interests. Our loss function encourages continuous
distances and consistent gradients in the field, which makes up the lack of
continuousness brought by neural networks. We justify our effectiveness and
highlight our superiority by numerical and visual comparisons with the latest
methods on the widely used benchmarks. Our contributions are listed below.

  1. i)

We propose GridPull to reconstruct surfaces from large scale point clouds
without using neural networks. GridPull speeds up the learning of implicit
function, which addresses the scalability challenge in surface reconstruction.

  2. ii)

We introduce a loss function to directly infer a discrete distance field
defined on grids but achieving continuous distances and consistent gradients
in the field.

  3. iii)

Our method outperforms state-of-the-art methods in surface reconstruction in
terms of speed and accuracy on the widely used benchmarks.

##  2 Related Work

Neural implicit representations have made a huge progress in various tasks
[59, 62, 26, 10, 101, 91, 77, 54, 68, 53, 39, 36, 60, 28, 7, 41, 76]. We can
use different supervision including 3D supervision [58, 64, 63, 56, 13, 92,
87], multi-view [72, 45, 34, 99, 44, 90, 61, 42, 94, 93, 19, 84, 98, 86, 81,
82, 25, 22, 29, 69, 74, 33, 55], and point clouds [89, 43, 57, 21] to learn
neural implicit representations. In the following, we focus on reviewing works
on surface reconstruction by learning implicit representations point clouds
below.

Learning with Priors. With neural networks, one intuitive strategy is to use
neural networks to learn priors from a training set and then generalize the
learned priors to unseen samples. We can learn either global priors to map a
shape level point cloud into a global implicit function [57, 21, 31, 46, 78,
73, 17] or local priors for local implicit functions to represent parts or
patches [89, 80, 8, 32, 6, 50] which are further used to approximate a global
implicit function.

These methods require large scale datasets to learn priors. However, the
learned priors may not generalize well to unseen point clouds that have large
geometric variations compared to training samples. Our method does not require
priors, and falls into the following category.

Learning with Overfitting. To improve the generalization, we can learn
implicit functions by overfitting neural networks on single point clouds.
Methods using this strategy introduce novel constraints [23, 2, 100, 3, 95,
4], ways of leveraging gradients [49, 14], differentiable poisson solver [65]
or specially designed priors [50, 51] to learn signed [49, 23, 2, 100, 3, 11]
or unsigned distance functions [14, 102, 47].

These methods rely on neural networks to infer implicit functions without
learning priors. However, the slow convergence of neural networks is a big
limit for them to scale up to large scale point clouds. Our method addresses
the scalability challenge by directly inferring a discrete distance field
without using neural networks. Our novel loss leads to less complexity and
higher accuracy than the overfitting based methods.

Learning with Grids. Learning implicit functions with grids has been used in
prior-based methods [77, 54, 66, 79, 46, 34, 85]. Using neural networks, these
methods learn features at vertices of grids and further map the interpolated
features into a distance or occupancy field. Some methods [70, 75, 96, 38]
inferred a discrete radiance field defined on grids to speed up the learning
of a radiance field, while they cared more about the quality of synthesized
views than the underlying geometry. Without using neural components, some
methods [67, 65] directly infer discrete implicit functions on grids. To make
up the continuousness of neural networks, they employed different constraints,
such as Possion equations or Viscosity priors, to pursue a continuous field
from a single point cloud. However, these methods are slowly converged, which
makes them hard to scale up to large scale point clouds. Although we also use
grids to speed up the inference, we introduce a more efficient training
strategy to decrease optimization space and more effective losses to increase
inference efficiency.

##  3 Method

Overview. GridPull aims to achieve a fast inference of a distance field, such
as a signed distance field, from a 3D point cloud $S=\\{s_{j}|j\in[1,J]\\}$
without using any neural components. As shown in Fig. 1 (b), we represent the
distance field by a signed distance function (SDF) $f$ defined on a set of
discrete grids $G$ with a resolution of $R^{3}$, where the vertices
$V=\\{v_{i}|i\in[1,I],I=(R+1)^{3}\\}$ shared among these grids hold learnable
signed distances $D=\\{d_{i}\\}$. For an arbitrary location $q$ in Fig. 1 (e),
such as a randomly sampled query or a surface point, we use trilinear
interpolation to approximate the signed distance $f(q)$ from the eight nearest
grid vertices below,

$f(q)=triInter(q,D).$ (1)

Our goal is to conduct direct optimization in Fig. 1 (c) to infer the signed
distances $D$ on grid vertices by minimizing our loss function $L$ on queries
$q$ below,

$\min_{D}\ L(f(q)).$ (2)

With the learned function $f$, we run the marching cubes algorithm [48] to
reconstruct a mesh as the surface as shown in Fig. 1 (d).

Figure 1: We proposed GridPull for fast inference of SDF defined on discrete
grids using effective losses.

Discrete Grids. We split the space occupied by the point cloud $S$ into
discrete grids $G$ with a resolution of $R$ in Fig. 1 (b). GridPull can work
with high resolutions, such as $R=512$, because of our efficient and effective
loss function and training strategy. Within each grid, we interpolate the
signed distance $f(q)$ of query $q$ using the trilinear interpolation in Eq.
1. We organize surface points in a tree structure to speed up the search of
the nearest surface point for queries $q$. Since we need to extensively
conduct trilinear interpolation on signed distances for queries $q$, we use
uniform grids for fast nearest grid search for queries. We did not use an
Octree to organize grids, due to its inefficient and inaccurate grid search
for queries.

Geometric Initialization. We initialize the signed distances $d_{i}$ on grid
vertices which represent a sphere in the field. The initialization can speed
up the convergence during optimization. Our preliminary results show that a
small sphere with a radius of one or two grids works well. Since enlarging a
small sphere to the target shape can calculate the continuous distance loss
over smaller regions than shrinking a large sphere.

Signed Distance Inference. For a query $q$, we infer its signed distance
$f(q)$ through pulling it onto its nearest surface point in the point cloud
$S$. As shown in Fig. 1 (e), we first obtain $f(q)$ as the interpolation of
the signed distances $d_{i}$ on the nearest eight vertices of $q$. We
implement the trilinear interpolation by solving $\bm{a}=[a_{0},..,a_{7}]$ in
a linear regression system on the eight vertices $v_{i}$ below,

$f(q)=a_{0}+a_{1}x+a_{2}y+a_{3}z+a_{4}xy+a_{5}xz+a_{6}yz+a_{7}xyz,$ (3)

where $[x,y,z]$ is the coordinate of $q$ and the eight constraints are
$\\{f(v_{i})=d_{i}\\}$.

With the predicted signed distance $f(q)$ and the gradient $\nabla
f(q)=\partial f/\partial q$, we pull a query $q$ onto the surface and obtain a
pulled query $p$ by $p=q-f(q)\times\nabla f(q)/||\nabla f(q)||_{2}$ in Fig. 1
(f).

Our goal is to minimize the distance between the pulled query $p$ and the
nearest surface point $NN(q)\in S$ of query $q$ below,

$\displaystyle L_{Pull}=||p-NN(q)||_{2}.$ (4)

We adopt this pulling loss from [49]. Instead, we do not rely on neural
network in the pulling procedure. We use the trilinear interpolation on signed
distances $d_{i}$ on grid vertices to obtain signed distances and gradients
according to Eq. 3.

We sample queries $q$ around surface points in each epoch. For each randomly
sampled surface point, we sample queries using a gaussian distribution with
the surface point as the center and a two-grid length variance. We build a KD-
tree on surface points so that we can speed up the searching of the nearest
neighbor from a large number of surface points for each query.

To speed up the optimization, we only calculate the pulling loss on grids near
the surface. We regard the grids containing surface points as the base and
find the union of their $M_{1}$ nearest neighboring grids. These grids form a
$M_{1}$-bandwidth near the surface in Fig. 1 (c).

Continuous Distance Fields. Although the pulling loss can infer correct signed
distances at grid vertices, it fails to learn continuous signed distances
across the field, especially near the surface. Fig. 2 (a) shows severe
artifacts on the surface and gradients near the surface are messy and
inconsistent, which are caused by the discontinuous signed distances across
neighboring regions.

Neural network based methods [49, 51] do not have this issue, since the
continuous character of neural networks makes the sudden change of signs on
the same shape side hard. While our method uses signed distances $d_{i}$ on
grid vertices to represent a discrete field, this makes that one $d_{i}$ can
change separately without considering the change of neighboring $d_{i}$, which
leads to a poor continuousness in the field.

To resolve this issue, we impose a total variation (TV) loss as a continuous
constraint on our signed distances $d_{i}$ on grid vertices. Our key idea is
to make the change of one $d_{i}$ affect the change of its neighboring
$NN(d_{i})$, which smoothes the local distance field and propagates local
updates to the rest grids. Hence, as shown in a 2D case in Fig. 1 (g), we
approximate one $d_{i}$ to each one of its six neighbors by minimizing the
difference between each two below,

$\displaystyle
L_{TV}=\sqrt{\sum\nolimits_{a=\\{x,y,z\\}}e_{a+1}(d_{i})^{2}+e_{a-1}(d_{i})^{2}},$
(5)

where $e_{a+1}(d_{i})$ is the difference between $d_{i}$ and the signed
distance along the $a$ axis, i.e., $x,y,z$ axis. Fig. 2 (b) shows that the
continuous constraints can significantly improve the signed distance field in
terms of continuousness and gradients.

One remaining issue is its computational complexity. Since we are targeting
grids at a high resolution, such as $R=512$, it is very time consuming to
impose this constraint on all $d_{i}$ across the field.

To resolve this issue, we focus on grids near the surface. Similar to imposing
the pulling loss on grids in the $M_{1}$-bandwidth near the surface, we impose
our continuous constraints on the grids in the $M_{2}$-bandwidth near the
surface, which significantly reduces the number of learnable parameters to
optimize.

Signed Distance Supervision. For signed distances, one supervision we can
directly use is the input point cloud $S$. It is a discrete surface indicating
the zero level-set of the signed distance function. Hence, as introduced in
Fig. 1 (h), we can constrain the signed distances at the points $s$ on the
surface below,

$\displaystyle L_{Surface}=||f(s)||_{2}.$ (6)

Although Fig. 2 (c) indicates that $L_{Surface}$ slightly improves the signed
distance field and leads to a little bit more continuous surface, it is
helpful to determine the zero level set with other losses. For noises on $S$,
our continuous constraint can relieve the impact of noises on this
supervision. We will show this in experiments.

