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#  Slopes of modular forms and reducible Galois representations
An oversight in the ghost conjecture

John Bergdall and Robert Pollack  John Bergdall
Bryn Mawr College
Bryn Mawr, PA 19010
USA<EMAIL_ADDRESS>http://jbergdall.digital.brynmawr.edu Robert
Pollack
Department of Mathematics and Statistics
Boston University
111 Cummington Mall
Boston, MA 02215
USA<EMAIL_ADDRESS>http://math.bu.edu/people/rpollack

###### Abstract.

The ghost conjecture, formulated by this article’s authors, predicts the list
of $p$-adic valuations of the non-zero $a_{p}$-eigenvalues (“slopes”) for
overconvergent $p$-adic modular eigenforms in terms of the Newton polygon of
an easy-to-describe power series (the “ghost series”). The prediction is
restricted to eigenforms whose Galois representation modulo $p$ is reducible
on a decomposition group at $p$. It has been discovered, however, that the
conjecture is not formulated correctly. Here we explain the issue and propose
a salvage.

###### 2000 Mathematics Subject Classification:

11F33 (11F85)

##  1\. Introduction

Let $p$ be a prime number. The authors previously made a conjecture on the
$p$-adic slopes of modular eigenforms with a fixed Galois representation
modulo $p$ subject to that representation being locally reducible at $p$. See
Conjecture 1.1 below and also [6, Conjecture 7.1]. The local reducibility
condition has turned out to be too strong — one reducible situation (and its
twists) must also be excluded. This article will explain the exclusion and how
to salvage the conjecture. We provide both computational and theoretical
evidence for the corrected conjecture.

It is important to correct false conjectures. In this case, however, it is
specifically important because there is ongoing progress toward a proof of the
conjecture, for Galois representations that are locally reducible with
sufficiently generic Serre weights, by Liu, Truong, Xiao, and Zhao. A paper
explaining their strategy is available as a preprint [19]. The proof itself
has also been announced [18].

A general proof, however, will remain out of reach until faulty hypotheses are
removed! Addressing the prior misconception further allows us the opportunity
to address, in writing, the special case of eigenforms that are twists of
$E_{2}$ modulo $p$. We had previously not tested that case out of convenience.

In [6], the authors called the reducibility assumption “Buzzard regular”,
based on Buzzard’s work [11]. It is unfortunate that Buzzard’s name was
attached, by us, to the faulty condition. Our strongest apologies.

###  1.1. Recollection of the ghost conjecture

Fix algebraic closures $\overline{{{\mathbb{Q}}}}$ of ${\mathbb{Q}}$,
$\overline{{{\mathbb{Q}}}}_{p}$ of ${\mathbb{Q}}_{p}$, and
$\overline{{\mathbb{F}}}_{p}$ of ${\mathbb{F}}_{p}$. Let
$G_{{\mathbb{Q}}}=\operatorname{Gal}(\overline{{{\mathbb{Q}}}}/{\mathbb{Q}})$
and $N\geq 1$ be an integer that is co-prime to $p$. We then consider
continuous, semi-simple, representations
$\overline{\rho}:G_{{\mathbb{Q}}}\rightarrow\operatorname{GL}_{2}(\overline{{\mathbb{F}}}_{p})$
that are modular of level $N$. Recall that if $f$ is a cuspidal, normalized,
eigenform of level $\Gamma_{1}(N)$ with coefficients in
$\overline{{{\mathbb{Q}}}}_{p}$, then there is a continuous, semi-simple,
Galois representation
$\overline{\rho}_{f}:G_{{\mathbb{Q}}}\rightarrow\operatorname{GL}_{2}(\overline{{\mathbb{F}}}_{p})$
such that if $\ell\nmid Np$ is a prime, then $\overline{\rho}_{f}$ is
unramified at $\ell$ and the trace of a Frobenius element at $\ell$ is equal
to $a_{\ell}(f)$, the $\ell$-th Fourier coefficient of $f$. Saying
$\overline{\rho}$ is modular of level $N$ means
$\overline{\rho}\cong\overline{\rho}_{f}$ for such an $f$.