Consistent Gradients. Fig. 2 (a) indicates that the pulling loss produces
gradients that are messy and non-orthogonal to the surface. This leads to
inaccurate signed distance inference during the pulling procedure.

Figure 2: The effect of losses on reconstruction.

We additionally introduce a constraint on gradients to improve the field near
the surface. As shown in Fig. 1 (i), we aim to encourage the gradient at a
query $q$ to point to the same direction of the gradient at its nearest
surface point $NN(q)$. We use a cosine distance below to achieve more
consistent gradients,

$\displaystyle L_{\nabla}=1-cos(\nabla f(q),\nabla f(NN(q))).$ (7)

$L_{\nabla}$ constrains the gradients in the field and corrects inaccurate
gradients. Fig. 2 (d) shows that more consistent gradients can improve the
surface to be more continuous using small planes on the surface which however
the surface not smooth at all.

Loss Function. We learn signed distances on grid vertices $D$ by minimizing
the loss function below,

$L=L_{Pull}+\alpha L_{TV}+\beta L_{Surface}+\gamma L_{\nabla}.$ (8)

where $\alpha$, $\beta$, and $\gamma$ are balance weights to make each term
contribute equally. Fig. 3 shows the signed distance field on a slide of $D$
during optimization. Our optimization starts from an initial sphere and only
optimizes variables in the bandwidth. We skip the grids that are not optimized
after the initialization during the marching cubes for surface reconstruction.
Please watch our video for more details.

Figure 3: Visualization of signed distances in optimization.

##  4 Experiments and Analysis

We evaluate GridPull by numerical and visual comparisons with the latest
methods on synthetic and real datasets in surface reconstruction.

Datasets and Metrics. We use benchmarks for shapes and scenes in evaluations.
For shapes, we conduct evaluations on five datasets including a subset of
ShapeNet [1], FAMOUS [17], Thingi10k [103], Surface Reconstruction Benchmark
(SRB) [89] and D-FAUST [5]. For scenes, we also report comparisons on five
datasets including 3DScene [104], SceneNet [27], 3DFRONT [18], Matterport [9]
and KITTI [20].

We use L2 Chamfer distance ($CD_{L2}$), $CD_{L1}$ and Hausdorff distance(HD)
to measure the error between the reconstructed surface and the ground truth.
Moreover, we use normal consistency (NC) and F-score to evaluate the accuracy
of normal on the reconstructed surface. We also report our time and storage
complexity to highlight our advantage towards scalability. We also use the
intersection over union (IoU) to measure the reconstruction for fair
comparison with the latest methods.

Details. We use $R=256$ to infer a distance field. We calculate the pulling
loss on grids in a bandwidth with $M_{1}=3$. We impose the continuous
constraint on grids in a bandwidth with $M_{2}=14$ for shapes and $M_{2}=25$
for scenes, since scenes contain open surfaces with larger empty spaces. We
run the marching cubes for surface reconstruction at a resolution of 256.

We use the Adam optimizer with an initial learning rate of 1.0. We decrease
the learning rate with a decay rate of 0.3 every 400 iterations. We use
$50,000$ queries in each iteration, and run 1600 iterations for each point
cloud during optimization. We set the balance weights $\alpha=1$, $\beta=1$,
and $\gamma=0.005$ for equal contribution from the three terms.

Figure 4: Visual comparison under ShapetNet dataset.

###  4.1 Surface Reconstruction for Shapes

ShapeNet. Following predictive context prior (PCP) [51] and NeuralPull (NP)
[49], we report our results on a subset in ShapeNet in terms of $CD_{L2}$ in
Tab. 1, Normal Consistency (NC) in Tab. 2, and F-Score with thresholds of
$0.002$ and $0.004$ in Tab. 3 and Tab. 4, which show our superiority over PCP
and NP in numerical comparisons over all classes, even though PCP employs a
local shape prior learned from a large scale dataset. The visual comparison in
Fig. 4 shows that we can reveal more accurate and detailed geometry.

Class | NP [49] | PCP [51] | Ours
---|---|---|---
Display | 0.039 | 0.0087 | 0.0082
Lamp | 0.080 | 0.0380 | 0.0347
Airplane | 0.008 | 0.0065 | 0.0007
Cabinet | 0.026 | 0.0153 | 0.0112
Vessel | 0.022 | 0.0079 | 0.0033
Table | 0.060 | 0.0131 | 0.0052
Chair | 0.054 | 0.0110 | 0.0043
Sofa | 0.012 | 0.0086 | 0.0015
Mean | 0.038 | 0.0136 | 0.0086

Table 1: Reconstruction accuracy under ShapeNet in terms of $CD_{L2}\times
100$.

Class | NP [49] | PCP [51] | Ours
---|---|---|---
Display | 0.964 | 0.9775 | 0.9847
Lamp | 0.930 | 0.9450 | 0.9693
Airplane | 0.947 | 0.9490 | 0.9614
Cabinet | 0.930 | 0.9600 | 0.9689
Vessel | 0.941 | 0.9546 | 0.9667
Table | 0.908 | 0.9595 | 0.9755
Chair | 0.937 | 0.9580 | 0.9733
Sofa | 0.951 | 0.9680 | 0.9792
Mean | 0.939 | 0.9590 | 0.9723

Table 2: Reconstruction accuracy under ShapeNet in terms of NC.

Class | NP [49] | PCP [51] | Ours
---|---|---|---
Display | 0.989 | 0.9939 | 0.9963
Lamp | 0.891 | 0.9382 | 0.9455
Airplane | 0.996 | 0.9942 | 0.9976
Cabinet | 0.980 | 0.9888 | 0.9901
Vessel | 0.985 | 0.9935 | 0.9956
Table | 0.922 | 0.9969 | 0.9977
Chair | 0.954 | 0.9970 | 0.9979
Sofa | 0.968 | 0.9943 | 0.9974
Mean | 0.961 | 0.9871 | 0.9896

Table 3: Reconstruction accuracy under ShapeNet in terms of F-Score with a
threshold of 0.002.

Class | NP [49] | PCP [51] | Ours
---|---|---|---
Display | 0.991 | 0.9958 | 0.9963
Lamp | 0.924 | 0.9402 | 0.9538
Airplane | 0.997 | 0.9972 | 0.9989
Cabinet | 0.989 | 0.9939 | 0.9946
Vessel | 0.990 | 0.9958 | 0.9972
Table | 0.973 | 0.9985 | 0.9990
Chair | 0.969 | 0.9991 | 0.9990
Sofa | 0.974 | 0.9987 | 0.9992
Mean | 0.976 | 0.9899 | 0.9923

Table 4: Reconstruction accuracy under ShapeNet in terms of F-Score with a
threshold of 0.004. Figure 5: Visual comparison under FAMOUS dataset.

FAMOUS. We follow the experimental setting in PCP [51] and NP [49] to evaluate
GridPull on FAMOUS. We compare GridPull with the latest methods with priors
including PCP, GenSDF, FGC and no priors including NP and IGR. GridPull
outperforms these methods in terms of $CD_{L2}$ and running time. Although FGC
is fast, it does not reconstruct a watertight mesh but just a polygon soup
with no normal, which contains almost no geometry details. Worse than that, it
fails to reconstruct some point clouds in more than eight hours, hence we
remove these shapes from its results. We highlight our superiority in visual
comparison in Fig. 5. We can reveal details like hair and expressions more
clearly.

Method | $CD_{L2}\times 100$ | Min time | Mean time
---|---|---|---
IGR [24] | 1.650 | 402s | 407s
GenSDF [15] | 0.668 | 230s | 232s
NP [49] | 0.220 | 593s | 605s
FGC [97] | 0.055 | 10s | Timeout
PCP [51] | 0.044 | 3416s | 3534s
Ours | 0.040 | 198s | 201s

Table 5: Reconstruction accuracy under FAMOUS. Figure 6: Visual comparison
under SRB dataset.

SRB. We follow the experimental setting in VisCo [67] to evaluate GridPull on
real scans in SRB. We report the comparison with the latest overfitting based
methods in terms of $CD_{L1}$ in Tab. 6. Meanwhile, we also report their one-
sided distances between the reconstructed mesh and the input noisy point
cloud. The numerical comparison shows that our method significantly
outperforms the latest. Moreover, we also report our results of learning
unsigned distances in Tab. 7. We replace our pulling loss with the pulling
loss introduced for unsigned distances in CAP-UDF [102], and remove the
gradient loss due to the nondifferentiable character of UDF on surface. Visual
comparison in Fig. 6 shows that our method can reconstruct surfaces that are
more compact than the methods learning SDF like SIREN or more smooth than the
methods learning UDF like CAP-UDF.

|  | Poisson [37] | IGR [24] | SIREN [71] | VisCo [67] | SAP [65] | Ours
---|---|---|---|---|---|---|---
Anchor | $CD_{L1}$ | 0.60 | 0.22 | 0.32 | 0.21 | 0.12 | 0.093
$HD$ | 14.89 | 4.71 | 8.19 | 3.00 | 2.38 | 1.804
$d_{\overrightarrow{C}}$ | 0.60 | 0.12 | 0.10 | 0.15 | 0.08 | 0.066
$d_{\overrightarrow{H}}$ | 14.89 | 1.32 | 2.43 | 1.07 | 0.83 | 0.460
Daratech | $CD_{L1}$ | 0.44 | 0.25 | 0.21 | 0.26 | 0.07 | 0.062
$HD$ | 7.24 | 4.01 | 4.30 | 4.06 | 0.87 | 0.648
$d_{\overrightarrow{C}}$ | 0.44 | 0.08 | 0.09 | 0.14 | 0.04 | 0.039
$d_{\overrightarrow{H}}$ | 7.24 | 1.59 | 1.77 | 1.76 | 0.41 | 0.293
DC | $CD_{L1}$ | 0.27 | 0.17 | 0.15 | 0.15 | 0.07 | 0.066
$HD$ | 3.10 | 2.22 | 2.18 | 2.22 | 1.17 | 1.103
$d_{\overrightarrow{C}}$ | 0.27 | 0.09 | 0.06 | 0.09 | 0.04 | 0.036
$d_{\overrightarrow{H}}$ | 3.10 | 2.61 | 2.76 | 2.76 | 0.53 | 0.539
Gargoyle | $CD_{L1}$ | 0.26 | 0.16 | 0.17 | 0.17 | 0.07 | 0.063
$HD$ | 6.80 | 3.52 | 4.64 | 4.40 | 1.49 | 1.129
$d_{\overrightarrow{C}}$ | 0.26 | 0.06 | 0.08 | 0.11 | 0.05 | 0.045
$d_{\overrightarrow{H}}$ | 6.80 | 0.81 | 0.91 | 0.96 | 0.78 | 0.700
Lord Quas | $CD_{L1}$ | 0.20 | 0.12 | 0.17 | 0.12 | 0.05 | 0.047
$HD$ | 4.61 | 1.17 | 0.82 | 1.06 | 0.98 | 0.569
$d_{\overrightarrow{C}}$ | 0.20 | 0.07 | 0.12 | 0.07 | 0.04 | 0.031
$d_{\overrightarrow{H}}$ | 4.61 | 0.98 | 0.76 | 0.64 | 0.51 | 0.370

Table 6: Reconstruction accuracy under SRB.

| NDF [14] | CAP-UDF [102] | Ours(UDF)
---|---|---|---
$CD_{L1}$ | 0.238 | 0.073 | 0.070
F-Score | 68.6 | 84.5 | 85.1

Table 7: Accuracy of reconstruction with UDF under SRB. Figure 7: Visual
comparison under Thingi10k and D-FAUST.

Thingi10K. We follow the experimental setting in SAP [65] to evaluate GridPull
in Thingi10K. Tab. 8 indicates our superiority over the overfitting based
methods in the numerical comparison. Visual comparisons in Fig. 7 show our
more accurate surfaces with complex details.

|  | IGR [24] | SAP [65] | Ours
---|---|---|---|---
Thing10k | $CD_{L1}$ | 0.440 | 0.054 | 0.051
F-Score | 0.505 | 0.940 | 0.948
NC | 0.692 | 0.947 | 0.965
DFAUST | $CD_{L1}$ | 0.235 | 0.043 | 0.015
F-Score | 0.805 | 0.966 | 0.975
NC | 0.911 | 0.959 | 0.978

Table 8: Reconstruction accuracy under Thingi10k and D-FAUST in terms of
$CD_{L1}$ and F-Score with a threshold of 0.01.

D-FAUST. We also follow the experimental setting in SAP [65] to evaluate
GridPull in D-FAUST. Tab. 8 shows that we outperform the overfitting based
methods with more surface details. As shown in Fig. 7, we recover more
accurate human bodies and more detailed expressions on faces.

Figure 8: Visual comparison under 3DScene dataset.

###  4.2 Surface Reconstruction for Scenes

3DScene. We follow PCP [51] to report $CD_{L1}$, $CD_{L2}$ and Normal
Consistency(NC) for evaluation. We report the comparisons with the latest
methods in Tab. 9. We outperform both kinds of methods with learned priors
such as ConvOcc [73] and PCP [51] and overfitting based NP [49] in all scenes.
The visual comparisons 8 shows that our method can work well with real scans
on scenes and reveal more geometry details on surfaces.

|  | ConvOcc [73] | NP [49] | PCP[51] | Ours
---|---|---|---|---|---
Burghers | $CD_{L2}\times 100$ | 26.69 | 1.76 | 0.267 | 0.246
$CD_{L1}$ | 0.077 | 0.010 | 0.008 | 0.008
NC | 0.865 | 0.883 | 0.914 | 0.926
Lounge | $CD_{L2}\times 100$ | 8.68 | 39.71 | 0.061 | 0.055
$CD_{L1}$ | 0.042 | 0.059 | 0.006 | 0.005
NC | 0.857 | 0.857 | 0.928 | 0.922
Copyroom | $CD_{L2}\times 100$ | 10.99 | 0.51 | 0.076 | 0.069
$CD_{L1}$ | 0.045 | 0.011 | 0.007 | 0.006
NC | 0.848 | 0.884 | 0.918 | 0.929
Stonewall | $CD_{L2}\times 100$ | 19.12 | 0.063 | 0.061 | 0.058
$CD_{L1}$ | 0.066 | 0.007 | 0.0065 | 0.006
NC | 0.866 | 0.868 | 0.888 | 0.893
Totepole | $CD_{L2}\times 100$ | 1.16 | 0.19 | 0.10 | 0.093
$CD_{L1}$ | 0.016 | 0.010 | 0.008 | 0.007
NC | 0.925 | 0.765 | 0.784 | 0.847

Table 9: Reconstruction accuracy under 3DScene. Figure 9: Visual comparison
under SceneNet dataset.

SceneNet. Following the experimental setting in PCP [51], we report our
$CD_{L1}$, NC, and F-score in Tab. 10. The comparison indicates that we
outperform these methods by producing more accurate and smooth surfaces. This
is also verified by our visual comparison in Fig. 9, where the latest methods
reveal either no or inaccurate indoor geometry.

|  | NP [49] | PCP [51] | Ours
---|---|---|---|---
Livingroom | $CD_{L1}$ | 0.088 | 0.017 | 0.015
$NC$ | 0.881 | 0.933 | 0.922
FScore | 0.801 | 0.966 | 0.953
Bathroom | $CD_{L1}$ | 0.036 | 0.016 | 0.013
$NC$ | 0.912 | 0.945 | 0.952
FScore | 0.860 | 0.977 | 0.974
Bedroom | $CD_{L1}$ | 0.034 | 0.014 | 0.014
$NC$ | 0.905 | 0.948 | 0.951
FScore | 0.876 | 0.980 | 0.986
Kitchen | $CD_{L1}$ | 0.049 | 0.015 | 0.014
$NC$ | 0.900 | 0.945 | 0.952
FScore | 0.825 | 0.976 | 0.984
Office | $CD_{L1}$ | 0.062 | 0.024 | 0.022
$NC$ | 0.879 | 0.919 | 0.931
FScore | 0.729 | 925 | 0.940
Mean | $CD_{L1}$ | 0.054 | 0.017 | 0.0156
$NC$ | 0.895 | 0.938 | 0.9416
FScore | 0.818 | 965 | 0.9674

Table 10: Reconstruction accuracy under SceneNet. FScore is with a threshold
of 0.025. Figure 10: Visual comparison under 3Dfront dataset.

3D-FRONT.  We follow the experimental setting in NeuralPoisson [16] to report
$CD_{L1}$, IoU and L2 distance over the voxelization of meshes. IoU and L2
distances are calculated as the error between the reconstructed surfaces and
the ground truth over the voxelization at a resolution of 128. The comparison
in Tab. 11 shows that our results are more accurate than the latest method.
Our visual comparison with SIREN in Fig. 10 shows that GridPull can infer an
accurate zero level set which leads to very sharp corners.

Method | CD | IOU | L2
---|---|---|---
SIREN [71] | 0.0183 | 0.524 | 2.373
NeuralPoisson [16] | 0.0161 | 0.545 | 2.073
Ours | 0.0141 | 0.562 | 1.984

Table 11: Reconstruction accuracy under 3DFRONT. Figure 11: Visual comparison
under Matterport3D dataset.

Matterport. We follow the experimental setting in NGS [30], and report the
comparison in terms of $CD_{L1}$, NC, and F-score in Tab. 12. We achieve the
best performance in terms of accuracy. NGS [30] has a pretty fast inference
but it costs about four to five days to learn priors from the training set in
Matterport. Visual comparison in Fig. 11 shows that we can reconstruct more
accurate surfaces while NGS is poorly generalized to unseen scenes.

Figure 12: Visual comparison under Kitti dataset.

KITTI. We evaluate GridPull on a large scale real scan from KITTI odometry
(Sequence00, frame 3000 to 4000) which contains about $13.8$ million points.
The comparison in Fig. 12 shows that we can reconstruct a more accurate
surface in much less time ($3810$s) than the overfitting based method SAP
($13005$s) [65].

Method | $CD_{L1}\times 1000$ | F-Score(0.002) | NC | Time
---|---|---|---|---
ConvOcc [73] | 6.21 | 88.9 | 92.0 | 34s
SAP [65] | 4.17 | 91.2 | 91.4 | 643s
NGS [30] | 3.04 | 97.3 | 95.7 | 2.1s
Ours | 2.93 | 97.7 | 96.3 | 223s

Table 12: Reconstruction accuracy under Matterport3D.

###  4.3 Ablation Studies

We conduct ablation studies on FAMOUS [17] to justify the effectiveness of
modules in GridPull.

Loss. We justify the effectiveness of our loss in terms of $CD_{L2}$ in Tab.
13. For the pulling loss, it is able to estimate a coarse shape but with
discontinuous meshes and artifacts. The gradient loss can improve the
continuousness on the surface. Using the TV loss, we obtained a more
continuous field, which removes most of artifacts and significantly improve
the accuracy. The surface loss encourages a more accurate zero level-set,
which also improves the accuracy.

Loss | $CD_{L2}\times 100$
---|---
$L_{Pull}$ | 0.184
$L_{Pull}+L_{\nabla}$ | 0.126
$L_{Pull}+L_{\nabla}+L_{TV}$ | 0.056
$L_{Pull}+L_{\nabla}+L_{TV}+L_{Surface}$ | 0.040

Table 13: Effect of losses. Figure 13: Effect of Resolution $R$.

Resolution $R$. We compare the effect of resolution $R$ on the reconstructed
surfaces. We try several candidates $\\{32,64,128,256,512\\}$ to learn the
discrete distance field. We use the same bandwidth for the calculation of the
pulling loss and the TV loss, and reconstruct meshes by running the marching
cubes at the same resolution. The numerical and visual comparisons show that
higher resolutions can infer more geometry details but also cost more
inference time.

| 32 | 64 | 128 | 256 | 512
---|---|---|---|---|---
$CD_{L2}\times 100$ | 0.186 | 0.097 | 0.048 | 0.040 | 0.035
Time | 50s | 67s | 84s | 201s | 645s

Table 14: Effect of Resolution $R$.

Point Number. We compare our performance on different numbers of points. We
use the same set of shapes but with different numbers of points including
$\\{10K,100K,1000K\\}$. The numerical comparison in Tab. 15 shows that our
method can estimate a discrete distance field from different numbers of points
well, which leads to more accurate reconstructed surfaces than the latest
overfitting based methods. In addition, our time complexity is not affected by
the number of points a lot, while the latest methods require more time to
converge on more points.