For $k\geq 2$ we define $S_{k}(\overline{\rho})\subseteq
S_{k}(\Gamma_{1}(N),\overline{\mathbb{Q}}_{p})$ to be the
$\overline{\mathbb{Q}}_{p}$-span of the eigenforms $f$ of weight $k$ and level
$\Gamma_{1}(N)$ such that $\overline{\rho}_{f}\cong\overline{\rho}$. So,
$\overline{\rho}$ is modular of level $N$ exactly when
$S_{k}(\overline{\rho})\neq\\{0\\}$ for some $k$. Define
$\Gamma_{0}=\Gamma_{1}(N)\cap\Gamma_{0}(p)$ and then
$S_{k}^{0}(\overline{\rho})\subseteq
S_{k}(\Gamma_{0},\overline{\mathbb{Q}}_{p})$ in a similar fashion. We write
the dimensions of these spaces as:

$\displaystyle d(k,\overline{\rho})$
$\displaystyle=\dim_{\overline{\mathbb{Q}}_{p}}S_{k}(\overline{\rho});$
$\displaystyle d_{p}(k,\overline{\rho})$
$\displaystyle=\dim_{\overline{\mathbb{Q}}_{p}}S_{k}^{0}(\overline{\rho}).$

Set $d_{p}^{\operatorname{new}}=d_{p}-2d$. (The function
$d_{p}^{\operatorname{new}}$ counts the eigenforms that are $p$-new.)

The ghost conjecture is based on the definition of a sequence of monic
polynomials $g_{1}(w),g_{2}(w),\dotsc\in{\mathbb{Z}}_{p}[w]$ as follows:

  1. (1)

The zeros of $g_{i}(w)$ are $w_{k}=(1+2p)^{k}-1$, for the finite list of
integers $k$ such that

$d(k,\bar{\rho})<i<d(k,\bar{\rho})+d^{\operatorname{new}}(k,\bar{\rho}).$

  2. (2)

Fix $k$ now. If $i_{1},i_{2},\dotsc,i_{m}$ is the list of consecutive indices
$i$ for which $g_{i}(w_{k})=0$ (the $i$ satisfying the prior inequalities),
then the multiplicities of $w_{k}$ as a zero are: for $i_{1},i_{m}$ the
multiplicity is one; for $i_{2},i_{m-1}$ the multiplicity is two; and so on.

In other words, the multiplicity of $w_{k}$ as a zero of $g_{i}$ for
$i=1,2,3,\dotsc$ has the form

$[0,0,\dotsc,0,1,2,3,\dotsc,3,2,1,0,0,\dotsc],$

where the first string of zeros ends at index $d(k,\overline{\rho})$ and there
are $(d^{\operatorname{new}}(k,\overline{\rho})-1)$-many non-zero numbers.

Having defined the polynomials $g_{i}(w)$, we set

$G_{\bar{\rho}}(w,t)=1+\sum_{i\geq 1}g_{i}(w)t^{i}\in
1+t{\mathbb{Z}}_{p}[\\![w,t]\\!].$

The ghost conjecture predicts that this series models the characteristic power
series of the $U_{p}$-operator acting on overconvergent $p$-adic cuspforms
whose eigensystem modulo $p$ matches $\overline{\rho}$.

More precisely, let $D\subseteq G_{{\mathbb{Q}}}$ be a decomposition group at
$p$ and $I\subseteq D$ be the inertia subgroup. Write
$\omega:G_{{\mathbb{Q}}}\rightarrow{\mathbb{F}}_{p}^{\times}$ for the
cyclotomic character modulo $p$. Then, denote by $b(\overline{\rho})$ the
unique integer $2\leq b(\overline{\rho})\leq p$ such that
$\det(\overline{\rho})|_{I}=\omega^{b(\overline{\rho})-1}$.