Number | Method | $CD_{L2}\times 100$ | Time | Memory
---|---|---|---|---
10K | NP | 0.351 | 548s | 4.5G
SAP | 0.275 | 147s | 3.5G
Ours | 0.122 | 82s | 3.3G
100K | NP | 0.224 | 615s | 4.6G
SAP | 0.076 | 354s | 3.6G
Ours | 0.040 | 84s | 3.3G
1000K | NP | 0.175 | 1304s | 4.8G
SAP | 0.063 | 841s | 3.7G
Ours | 0.039 | 90s | 3.4G

Table 15: Effect of point numbers.

Initialization. We highlight our geometric initialization of distances on grid
vertices by comparing it with random initialization. The comparison in Tab. 16
shows that random initialization makes the optimization converge hard, and
cannot work well with the TV loss, especially with a bandwidth, which produces
lots of artifacts in empty spaces. Moreover, we try an Eikonal constraint on
gradients based on the geometric initialization, but did not get improvement.

| Random | Geometric | $||\nabla||=1$
---|---|---|---
$CD_{L2}\times 100$ | 0.155 | 0.040 | 0.054

Table 16: Effect of Initialization.

Bandwidth. We report the effect of bandwidth $M_{1}$ and $M_{2}$ in the
pulling loss and the TV loss in Tab. 17. For both losses, the bandwidth
controls the scope of query sampling and continuous constraints. A larger
bandwidth would cover more parameters which require more time and make the
optimization converge slowly. For instance, if we sample queries and impose
the pulling in all $R^{3}$ grids, the speed is even slower than NeuralPull
which uses neural network for the pulling, since neural networks can
generalize signed distances nearby and do not need to infer in all grids.
Although a small bandwidth would save time, a too small bandwidth would cover
too few grids around the surface, which leads to discontinuousness and
artifacts. While if the bandwidth is large enough, we may not see surface
improvement but just cost more time. Moreover, even with the same loss,
NeuralPull does not produce better results than ours. Comparing with methods
using neural networks, our discrete SDF can have faster convergence, as
indicated by the angular error of gradients in Fig. 14.

Bandwidth | $CD_{L2}\times 100$ | Time
---|---|---
$L_{Pull}$ ($M_{1}=3$) | 0.040 | 84s
$L_{Pull}$ (All Grids) | 0.043 | 628s
$L_{TV}$ ($M_{2}=2$) | 0.079 | 83s
$L_{TV}$ ($M_{2}=14$) | 0.040 | 84s
$L_{TV}$ ($M_{2}=30$) | 0.041 | 89s
$L_{TV}$ ($M_{2}=50$) | 0.040 | 96s
$L_{TV}$ (All Grids) | 0.040 | 101s
NeuralPull | 0.220 | 552s
NeuralPull+$L_{\nabla}$+$L_{Surface}$ | 0.203 | 554s

Table 17: Effect of Bandwidth. Figure 14: Gradients comparison with neural
networks.

Noises. We report our performance on point clouds with noise. We follow the
experimental setting in PCP, and report our results with middle and large
level noise in Tab. 18. To handle noise, we weight more on the continuous
constraint $L_{TV}$ using a larger $\alpha=2$ to decrease the impact of noise.
The comparison shows that our method can handle noise better than the neural
network based methods. Since the continuousness of neural networks make
gradients near the surface over smooth to overfit all noise, this makes it
hard to infer accurate zero level-set.

Noise level | NP [49] | PCP [51] | Ours
---|---|---|---
F-med-noise | 0.280 | 0.071 | 0.044
F-max-noise | 0.310 | 0.298 | 0.060

Table 18: Effect of noises.

Normals. We report our performance on point clouds with normals. If GT normal
is available, we could add a supervision on normals at points on the surface.
What we do is to supervise the gradients $\lambda f$ at points on the surface
using their ground truth normals. The comparisons in Tab. 19 show that the
normal supervision improves the performance a little. This indicates that our
method can infer pretty accurate gradients on the surface or near the surface,
where gradients on the surface can be highly accurate to the normal ground
truth on surfaces.

Class | Ours | Ours(Normals)
---|---|---
$CD_{L2}$ | 0.0401 | 0.0403

Table 19: Effect of normals.

##  5 Conclusion

We introduce GridPull for the fast inference of distance fields from large
scale point clouds without using neural components. We infer a distance field
using learnable parameters defined on discrete grids, which we can directly
optimize in a more efficient way than neural networks. Besides the complexity
advantage, our loss function manages to achieve more continuous distance
fields with more consistent gradients, which leads to higher accuracy in
surface reconstruction. We evaluate GridPull and justify its effectiveness on
benchmarks. Numerical and visual comparisons show that GridPull outperforms
the latest methods in terms of accuracy and complexity, even without using
priors, normal, and neural networks.

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# A remark on prime (non)congruent numbers

Tim Evink  Institute of Algebra and Number Theory, Ulm University,
Helmholtzstr. 18, 89081 Ulm, Germany<EMAIL_ADDRESS>,  Jaap Top
Bernoulli Institute, University of Groningen, Nijenborgh 9, 9747 AG Groningen,
the Netherlands<EMAIL_ADDRESS>and  Jakob Dirk Top  Bernoulli Institute,
University of Groningen, Nijenborgh 9, 9747 AG Groningen, the Netherlands.
<EMAIL_ADDRESS>
###### Abstract.

This paper discusses prime numbers that are (resp. are not) congruent numbers.
Particularly the only case not fully covered by earlier results, namely primes
of the form $p=8k+1$, receives attention.

##  1\. Introduction

A congruent number is, by definition, a positive integer $n$ that occurs as
the area of a rectangular triangle with rational side lengths. In other words,
nonzero $a,b,c\in\mathbb{Q}$ should exist such that $a^{2}+b^{2}=c^{2}$ and
$ab/2=n$. A classical, equivalent definition is that $n>0$ is congruent
precisely when an arithmetic progression $(x-n,x,x+n)$ consisting of three
rational squares exists. There is an abundance of literature on congruent
numbers, particularly because of their connection to the arithmetic of
elliptic curves and to modular forms. For example, the textbook [13] describes
many of the spectacular results on congruent numbers. Part of it is also
discussed in [29], and various centuries old results including some that
already appeared in Leonardo Pisano’s Liber Quadratorum published in 1225, can
be found in [8, Chapter XVI]. Many investigations regarding (non)congruent
numbers start with the standard and well-known observation
$(1)\Leftrightarrow(2)\Leftrightarrow(3)$, with

  1. (1)

$n$ is congruent;

  2. (2)

$E^{(n)}\colon y^{2}=x^{3}-n^{2}x$ contains a point $(a,b)\in
E^{(n)}(\mathbb{Q})$ with $b\neq 0$;

  3. (3)

$E^{(n)}\colon y^{2}=x^{3}-n^{2}x$ contains $P\in E^{(n)}(\mathbb{Q})$ of
infinite order.

In the present note we collect some information regarding prime numbers that
are congruent numbers or noncongruent numbers. Concretely:

  * •

We review known results, including detailed references to in some cases very
classical sources (the remainder of the present section).

  * •

We recall the equivalence of various conditions that imply a prime $p\equiv
1\bmod 8$ to be noncongruent and reformulate this in a way allowing one to
obtain a density result on the number of such primes (Proposition 2.1 and
Corollary 2.6).

  * •

By comparing a $2$-descent over $\mathbb{Q}$ with a $2$-descent over
$\mathbb{Q}(\sqrt{p})$ we obtain a new and relatively general approach towards
obtaining classical results on a class of primes $p\equiv 1\bmod 8$ not being
congruent (Propositions 2.3 and 2.5 as well as Corollary 2.6).

  * •

The only primes for which no unconditional general statement regarding
congruence is known to date, are the $p\equiv 1\bmod 8$ ones not in the class
referred to above. We present some data useful for predicting density
statements in this case (Section 3).

  * •

We show how a conjecture of Bouniakowsky implies the existence of infinitely
many primes $p\equiv 1\bmod 8$ that are congruent numbers. Moreover we
estimate the number of primes below any bound $b$ that our construction is
predicted to give (Proposition 3.1 and Remark 3.2).

Recognizing whether a prime is congruent or not, is also the topic of the 2006
master’s thesis [11]. The detailed explanation given there allows us to remain
brief on some issues here.

Historically, $p=2$ is the subject of [1, Prop. XII, pp. 114–116]: Peter
Barlow already in 1811 showed that $2$ is not congruent. As for the odd primes
$p$, probably the earliest result not restricted to a single prime is due to
A. Genocchi (1855, [10, pp. 314–315], see also T. Nagell’s 1929 expository
text [16, pp. 16–17]): no prime $p\equiv 3\bmod 8$ is congruent. The same
result with a totally different and much less elementary proof is presented in
[30, Prop. 5]. In contrast, every prime $p\equiv 5,7\bmod 8$ turns out to be
congruent. This is observed in a short paper by N.M. Stephens [23, bottom of
p. 183]; a detailed proof can be found in [15].

The situation for $p\equiv 1\bmod 8$ is less complete. L. Bastien [3] in 1915
announced that, writing $p=a^{2}+b^{2}$ as a sum of two squares, if the
Legendre symbol $\genfrac{(}{)}{}{}{a+b}{p}$ equals $-1$ then $p$ is
noncongruent. The same result can be deduced from a theorem by Michael J.
Razar (1974): although his condition on $p$ looks different at first sight and
the notion “congruent number” does not occur in the paper, from [18, Thm 2]
and the remarks on “first descent” contained in the text it is immediate that
if one of the following conditions holds:

  * (a)

$p\equiv 1\bmod 16$ and $2$ is not a $4$th power modulo $p$;

  * (b)

$p\equiv 9\bmod 16$ and $2$ is a $4$th power modulo $p$,

then $p$ is not a congruent number. Yet another way to formulate this result,
is presented in J.B. Tunnell’s 1983 paper [30, Prop. 6]: if the prime $p\equiv
1\bmod 8$ is written as $p=a^{2}+4b^{2}$ for integers $a,b$, then $16\nmid
p-1+4b\Rightarrow p\;\text{is not congruent}$. Tunnell’s proof is very
different from Razar’s. Again an alternative way to formulate and prove the
same result one finds in the 2006 master’s thesis of Brett Hemenway [11, p.
41]. He shows, writing $p\equiv 1\bmod 8$ as $p=a^{2}+b^{2}$ for integers
$a,b$, that if $(a+b)^{2}\equiv 9\bmod 16$ then $p$ is not congruent. His
proof (in fact similar to Razar’s but more detailed) consists of a $2$-descent
on the elliptic curve given by $y^{2}=x^{3}+4p^{2}x$; this curve is
$2$-isogenous to the one with equation $y^{2}=x^{3}-p^{2}x$. Although
involving different curves, this is very much in the spirit of [28]. The last
way we mention of formulating an equivalent result is a special case of [17,
Thm. 1.1(1)]: if for a prime $p\equiv 1\bmod 8$ the Legendre symbol
$\genfrac{(}{)}{}{}{1+i}{p}=-1$ (here $i\in\mathbb{F}_{p}$ is a square root of
$-1$), then $p$ is noncongruent.

In the next section we discuss the equivalence of the criteria above. In
particular this allows one to find the density of the involved set of primes.
We also sketch how, apart from the fact that all primes $\equiv 5,7\bmod p$
are congruent, the results mentioned here can be verified by comparing
$2$-descents over $\mathbb{Q}$ and over $\mathbb{Q}(\sqrt{p})$; this unified
approach, inspired by [9] and in particular by Propositions 5.5-5.6 in loc.
sit., is somewhat different from what one finds in earlier literature. The
last section contains results and some data regarding primes $p\equiv 1\bmod
8$ not satisfying the equivalent conditions described above.

##  2\. Equivalences and Selmer groups

Consider a prime number $p\equiv 1\bmod 8$. The congruence condition implies
that $-1$ is a fourth power modulo $p$ and that $2$ is a square modulo $p$.
Moreover, one can write $p=a^{2}+b^{2}$ for integers $a,b$ of which one is odd
and the other is divisible by $4$. The equivalences mentioned in the
introduction are included in the next result. It uses a certain Galois
extension $N/\mathbb{Q}$, namely the splitting field over $\mathbb{Q}$ of the
minimal polynomial $x^{4}-2x^{2}+2$ of $\alpha:=\sqrt{1+i}$; this minimal
polynomial has zeroes $\pm\alpha$ and $\pm\beta$ with $(\alpha\beta)^{2}=2$.
Hence $N=\mathbb{Q}(\sqrt{2},\alpha)$ and the Galois group
$\mathrm{Gal}(N/\mathbb{Q})$ is the dihedral group of order $8$, with
generators $\rho$ (of order $4$) defined by $\alpha\mapsto\beta\mapsto-\alpha$
and $\sigma$ (of order $2$) defined by $\alpha\mapsto\beta\mapsto\alpha$. Note
$\mathrm{Gal}(N/\mathbb{Q}(\sqrt{-2}))=\langle\rho\rangle$ (cyclic group of
order $4$; $(\alpha^{3}-\alpha)\beta$ is a square root of $-2$ and is fixed by
$\rho$). The element $\rho^{2}$ is the unique nontrivial element in the center
of $\mathrm{Gal}(N/\mathbb{Q})$; the field of invariants under $\rho^{2}$
equals $\mathbb{Q}(i,\sqrt{2})$ which is the $8$-th cyclotomic field. In
particular, $\rho^{2}$ being in the center implies that a Frobenius element as
appearing in part (7) of the next result, is well-defined (and not only
defined up to conjugacy).

######  Proposition 2.1.

For a prime $p\equiv 1\bmod 8$ the following statements are equivalent.

  1. (1.)

For $j\in\mathbb{Z}$ with $j^{2}\equiv-1\bmod p$ one has
$\genfrac{(}{)}{}{}{1+j}{p}=-1$;

  2. (2.)

For $r\in\mathbb{Z}$ with $r^{2}\equiv 2\bmod p$ one has
$\genfrac{(}{)}{}{}{1+r}{p}=-1$;

  3. (3.)

For $r\in\mathbb{Z}$ with $r^{2}\equiv 2\bmod p$ one has
$(-1)^{(p-1)/8}\cdot\genfrac{(}{)}{}{}{r}{p}=-1$;

  4. (4.)

Writing $p=a^{2}+b^{2}$ with $a,b\in\mathbb{Z}$, one has
$\genfrac{(}{)}{}{}{a+b}{p}=-1$;

  5. (5.)

Writing $p=a^{2}+b^{2}$ with $a,b\in\mathbb{Z}$, one has $(a+b)^{2}\equiv
9\bmod 16$;

  6. (6.)

Writing $p=a^{2}+4c^{2}$ with $a,c\in\mathbb{Z}$, one has $16\nmid p-1+4c$;

  7. (7.)

The (unique) Frobenius element in
$G:=\mathrm{Gal}(\mathbb{Q}(\sqrt{2},\sqrt{1+i})/\mathbb{Q})$ corresponding to
$p$ equals the generator of the center of $G$.

  8. (8.)

With $M$ the normal closure of $\mathbb{Q}(\sqrt{1+\sqrt{2}})$ over
$\mathbb{Q}$ which is a dihedral extension of degree $8$, the (unique)
Frobenius element in $\mathrm{Gal}(M/\mathbb{Q})$ corresponding to $p$ equals
the generator of the center.

###### Proof.

(1.)$\Leftrightarrow$(2.): this is also shown in [18, Lemma 2]. If
$i,r\in\mathbb{F}_{p}$ satisfy $i^{2}=-1$ and $r^{2}=2$, then one has
$(1+i)(1+r)=(1+\zeta)^{2}$ for $\zeta\in\mathbb{F}_{p}^{\times}$ of order $8$.
The result follows.
(1.)$\Leftrightarrow$(3.): with $\zeta\in\mathbb{F}_{p}$ as above, note that
$i:=\zeta^{2}$ satisfies $i^{2}=-1$ and $r:=\zeta\cdot(1-i)$ has the property
$r^{2}=2$. Therefore

$(-1)^{(p-1)/8}\cdot\genfrac{(}{)}{}{}{r}{p}=\genfrac{(}{)}{}{}{\zeta}{p}\cdot\genfrac{(}{)}{}{}{r}{p}=\genfrac{(}{)}{}{}{\zeta^{2}\cdot(1-i)}{p}=\genfrac{(}{)}{}{}{1+i}{p}.$

(1.)$\Leftrightarrow$(4.): this is also shown in [11, §5.4.1], for
completeness we recall the proof. In $p=a^{2}+b^{2}$ we can and will assume
$a$ is odd. Using the Jacobi symbol and in particular reciprocity for it, one
finds the property
$\genfrac{(}{)}{}{}{a}{p}=\genfrac{(}{)}{}{}{p}{a}=\genfrac{(}{)}{}{}{a^{2}+b^{2}}{a}=1$,
hence with $i:=(b\bmod p)/(a\bmod p)\in\mathbb{F}_{p}$ a primitive $4$th root
of unity
$\genfrac{(}{)}{}{}{a+b}{p}=\genfrac{(}{)}{}{}{a}{p}\cdot\genfrac{(}{)}{}{}{1+i}{p}=\genfrac{(}{)}{}{}{1+i}{p}$.
(4.)$\Leftrightarrow$(5.): this also follows using [11, §5.4.1], we repeat the
calculation.

$\genfrac{(}{)}{}{}{a\\!+\\!b}{p}=\genfrac{(}{)}{}{}{p}{a\\!+\\!b}=\genfrac{(}{)}{}{}{2}{a\\!+\\!b}\genfrac{(}{)}{}{}{(a\\!+\\!b)^{2}+(a\\!-\\!b)^{2}}{a+b}=(-1)^{\left((a+b)^{2}-1)\right)/8},$

from which the result is immediate.
(5.)$\Leftrightarrow$(6.): as before, write $p=a^{2}+b^{2}$ with $a$ odd. The
assumption $p\equiv 1\bmod 8$ implies $4|b$. With $b=2c=4d$ one obtains
congruences $(a+b)^{2}\equiv a^{2}+8ad\equiv a^{2}+8d\equiv p+4c\bmod 16$ and
the result follows.
(1.)$\Leftrightarrow$(7.): using the notations preceding the statement of
Prop. 2.1, one has
$\mathbb{Q}\subset\mathbb{Q}(i)\subset\mathbb{Q}(\zeta)\subset\mathbb{Q}(\zeta,\alpha)=N$
where $\zeta$ is a primitive $8$th root of unity, and each consecutive
extension has degree $2$. Note that $\mathbb{Q}(\zeta)$ is the field of
invariants of the generator $\rho^{2}$ of the center of
$\mathrm{Gal}(N/\mathbb{Q})$. The condition $p\equiv 1\bmod 8$ is equivalent
to $p$ splitting completely in $\mathbb{Q}(\zeta)$. Hence $p$-adically one
obtains $\mathbb{Q}_{p}(\zeta)=\mathbb{Q}_{p}$ with the extension obtained by
adjoining a square root of $1+i\in\mathbb{Q}_{p}$. This extension is
nontrivial precisely when a Frobenius at $p$ acts nontrivially on $N$, hence
as $\rho^{2}$. Since $1+i$ is a square in $\mathbb{Q}_{p}$ iff
$\genfrac{(}{)}{}{}{1+i}{p}=1$, this proves the result.
(2.)$\Leftrightarrow$(8.): the minimal polynomial of
$\gamma:=\sqrt{1+\sqrt{2}}$ over $\mathbb{Q}$ equals $x^{4}-2x^{2}-1$ with
zeroes $\pm\gamma$ and $\pm\delta$, where $(\gamma\delta)^{2}=-1$. The
splitting field $M$ fits in a tower
$\mathbb{Q}\subset\mathbb{Q}(i)\subset\mathbb{Q}(\zeta)\subset\mathbb{Q}(\zeta,\gamma)=M$
where $\zeta$ is a primitive $8$th root of unity and each consecutive
extension has degree $2$. The remainder of the argument is analogous to the
proof of (1.)$\Leftrightarrow$(7.), with now $1+\sqrt{2}$ taking the role of
$1+i$. ∎

######  Remark 2.2.

Note that the fields $N$ and $M$ appearing here are isomorphic; this provides
a direct proof of (7.)$\Leftrightarrow$(8.). This elementary observation is in
fact the special case $(Q,a,b,x,y,z)=(\mathbb{Q},-1,2,1,1,1)$ of [26, Cor.
5.2].