We consider the rigid analytic space $\mathcal{W}_{b(\overline{\rho})}$ whose
$\overline{{{\mathbb{Q}}}}_{p}$-points are the $p$-adic weights
$\kappa:{\mathbb{Z}}_{p}^{\times}\rightarrow\overline{{{\mathbb{Q}}}}_{p}^{\times}$
whose action on the the torsion subgroup of ${\mathbb{Z}}_{p}^{\times}$ is
raising to the $b(\overline{\rho})$-th power. Given
$\kappa\in\mathcal{W}_{b(\overline{\rho})}$, write $w_{\kappa}=\kappa(1+2p)-1$
and $S_{\kappa}^{\dagger}(\overline{\rho})$ for the $\overline{\rho}$-isotypic
component of the space of overconvergent $p$-adic cuspforms of level
$\Gamma_{1}(N)$ and weight $\kappa$. The zeros $w_{k}$ defined in (1) occur
only at integer weights $k\geq 2$ that lie in
$\mathcal{W}_{b(\overline{\rho})}$. (Recall,
$\det(\overline{\rho}_{f})|_{I}=\omega^{k-1}$, so if $d(k,\overline{\rho})\neq
0$ then $b(\overline{\rho})\equiv k\bmod p-1$.)

The ghost conjecture for $\overline{\rho}$ stated in [6] is:

######  Conjecture 1.1 (False).

Assume $p\geq 5$ and $\overline{\rho}\not\cong\omega^{j}\oplus\omega^{j+1}$
for any $j$. If $\overline{\rho}|_{D}$ is reducible, then, for each
$\kappa\in\mathcal{W}_{b(\overline{\rho})}$, the Newton polygon of
$G_{\bar{\rho}}(w_{\kappa},t)$ is the same as the Newton polygon of the
characteristic series of the $U_{p}$-operator acting on
$S_{\kappa}^{\dagger}(\bar{\rho})$.

###  1.2. Counter-examples

We give two counter-examples here.

####  1.2.1. Detailed example

Let $p=5$ and $N=3$. We choose $\bar{\rho}=\omega\oplus\omega\chi$ where
$\chi$ is the quadratic Dirichlet character of conductor 3. The dimension
functions $d$, $d_{p}$, and $d_{p}^{\operatorname{new}}$ can be determined
using a computer algebra system with built-in libraries for modular forms,
such as Magma [9] or Sage [21]. The result is compiled in Table 1. From the
dimensions, we see that

$\displaystyle g_{1}(w)$ $\displaystyle=w-w_{3};$ $\displaystyle g_{2}(w)$
$\displaystyle=w-w_{7};$ $\displaystyle g_{3}(w)$
$\displaystyle=(w-w_{11})(w-w_{15})\dotsb.$

Since $g_{1}(w_{7})$ has $5$-adic valuation 1, the least slope on the Newton
polygon of $G_{\overline{\rho}}(w_{7},t)$ is 1 or less. On the other hand, any
weight 7 overconvergent $p$-adic eigenform with slope at most 1 is classical
by Coleman’s classicality theorem [15, Theorem 6.1]. Yet, the slopes of
$U_{5}$ acting on $S_{7}^{0}(\overline{\rho})\subseteq S_{7}^{\dagger}$ are
$[\frac{5}{2},\frac{5}{2},3,3]$ (calculated using Magma). This is the
contradiction.

Table 1. Dimensions of $\overline{\rho}$-isotypic components in $S_{k}(\Gamma_{1}(3))$ and $S_{k}(\Gamma_{1}(3)\cap\Gamma_{0}(5))$, for $\overline{\rho}=\omega\oplus\omega\chi$. $k$ | $d(k,\bar{\rho})$ | $d_{p}(k,\bar{\rho})$ | $d_{p}^{\operatorname{new}}(k,\bar{\rho})$
---|---|---|---
3 | 0 | 2 | 2
7 | 1 | 4 | 2
11 | 2 | 6 | 2
15 | 2 | 8 | 4
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$

######  Remark 1.2.

Strictly speaking, neither Magma (nor Sage) currently has intrinsic commands
to determine the slopes in $S_{k}(\overline{\rho})$. The authors provide code
in the github repository [4].