Several of the conditions appearing in Prop. 2.1 play a role in various papers
on $8$-rank of class groups. For example, [12, Thm. 2] states for primes
$p\equiv 1\bmod 8$ that the class number of $\mathbb{Q}(\sqrt{-p})$ is not
divisible by $8$ precisely when condition (5.) holds. This equivalence was
already observed by Gauss in a letter to Dirichlet dated 30 May 1828.
Similarly, reformulating [6, Thm. 1] yields that for primes $p\equiv 1\bmod 8$
the class number of $\mathbb{Q}(\sqrt{-p})$ is not divisible by $8$ precisely
when condition (6.) holds. Included in [2, Main Theorem] as well as in part of
[25, Thm 1], the same class number criterion is shown to be equivalent to
(1.). A reason for these relations with class numbers can be found as a
special case of [31, Thm. 1.1]. Still other equivalent criteria as well as
many of the ones above are listed in [14, Lemma-Def. 1] and [31, Thm 4.2]. As
an additional, related and very accessible text in the same spirit we refer to
[7].

Alternatively, the conditions (1.)–(8.) can be expressed in terms of a Rédei
symbol (see, e.g., [26, Section 6]). They are equivalent to the statement
$[-1,2,p]=-1$.

We now provide some details regarding the $2$-descent calculations for the
elliptic curves involved. For any prime number $p$, write

$E^{(p)}\colon y^{2}=x^{3}-p^{2}x.$

The curve $E^{(p)}/\mathbb{Q}$ has good reduction away from $\\{2,p\\}$. With
notations as in, e.g., [21, Ch. X §1] take $S=\\{2,p,\infty\\}$ and denote
$\mathbb{Q}(S)=\mathbb{Q}(S,2):=\\{x\in\mathbb{Q}^{*}/{\mathbb{Q}^{*}}^{2}\;:\;v(x)\equiv
0\bmod 2\;\text{for all}\;v\not\in S\\}$ which is an elementary $2$-group
generated by the classes of $-1,2$, and $p$. With

$H:=\left\\{(c_{1},c_{2},c_{3})\in\mathbb{Q}(S)\times\mathbb{Q}(S)\times\mathbb{Q}(S)\;:\;c_{1}c_{2}c_{3}=1\right\\},$

one has the Kummer homomorphism $\delta\colon E^{(p)}(\mathbb{Q})\to H$ with
kernel $2E^{(p)}(\mathbb{Q})$ given by

$\delta(a,b)=\left\\{\begin{array}[]{ll}(a+p,a,a-p)&\text{for }b\neq 0;\\\
(2,-p,-2p)&\text{for }a=-p;\\\ (p,-1,-p)&\text{for }a=0;\\\ (2p,p,2)&\text{for
}a=p.\end{array}\right.$

One has local versions $\delta_{v}\colon E^{(p)}(\mathbb{Q}_{v})\to H_{v}$ of
this, obtained by replacing $\mathbb{Q}$ by the completion $\mathbb{Q}_{v}$.
The inclusion $\mathbb{Q}\subset\mathbb{Q}_{v}$ induces maps
$E^{(p)}(\mathbb{Q})/2E^{(p)}(\mathbb{Q})\to
E^{(p)}(\mathbb{Q}_{v})/2E^{(p)}(\mathbb{Q}_{v})$ as well as
$\mathbb{Q}(S)\to\mathbb{Q}^{*}_{v}/{\mathbb{Q}_{v}^{*}}^{2}$ and $H\to
H_{v}$; they will all be denoted as $\iota_{v}$. The $2$-Selmer group
$S^{2}(E^{(p)}/\mathbb{Q})\subset H$ is given by

$S^{2}(E^{(p)}/\mathbb{Q}):=\left\\{c=(c_{1},c_{2},c_{3})\in
H\;:\;\iota_{v}(c)\in\delta_{v}(E^{(p)}(\mathbb{Q}_{v}))\,\forall\,v\in
S\right\\}.$

It fits in a short exact sequence

$0\to
E^{(p)}(\mathbb{Q})/2E^{(p)}(\mathbb{Q})\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}S^{2}(E^{(p)}/\mathbb{Q})\longrightarrow\Sh(E^{(p)}/\mathbb{Q})[2]\to
0$

where $\Sh(E^{(p)}/\mathbb{Q})[2]$ is the $2$-torsion in the Shafarevich-Tate
group of $E^{(p)}/\mathbb{Q}$. To ease notation, for fields $F$ an element
$c{F^{*}}^{2}\in F^{*}/{F^{*}}^{2}$ will simply be denoted $c$. A short
calculation shows:

  * •

$\delta_{\infty}(E^{(p)}(\mathbb{R}))$ is generated by $(1,-1,-1)$;

  * •

$\delta_{p}(E^{(p)}(\mathbb{Q}_{p}))$ is generated by $(2,-p,-2p),\;(p,-1,-p)$
for $p\neq 2$;

  * •

$\delta_{2}(E^{(p)}(\mathbb{Q}_{2}))$ has generators
$(2,-p,-2p),\;(p,-1,-p),\;(5,1,5)$ for $p\neq 2$. Here, apart from $2$-torsion
one may use the point $(\frac{1}{4},\frac{1}{8}\sqrt{1-16p^{2}})\in
E^{(p)}(\mathbb{Q}_{2})$.

An immediate consequence is the following.

######  Proposition 2.3.

$\dim_{\mathbb{F}_{2}}\,S^{2}(E^{(p)}/\mathbb{Q})=\left\\{\begin{array}[]{ll}4&\text{if
}\;p\equiv 1\bmod 8;\\\ 2&\text{if }\;p\equiv 3\bmod 8;\\\ 3&\text{if
}\;p\equiv 5,7\bmod 8.\end{array}\right.$

###### Proof.

We only sketch the case $p\equiv 1\bmod 8$, the other cases are analogous.
Since $p\equiv 1\bmod 8$, one has that $-1,2$ are squares in $\mathbb{Q}_{p}$
hence the $\delta_{p}$-image is generated by $(1,p,p)$ and $(p,1,p)$.
Similarly, the $\delta_{2}$-image has generators
$(2,-1,-2),\;(1,-1,-1),\;(5,1,5)$. Considering the $\delta_{\infty}$-image one
observes that the first coordinate of any $c\in S^{2}(E^{(p)}/\mathbb{Q})$ is
one of $\\{1,2,p,2p\\}$. Now consider the possibilities:
if $c=(1,a,a)$, then the local conditions at $2,p$ imply $a\in\\{\pm 1,\pm
p\\}$, giving $4$ elements. For $c=(2,a,2a)$ again one finds $a\in\\{\pm 1,\pm
p\\}$, and the same conclusion holds for $c=(p,a,ap)$ and $c=(2p,a,2ap)$. ∎

Using Monsky’s result (predicted by Stephens) asserting that primes $p\equiv
5,7\bmod 8$ are congruent, the following is a consequence.

######  Corollary 2.4.

For any odd prime $p\not\equiv 1\bmod 8$ one has

$\Sh(E^{(p)}/\mathbb{Q})[2]=(0)\;\;\textit{and}\;\;\textit{rank}\,E^{(p)}(\mathbb{Q})=\left\\{\begin{array}[]{ll}0&\textit{if
}\;p\equiv 3\bmod 8,\\\ 1&\textit{if }\;p\equiv 5,7\bmod 8.\end{array}\right.$

###### Proof.

Since
$\text{rank}\,E^{(p)}(\mathbb{Q})=\dim_{\mathbb{F}_{2}}E^{(p)}(\mathbb{Q})/2E^{(p)}(\mathbb{Q})-2$,
the exact sequence for $S^{2}$ implies

$\text{rank}\,E^{(p)}(\mathbb{Q})+\dim_{\mathbb{F}_{2}}\Sh(E^{(p)}/\mathbb{Q})[2]=\dim_{\mathbb{F}_{2}}\,S^{2}(E^{(p)}/\mathbb{Q})-2.$

Together with Proposition 2.3 this implies the corollary for the primes
$p\equiv 3\bmod 8$. Since Monsky’s result shows that $E^{(p)}/\mathbb{Q}$ has
positive rank for $p\equiv 5,7\bmod 8$, the corollary also follows in those
cases. ∎

In the remainder of this section let $K=\mathbb{Q}(\sqrt{p})$. Over $K$ the
curve $E^{(p)}$ is isomorphic to

$E\colon y^{2}=x^{3}-x.$

Moreover
$\text{rank}\,E(K)=\text{rank}\,E(\mathbb{Q})+\text{rank}\,E^{(p)}(\mathbb{Q})$,
corresponding to the decomposition of $E(K)\otimes\mathbb{Q}$ into eigenspaces
for $\mathrm{Gal}(K/\mathbb{Q})$. It is well known that
$\text{rank}\,E(\mathbb{Q})=0$ (equivalently, $1$ is not a congruent number, a
fact already stated by Fibonacci in 1225 and a consequence of a result of
Fermat; a detailed proof is also presented in [1]). Hence

$\text{rank}\,E^{(p)}(\mathbb{Q})=\text{rank}\,E(K).$

In some cases $p\equiv 1\bmod 8$, the bound
$\text{rank}\,E(K)\leq\dim_{\mathbb{F}_{2}}S^{2}(E/K)-2$ involving the
$2$-Selmer group of $E$ over $K=\mathbb{Q}(\sqrt{p})$ gives a stronger bound
for $\text{rank}\,E^{(p)}(\mathbb{Q})$ than the one using
$S^{2}(E^{(p)}/\mathbb{Q})$. In such a case it follows that
$\Sh(E^{(p)}/\mathbb{Q})[2]$ is nontrivial.

######  Proposition 2.5.

Let $p\equiv 1\bmod 8$ be prime, and let $j\in\mathbb{Z}$ satisfy
$j^{2}\equiv-1\bmod p$. Then
$\dim_{\mathbb{F}_{2}}S^{2}(E/\mathbb{Q}(\sqrt{p}))=\left\\{\begin{array}[]{ll}4&\textit{if}\;\genfrac{(}{)}{}{}{1+j}{p}=1,\\\
2&\textit{if}\;\genfrac{(}{)}{}{}{1+j}{p}=-1.\end{array}\right.$

The result already announced in Bastien’s 1915 paper is by the method sketched
above a consequence of this:

######  Corollary 2.6.

Let $p\equiv 1\bmod 8$ be a prime satisfying the equivalent conditions given
in Proposition 2.1. Then $\textit{rank}\,E^{(p)}(\mathbb{Q})=0$ and
$\Sh(E^{(p)}/\mathbb{Q})[2]\cong\left(\mathbb{Z}/2\mathbb{Z}\right)^{2}$. In
particular, $p$ is not congruent.

The set of primes discussed here has natural density $1/8$.

###### Proof.

Most of this is immediate from Propositions 2.3 and 2.5 by the argument
sketched above. The assertion about the density is obtained by applying
Chebotarëv’s density theorem (see, e.g. [24]) in the situation of Proposition
2.1(7). ∎

###### Proof.

(of Proposition 2.5). The elliptic curve $E$ has good reduction away from $2$,
so we take $S=\\{\mathfrak{p}_{2},\mathfrak{q}_{2},\infty_{1},\infty_{2}\\}$,
with $\mathfrak{p}_{2},\mathfrak{q}_{2}$ the primes over $2$ in
$K=\mathbb{Q}(\sqrt{p})$ and $\infty_{1},\infty_{2}$ the two real embeddings.
As with $E^{(p)}/\mathbb{Q}$, set $K(S)=\\{x\in K^{*}/K^{*2}:v(x)\equiv 0\bmod
2\text{ for all }v\notin S\\}$ and $H=\\{(c_{1},c_{2},c_{3}):K(S)\times
K(S)\times K(S):c_{1}c_{2}c_{3}=1\\}$. The Kummer homomorphism $\delta\colon
E(K)\to H$ is then given by

$\delta(a,b)=\left\\{\begin{array}[]{ll}(a+1,a,a-1)&\text{for }b\neq 0;\\\
(2,-1,-2)&\text{for }a=-1;\\\ (1,-1,-1)&\text{for }a=0;\\\ (2,1,2)&\text{for
}a=1.\end{array}\right.$

Regarding the local images $\delta_{v}(E(K_{v}))$, note that the completion
$K_{v}$ equals $\mathbb{Q}_{2}$ for
$v\in\\{\mathfrak{p}_{2},\mathfrak{q}_{2}\\}$ and $K_{v}=\mathbb{R}$ for
$v\in\\{\infty_{1},\infty_{2}\\}$. Observe:

  * •

$\delta_{v}(E(\mathbb{R}))$ is generated by $(1,-1,-1)$ for
$v\in\\{\infty_{1},\infty_{2}\\}$,

  * •

$\delta_{v}(E(\mathbb{Q}_{2}))$ has generators
$(2,-1,-2),\;(1,-1,-1),\;(-3,1,-3)$ for
$v\in\\{\mathfrak{p}_{2},\mathfrak{q}_{2}\\}$.

The group $K(S)$ fits in an exact sequence

$0\to R_{S}^{*}/R_{S}^{*2}\to K(S)\to\mathrm{Cl}(R_{S})[2]\to 0,$

where $R_{S}$ is the ring of $S$-integers of $K$. We have
$\mathrm{Cl}(R_{S})[2]=0$ since $\mathrm{Cl}(R_{S})$ is a quotient of
$\mathrm{Cl}_{K}$ and $K$ has odd class number. As $R_{S}^{*}$ has rank $3$
and torsion subgroup $\\{\pm 1\\}$ it follows that $K(S)$ has
$\mathbb{F}_{2}$-dimension $4$. We now describe a convenient basis of $K(S)$.

Given $c\in K(S)$ and $v\in S$, write $\mathrm{im}_{v}(c)$ for the image of
$c$ under $K(S)\to K_{v}^{*}/K_{v}^{*2}$. Note that a fundamental unit of $K$
has negative norm (the narrow Hilbert class field of $K$ has odd degree).
Choose a fundamental unit $\varepsilon\in\mathcal{O}_{K}^{*}$ with
$\infty_{1}(\varepsilon)>0$, so $\infty_{2}(\varepsilon)<0$. Since
$\varepsilon\overline{\varepsilon}=-1$,
$\mathrm{im}_{\mathfrak{p}}(\varepsilon)=\mathrm{im}_{\mathfrak{q}}(\overline{\varepsilon})$
for $\mathfrak{p}$ and $\mathfrak{q}$ conjugate, and
$\mathrm{im}_{\mathfrak{p}_{2}}(\varepsilon)\subset\langle-1,3\rangle\subset\mathbb{Q}_{2}^{*}/\mathbb{Q}_{2}^{*2}$,
precisely one of $\mathfrak{p}_{2},\,\mathfrak{q}_{2}$ is unramified in
$K(\sqrt{\varepsilon})$. By interchanging the two if necessary we may and will
assume that $\mathfrak{p}_{2}$ is unramified in $K(\sqrt{\varepsilon})$.
Equivalently, this means
$\mathrm{im}_{\mathfrak{p}_{2}}(\epsilon)\in\\{1,-3\\}$, and of course
$\mathrm{im}_{\mathfrak{q}_{2}}(\epsilon)=-\mathrm{im}_{\mathfrak{p}_{2}}(\epsilon)$.
In terms of a Rédei symbol as in [26], in fact
$\mathrm{im}_{\mathfrak{p}_{2}}(\epsilon)=1\Leftrightarrow[p,-1,2]=1$.
Take $k$ any odd multiple of the order of
$[\mathfrak{p}_{2}]\in\mathrm{Cl}_{K}$ and write
$\mathfrak{p}_{2}^{k}=(x_{2})$ for some $x_{2}\in K$. Multiplying $x_{2}$ by
$\pm\varepsilon$ if necessary we may assume $x_{2}$ has positive norm and that
$\mathfrak{q}_{2}$ is unramified in $K(\sqrt{x_{2}})$, so
$x_{2}\cdot\overline{x_{2}}=2^{k}$ and
$\mathrm{im}_{\mathfrak{q}_{2}}(x_{2})\in\\{1,-3\\}$. These choices yield
$K(S)=\langle-1,\varepsilon,x_{2},y_{2}\rangle$, with
$y_{2}=\overline{x_{2}}$. Moreover
$\mathrm{im}_{\infty_{m}}(x_{2})=\mathrm{im}_{\infty_{m}}(y_{2})$ is
independent of $m\in\\{1,2\\}$ and it equals the Rédei symbol $[p,2,-1]$.
Rédei reciprocity [26, Thm. 1.1] implies in particular $[p,2,-1]=[p,-1,2]$, so
$\mathrm{im}_{\infty_{1}}(x_{2})=1\Leftrightarrow\mathrm{im}_{\mathfrak{p}_{2}}(\epsilon)=1$.
Again using Rédei reciprocity one has, for $j\in\mathbb{Z}$ such that
$j^{2}\equiv-1\bmod p$, the equality
$[p,2,-1]=[2,-1,p]=\genfrac{(}{)}{}{}{1+j}{p}$ (note that this is exactly the
example discussed in [26, Section 6]). For $c$ any of these generators and
$v\in S$, this means that the following tables give $\mathrm{im}_{v}(c)\in
K_{v}^{*}/K_{v}^{*2}$.

Case $\genfrac{(}{)}{}{}{1+j}{p}=1$ (with $j\in\mathbb{Z}$ such that
$j^{2}\equiv-1\bmod p$):

$\begin{array}[]{c|c c c
c}&\mathfrak{p}_{2}&\mathfrak{q}_{2}&\infty_{1}&\infty_{2}\\\
\hline\cr-1&-1&-1&-1&-1\\\
\varepsilon&\pagecolor[gray]{0.8}1&\pagecolor[gray]{0.8}-1&1&-1\\\
x_{2}&\pagecolor[gray]{0.65}2a&\pagecolor[gray]{0.65}a&\pagecolor[gray]{0.8}1&\pagecolor[gray]{0.8}1\\\
y_{2}&\pagecolor[gray]{0.65}a&\pagecolor[gray]{0.65}2a&\pagecolor[gray]{0.8}1&\pagecolor[gray]{0.8}1\\\
\end{array}$

Here
$a=\mathrm{im}_{\mathfrak{q}_{2}}(x_{2})\in\langle-3\rangle\subset\mathbb{Q}_{2}^{*}/\mathbb{Q}_{2}^{*2}$.

To compute $S^{2}(E/K)=\\{c\in
H:\iota_{v}(c)\in\delta_{v}(E(K_{v}))\;\forall\,v\in S\\}$ in this situation,
write $S^{2}(E/K)=\delta(E(K)[2])\oplus A$ with

$A=\\{c\in
S^{2}(E/K)\;:\;\iota_{\mathfrak{p}_{2}}(c)\in\langle(-3,1,-3)\rangle\\}.$

Let $c=(c_{1},c_{2},c_{3})\in A$. As $c_{1}$ is totally positive it follows
that $c_{1}\in\langle x_{2},y_{2}\rangle$. Considering both possibilities for
$\mathrm{im}_{\mathfrak{p}_{2}}(x_{2})$ one finds that
$\mathrm{im}_{\mathfrak{p}_{2}}(c_{1})\subset\langle-3\rangle$ implies
$c_{1}\in\langle y_{2}\rangle$. The $\mathfrak{p}_{2}$-adic and
$\mathfrak{q}_{2}$-adic images of $c_{2}$ result in
$c_{2}\in\langle\varepsilon\rangle$. Hence $A$ consists of at most four
elements. One checks that $(y_{2},1,y_{2}),(1,\varepsilon,\varepsilon)\in A$
regardless of $a=\mathrm{im}_{\mathfrak{q}_{2}}(x_{2})\in\\{1,-3\\}$. As a
consequence $\dim_{\mathbb{F}_{2}}A=2$ and therefore
$\dim_{\mathbb{F}_{2}}S^{2}(E/K)=4$ in this situation.