####  1.2.2. Another example

The reader might wonder if the fundamental issue in the prior example is that
either $\overline{\rho}$ is Eisenstein or
$\overline{\rho}\cong\overline{\rho}\otimes\chi$, both somehow global
phenomena. This is not the case.

Explicitly, continue to let $p=5$ and also let $\psi$ be the quadratic
character modulo $43$. Then, in $S_{7}(\Gamma_{1}(43),\psi)$ there is a
20-dimensional Galois orbit of newforms in which the $5$-adic slope $2$ occurs
twice, once for an absolutely irreducible $\overline{\rho}$ and once more for
$\overline{\rho}\otimes\psi$ (which is a new representation). Further,
$S_{7}(\overline{\rho})$ is 1-dimensional and thus 2 is the lowest slope
occurring for $\overline{\rho}$. Yet, as in Section 1.2.1, one can check that
the ghost series formalism would predict the lowest slope in
$S_{7}(\overline{\rho})$ is 1.

In fact, $\overline{\rho}\otimes\omega^{-1}$ lifts to an ordinary eigenform of
weight $5$, and thus $\overline{\rho}$ is locally reducible at $p=5$. By
Proposition 2.2 and point (ii) in the proof of Theorem 2.5 below, we even have

(1.1)
$\overline{\rho}|_{D}\cong\begin{pmatrix}\operatorname{nr}(\alpha)\omega&\ast\\\
0&\operatorname{nr}(-\alpha)\omega\end{pmatrix}$

where $\alpha\in\overline{{\mathbb{F}}}_{5}^{\times}$. (For unfamiliar
notations or definitions, see Section 2.) The extension (1.1) is even non-
split! If it were split, then by [16, Theorem 4.5] the representation
$\overline{\rho}\otimes\omega^{-1}$ would arise from a weight one form over
$\overline{{\mathbb{F}}}_{5}$ of level 43 and character $\psi$. We verified
such a form does not exist using a computer package called Weight1, which
calculates weight one eigenforms modulo $p$, written by Wiese. (The code,
which requires Magma, is currently available on Wiese’s website [24].)

####  1.2.3. The general phenomenon

The common link between the examples is indicated by the equation (1.1). In
both cases, $\overline{\rho}\otimes\omega^{-1}$ is reducible at $p=5$ and its
semi-simplification is an unramified representation for which a Frobenius
element acts with trace zero. Generalizing these examples, the fundamental gap
we found in the ghost conjecture occurs for $\overline{\rho}$ that are twists
of representations which locally at $p$ are reducible, with semi-
simplification that is unramified with Frobenius trace zero. Indeed, in
Proposition 3.8 we will show that the ghost conjecture will always fail in
such examples. However, as long as we exclude such representations (which we
do in Section 2), we believe that the ghost conjecture will hold for remaining
$\overline{\rho}$’s.

###  1.3. A brief history

We first encountered the example in Section 1.2.1 in 2015-16, when we learned
by explicit computation that the ghost series formalism failed to predict the
5-adic slopes of eigenforms of level $\Gamma_{1}(3)$ (with no restriction on
the Galois representations modulo 5). We apparently overlooked that
computation when formulating the $\overline{\rho}$-version of our conjecture.
Later, in 2018, James Newton asked us about the $\overline{\rho}$-conjecture
for examples where $\overline{\rho}$ is unramified at $p$ (up to a twist). We
mistakenly missed the counter-examples once more. We realized our mistake only
in early 2021 after our attention was diverted toward questions on reductions
of crystalline Galois representations.

###  1.4. Plan for the remainder of the article

In Section 2 we revisit the reducible versus irreducible dichotomy and replace
it with a new one, which we name regular versus irregular. We study this new
dichotomy in both local and global terms, relating it to crystalline lifts on
the one hand and slopes of modular forms in low weight on the other. The
salvaged conjecture and evidence is given Section 3. Concluding, and open-
ended, remarks occupy Section 4.

###  1.5. Acknowledgements

We thank Liang Xiao for keeping us updated on the progress of his
collaboration with Liu, Truong, and Zhao, which aims to prove the ghost
conjecture. Both authors also thank the Max Planck Institute for Mathematics
in Bonn, Germany, for its hospitality in July 2021, when this paper was
largely written. The research was also supported by a Simons Collaboration
Grant (PI: J.B. Award #713782) and an NSF grant (PI: R.P. DMS-1702178).

##  2\. Salvaging the ghost conjecture: redefining regularity

To salvage Conjecture 1.1, we redefine regularity, replacing the reducible vs.
irreducible dichotomy with a slightly different one. In this section we let
$p$ be any prime, unless noted.

We first fix notations for local Galois representations modulo $p$. Let
${\mathbb{Q}}_{p^{2}}$ be the unramified quadratic extension of
${\mathbb{Q}}_{p}$ and write
$\omega_{2}:G_{{\mathbb{Q}}_{p^{2}}}\rightarrow\overline{\mathbb{F}}_{p}^{\times}$
for a niveau 2 fundamental character. Let
$G_{{\mathbb{Q}}_{p}}=\operatorname{Gal}(\overline{{{\mathbb{Q}}}}_{p}/{\mathbb{Q}}_{p})$
and
$G_{{\mathbb{Q}}_{p^{2}}}=\operatorname{Gal}(\overline{{{\mathbb{Q}}}}_{p}/{\mathbb{Q}}_{p^{2}})\subseteq
G_{{\mathbb{Q}}_{p}}$. We then write
$\operatorname{ind}(\omega_{2}^{s})=\operatorname{Ind}_{G_{{\mathbb{Q}}_{p}^{2}}}^{G_{{\mathbb{Q}}_{p}}}(\omega_{2}^{s}\eta)$
where $\eta$ is the quadratic character of ${\mathbb{Q}}_{p^{2}}$. Thus
$\operatorname{ind}(\omega_{2}^{s})$ has determinant $\omega^{s}$ and
$\operatorname{ind}(\omega_{2}^{s})|_{I}=\omega_{2}^{s}\oplus\omega_{2}^{sp}$.
Given $\alpha\in\overline{{\mathbb{F}}}_{p}^{\times}$ we also write
$\operatorname{nr}(\alpha)$ for the character on $G_{{\mathbb{Q}}_{p}}$ that
is unramified and whose value on a Frobenius element is $\alpha$. In this
notation,
$\operatorname{ind}(1)=\operatorname{nr}(\sqrt{-1})\oplus\operatorname{nr}(-\sqrt{-1})$.

The continuous, semi-simple, representations
$\overline{r}:G_{{\mathbb{Q}}_{p}}\rightarrow\operatorname{GL}_{2}(\overline{\mathbb{F}}_{p})$
are completely classified. Up to a twist by a character, any $\overline{r}$ is
isomorphic to either $\operatorname{ind}(\omega_{2}^{s})$ for some $0\leq
s\leq p-1$ or
$\operatorname{nr}(\alpha)\oplus\operatorname{nr}(\beta)\omega^{t}$ with
$\alpha,\beta\in\overline{{\mathbb{F}}}_{p}^{\times}$ and $0\leq t\leq p-2$.
The irreducible $\overline{r}$ occur for twists of
$\operatorname{ind}(\omega_{2}^{s})$ with $1\leq s\leq p-1$. The
representation $\operatorname{ind}(\omega_{2}^{s})$ is reducible when $s\equiv
0\bmod p+1$.

######  Definition 2.1.

A continuous, semi-simple, representation
$\overline{r}:G_{{\mathbb{Q}}_{p}}\rightarrow\operatorname{GL}_{2}(\overline{\mathbb{F}}_{p})$
is called irregular if $\overline{r}$ is a twist of
$\operatorname{ind}(\omega_{2}^{s})$ for some $s\in{\mathbb{Z}}$.

Naturally, we say $\overline{r}$ is regular when it is not irregular. Note
that regular representations are always reducible, but the regularity of
$\overline{r}$ depends on the entire action of $D$, whereas the reducibility
depends only on $\overline{r}|_{I}$.