The other case is $\genfrac{(}{)}{}{}{1+j}{p}=-1$ (again $j^{2}\equiv-1\bmod
p$). Here one has

$\begin{array}[]{c|c c c
c}&\mathfrak{p}_{2}&\mathfrak{q}_{2}&\infty_{1}&\infty_{2}\\\
\hline\cr-1&-1&-1&-1&-1\\\
\varepsilon&\pagecolor[gray]{0.8}-3&\pagecolor[gray]{0.8}3&1&-1\\\
x_{2}&\pagecolor[gray]{0.65}2a&\pagecolor[gray]{0.65}a&\pagecolor[gray]{0.8}-1&\pagecolor[gray]{0.8}-1\\\
y_{2}&\pagecolor[gray]{0.65}a&\pagecolor[gray]{0.65}2a&\pagecolor[gray]{0.8}-1&\pagecolor[gray]{0.8}-1\\\
\end{array}$

with
$a=\mathrm{im}_{\mathfrak{q}_{2}}(x_{2})\in\langle-3\rangle\subset\mathbb{Q}_{2}^{*}/\mathbb{Q}_{2}^{*2}$.
As before, one uses the decomposition $S^{2}(E/K)=\delta(E(K)[2])\oplus A$ in
which one takes the subgroup $A=\\{c\in
S^{2}(E/K)\;:\;\iota_{\mathfrak{p}_{2}}(c)\in\langle(-3,1,-3)\rangle\\}$.

Let $c=(c_{1},c_{2},c_{3})\in A$. Considering the $\mathfrak{p}_{2}$-adic
image shows that either $c_{2}=y_{2}$ and $a=1$ which is inconsistent with the
possibilities for $\iota_{\mathfrak{q}_{2}}(c)$, or $c_{2}=\varepsilon y_{2}$
and $a=-3$ (again, inconsistent with $\iota_{\mathfrak{q}_{2}}(c)$), or
$c_{2}=1$. Since $c_{1}$ is totally positive, one has
$c_{1}\in\\{1,-x_{2},-y_{2},x_{2}y_{2}\\}$. The possibility $c_{1}=-x_{2}$ is
excluded by considering the $\mathfrak{p}_{2}$-image, and in the same way
$c_{1}=-y_{2}$ is impossible because of the $\mathfrak{q}_{2}$-image. Since
also $c_{1}=x_{2}y_{2}$ is not compatible with $c\in A$, it follows that
$c_{1}=1$ and therefore $A=(0)$, so $\dim_{\mathbb{F}_{2}}S^{2}(E/K)=2$,
completing the proof. ∎

##  3\. Examples

A remarkable result by J.B. Tunnell [30] (here restricted to odd integers $n$)
states, in a version proposed by N.D. Elkies, that

$\begin{array}[]{c}n\;\textit{is congruent}\\\ \Downarrow\\\
2x^{2}+y^{2}+8z^{2}=n\;\textit{has as many solutions
in}\;\mathbb{Z}^{3}\;\textit{with}\;2|z\;\textit{as with}\;2\nmid
z.\end{array}$

This gives an easy method for showing that certain integers are not congruent.
Applying this to the primes $p\equiv 1\bmod 8$ below some bound $b$ such that
$p$ does not satisfy the conditions given in Proposition 2.1 one obtains (we
used Magma [4] for this and further calculations) the following table.

$\begin{array}[]{rcccccc}b:&2000&4000&6000&8000&10000&12000\\\
\pi(b):&303&550&783&1007&1229&1438\\\ \\#p\equiv 1\bmod
8\;\&\;\genfrac{(}{)}{}{}{1+i}{p}=1:&30&62&93&116&146&172\\\
\\#p\text{noncongruent:}&19&41&62&78&99&122\end{array}$

As a consequence of [31, Thm. 1.1] the primes $p$ discussed here (i.e.,
$p\equiv 1\bmod 8$ and $\genfrac{(}{)}{}{}{1+i}{p}=1$ and $p$ is noncongruent)
have the property
$\Sh(E^{(p)}/\mathbb{Q})[2^{\infty}]\supsetneq\left(\mathbb{Z}/2\mathbb{Z}\right)^{2}$.
In particular this implies that $\Sh(E^{(p)}/\mathbb{Q})$ contains elements of
order $4$. For the primes contributing to the table (so $p<12000$, the set
looks like $\\{113,\,337,\,409,\ldots\ldots,\,11897,\,11969\\}$) a simple
Magma test confirms this as well:

      E:=EllipticCurve([-p^2,0]);
      MordellWeilShaInformation(E : ShaInfo);

The table suggests that at least $1/12$-th of the set of all primes has the
property described here. If the converse of Tunnell’s theorem holds (as would
be a corollary of the Birch and Swinnerton-Dyer conjecture), this would mean
that at most $1/6$-th of the set of primes $p\equiv 1\bmod 8$ is congruent. A
Magma test yields the following data.

$\begin{array}[]{rccccccc}\text{bound}\;b:&2000&4000&6000&8000&10000&12000&14000\\\
\\#\leq b,\;\equiv 1\bmod 8:&68&129&186&243&295&341&400\\\
\\#p\;\text{congruent}:&11&21&31&38&47&50&58\end{array}$

The $58$ primes
$\\{41,\,137,\,257,\,\ldots\ldots,9377,\ldots\ldots,13513,13841,13921\\}$ here
all result in $\text{rank}E^{(p)}(\mathbb{Q})=2$ and hence
$\Sh(E^{(p)}/\mathbb{Q})[2]=(0)$. Indeed, Magma quickly finds two independent
points except in one case: for $p=9377$ a point of infinite order with
$x=-\frac{6635776}{4225}$ is easily found. To obtain a second, independent
rational point one needs to increase the Effort parameter in
MordellWeilShaInformation, resulting in the points with

$x=\frac{9377\cdot(46111436236957655053256122338300576234143337)^{2}}{28100967414057580762568605652421621908428616^{2}}.$

Not surprisingly, our data is consistent with the list of congruent numbers
$n<10000$ available from [22]. It is conceivable that using e.g. higher
descents, a density result for primes $p\equiv 1\bmod 8$ satisfying
$(\mathbb{Z}/4\mathbb{Z})^{2}\subset\Sh(E^{(p)}/\mathbb{Q})$ may be obtained.

In contrast, the question whether or not infinitely many primes $p\equiv
1\bmod 8$ are congruent numbers seems much harder. We finish this paper by
showing that a special case of Bouniakowsky’s conjecture implies the existence
of infinitely many such primes. Recall that Bouniakowsky’s conjecture from
1854 (see [5, p. 328]) states that if $g(x)\in\mathbb{Z}[x]$ is irreducible,
has leading coefficient $>0$, and the gcd of all values
$\\{g(m)\;:\;m\in\mathbb{Z}\\}$ equals $1$, then $g(m)$ is claimed to be a
prime number for infinitely many $m\in\mathbb{Z}$. Note that this is a very
early special case of Schinzel’s Hypothesis H formulated in [19, p. 188].

######  Proposition 3.1.

The polynomial $f(x):=8x^{4}+16x^{3}+12x^{2}+4x+1$ satisfies the conditions in
Bouniakowsky’s conjecture, so $f(\mathbb{Z})$ is supposed to contain
infinitely many prime numbers.

If $n\in f(\mathbb{Z})\setminus\\{1\\}$, then $n\equiv 1\bmod 8$ and $n$ is a
congruent number.

###### Proof.

It is not difficult to verify that $f(x)\in\mathbb{Z}[x]$ is irreducible (in
fact, up to a a factor $2$ it equals the $8$-th cyclotomic polynomial
evaluated in $2x+1$). Clearly the leading coefficient is positive, and since
$f(0)=1$ the gcd of the values $f(k)$ equals $1$. This shows the first part of
the proposition.

For integers $k$, note that $f(k)\equiv 4(k^{2}+k)+1\bmod 8=1\bmod 8$. Also,
$2f(k)=(2k+1)^{4}+1$ which implies $f(k)\geq 1$ whenever $k\in\mathbb{Z}$ and
$f(k)=1\Leftrightarrow k\in\\{-1,0\\}$ (assuming $k$ is real). Hence to
complete the proof, take $n=f(k)$ with $k\in\mathbb{Z}\setminus\\{-1,0\\}$, so
that $n\geq f(1)=41$ (because $f(x)=f(-x-1)$ and $x\mapsto f(x)$ increases for
$x>-\frac{1}{2}$). Then

$\begin{array}[]{l}a:=2n\cdot(2k+1)^{2},\\\
b:=16\cdot(k^{2}+k)^{2}(2k^{2}+2k+1)^{2}=n^{2}-(2k+1)^{4},\\\
c:=n^{2}+(2k+1)^{4}\end{array}$

are positive integers satisfying $a^{2}+b^{2}=c^{2}$, hence they arise as
lengths of a right-angled triangle with area $n$ times a square. This implies
that $n$ is a congruent number. ∎

######  Remark 3.2.

Of the $58$ prime numbers $p\equiv 1\bmod 8$ with $p<14000$ that are
congruent, $4$ are in $f(\mathbb{Z})$. Note that $f(\mathbb{Z})$ contains only
$5$ integers in the interval $[2,14000]$. A special case of a conjecture of
P.T. Bateman and R.A. Horn predicts the number of integers $1\leq k\leq b$
such that $f(k)$ is prime: it should grow $\sim\frac{C(f)}{4}b/\ln(b)$ with
$C(f)=\prod_{p}\frac{p-\omega(p)}{p-1}$ and $\omega(p)=4$ for $p\equiv 1\bmod
8$ while $\omega(p)=0$ otherwise. An approximation $C(f)\approx 5.358$ one
obtains using [20].

Since
$f(\frac{u}{v})=\left((2\frac{u}{v}+1)^{4}+1\right)/2=\frac{1}{v^{4}}\left((2u+v)^{4}+v^{4}\right)/2$,
the proof of Proposition 3.1 shows that integers represented by the binary
form

$\left((2u+v)^{4}+v^{4}\right)/2=8u^{4}+16u^{3}v+12u^{2}v^{2}+4uv^{3}+v^{4}$

(except for $u=0$ and for $u=-v$) are congruent numbers. Such numbers are
$\equiv m^{4}\bmod 8$, so $\equiv 1\bmod 8$ whenever $m$ is odd. Below the
bound $14000$ this yields $9$ prime congruent numbers, namely

$41,\,313,\,353,\,1201,\,3593,\,4481,\,7321,\,8521,\,10601.$

Applying [27, Thm. 1] to the binary form given here yields a constant $C>0$
such that up to any bound $B$ it attains at least $C\sqrt{B}$ square-free
values; by construction these are congruent numbers $\equiv 1\bmod 8$ since
for $v$ even, the number would be divisible by $8$ hence it would not be
square-free. It seems that this is slightly stronger than what would be
obtained starting from $(2uv)^{2}+(u^{2}-v^{2})^{2}=(u^{2}+v^{2})^{2}$, i.e.,
from the binary form $uv(u+v)(u-v)$.

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