The next proposition illustrates the regular versus irregular dichotomy in
terms of crystalline lifts with small Hodge–Tate weights. The reader who is
unfamiliar with $p$-adic Hodge theory may prefer to skip to Theorem 2.5 or
reference the well-written introduction to [13]. The notation is as follows.
For $a_{p}\in\overline{{{\mathbb{Q}}}}_{p}$ such that $v_{p}(a_{p})>0$ and
$k\geq 2$, $V_{k,a_{p}}$ denotes the unique two-dimensional, irreducible,
crystalline representation of $G_{{\mathbb{Q}}_{p}}$ whose Hodge–Tate weights
are $\\{0,k-1\\}$ and such that the characteristic polynomial of the
crystalline Frobenius acting on $D_{\operatorname{crys}}(V_{k,a_{p}}^{\ast})$
is $t^{2}-a_{p}t+p^{k-1}$. Write $\overline{V}_{k,a_{p}}$ for the semi-
simplification of the reduction of $V_{k,a_{p}}$ modulo $p$.

######  Proposition 2.2.

Let
$\overline{r}:G_{{\mathbb{Q}}_{p}}\rightarrow\operatorname{GL}_{2}(\overline{{\mathbb{F}}}_{p})$
be a continuous, semi-simple, representation. The following conditions are
equivalent (the (a)’s and (b)’s are individually equivalent):

  1. (1)

The representation $\overline{r}$ is irregular:

    1. (a)

$\overline{r}$ is a twist of $\operatorname{ind}(\omega_{2}^{s})$ with
$s\not\equiv 0\bmod p+1$, or

    2. (b)

$\overline{r}$ is a twist of $\operatorname{ind}(\omega_{2}^{s})$ with
$s\equiv 0\bmod p+1$.

  2. (2)

Either:

    1. (a)

$\overline{r}$ irreducible, or

    2. (b)

$\overline{r}$ a twist of an unramified representation whose Frobenius trace
is zero.

  3. (3)

Either:

    1. (a)

$\overline{r}$ is a twist of $\overline{V}_{k,a_{p}}$ for some $2\leq k\leq
p+1$ and $v_{p}(a_{p})>0$, or

    2. (b)

$\overline{r}$ is a twist of $\overline{V}_{p+2,a_{p}}$ for any
$v_{p}(a_{p})>1$.

###### Proof.

The equivalence of (1) and (2) follows directly from the discussion prior to
Definition 2.1. The equivalence of (1) and (3) follows from calculations of
$\overline{V}_{k,a_{p}}$ when $k$ is small. Namely, if $2\leq k\leq p+1$ then
$\overline{V}_{k,a_{p}}\cong\operatorname{ind}(\omega_{2}^{k-1})$ (which is,
in particular, irreducible and thus irregular). When $k=p+2$, we have:

(2.1)
$\overline{V}_{p+2,a_{p}}=\begin{cases}\operatorname{ind}(\omega_{2}^{2})&\text{if
$0<v_{p}(a_{p})<1$;}\\\
\operatorname{nr}(\alpha)\oplus\operatorname{nr}(\alpha^{-1})&\text{if
$1=v_{p}(a_{p})$, where
$\alpha+\alpha^{-1}=\frac{a_{p}}{p}\in\overline{{\mathbb{F}}}_{p}^{\times}$;}\\\
\operatorname{ind}(1)\otimes\omega&\text{if $1<v_{p}(a_{p})$.}\end{cases}$

Thus $\overline{V}_{p+2,a_{p}}$ is irregular when $v_{p}(a_{p})\neq 1$. See
[7, Théorème 3.2.1] for these results. From them, we plainly see (3) implies
(1) and (1) implies (3) since any irregular $\overline{r}$ is a twist of
$\operatorname{ind}(\omega_{2}^{s})$ for some $0\leq s\leq p-1$. ∎

######  Remark 2.3.

The trivial representation mod 2 is irregular by Proposition 2.2(2), since it
is unramified with Frobenius trace $2=0$. See Section 4.1.

We now turn toward global considerations. Given a representation $\rho$, write
$\rho^{\operatorname{ss}}$ for the semi-simplification. (This depends on the
group that $\rho$ is representing, but the context will be clear any time we
use the notation.) Since the decomposition groups $D\subseteq
G_{{\mathbb{Q}}}$ at $p$ are all conjugate and isomorphic to
$G_{{\mathbb{Q}}_{p}}$, we can speak unambiguously about a global
representation $\overline{\rho}$ being regular on $D$ or not:

######  Definition 2.4.

We say $\overline{\rho}$ is regular if
$(\overline{\rho}|_{D})^{\operatorname{ss}}$ is regular.

The action of $D$ is reducible for every regular $\overline{\rho}$, so our new
regular condition implies the condition in Conjecture 1.1. The difference is
that we have further excluded one particular twist-stable family of reducible
representations.

We describe now a mechanism in terms of the arithmetic of modular forms to
determine whether or not a given $\overline{\rho}$ is regular. Write

$S_{k}(\\!(\overline{\rho})\\!)=\sum_{j=0}^{p-2}S_{k}(\overline{\rho}\otimes\omega^{j})\subseteq
S_{k}(\Gamma_{1}(N),\overline{{{\mathbb{Q}}}}_{p}).$

The space $S_{k}(\\!(\overline{\rho})\\!)$ is stable under the
$T_{p}$-operator. Recall, if $f$ is an eigenform then $v_{p}(a_{p}(f))\geq 0$.
We say $f$ is ordinary if $v_{p}(a_{p}(f))=0$.

######  Theorem 2.5.

Consider the following two conditions:

  1. (1)

The representation $\overline{\rho}$ is regular.

  2. (2)

For all $2\leq k\leq p+1$, the eigenforms in $S_{k}(\\!(\overline{\rho})\\!)$
are all ordinary, and the eigenforms in $S_{p+2}(\\!(\overline{\rho})\\!)$ are
either ordinary or satisfy $v_{p}(a_{p}(f))=1$.

Then, (1) always implies (2). If $p\geq 3$, then (2) implies (1).

More precisely:

  1. (a)

If $\overline{\rho}|_{D}$ is irreducible, then there exists a non-ordinary
eigenfrom $g$ of weight between $2$ and $\frac{p+3}{2}$ such that
$\overline{\rho}$ is a cyclotomic twist of $\overline{\rho}_{g}$.

  2. (b)

If $p\geq 3$ and $\overline{\rho}$ is irregular but $\overline{\rho}|_{D}$ is
reducible, then there exists a $j$ such that
$S_{p+2}(\overline{\rho}\otimes\omega^{j})\neq\\{0\\}$ and $v_{p}(a_{p}(g))>1$
for all $g\in S_{p+2}(\overline{\rho}\otimes\omega^{j})$.

###### Proof.

To start, we recall two fundamental facts. For any eigenform $f$, we let

$\rho_{f}:G_{{\mathbb{Q}}}\rightarrow\operatorname{GL}_{2}(\overline{{{\mathbb{Q}}}}_{p})$

be the corresponding global $p$-adic Galois representation. Then:

  1. (i)

If $f$ is ordinary, then $\rho_{f}|_{D}$ is reducible and
$(\overline{\rho}_{f}|_{I})^{\operatorname{ss}}\cong 1\oplus\omega^{k-1}$.

  2. (ii)

If $f$ is non-ordinary, then $\rho_{f}|_{D}$ is isomorphic to a twist of
$V_{k,a_{p}(f)}$.

It seems the first result was first proven by Deligne in the 1970’s but never
published. A common reference is [25, Theorem 2]. For the second, see [10,
Théorème 6.5].

Now we prove (1) implies (2). Suppose that $\overline{\rho}$ is regular and
$f$ is an eigenform of weight $2\leq k\leq p+2$ such that
$\overline{\rho}_{f}=\overline{\rho}\otimes\omega^{j}$. Then, either $f$ is
ordinary or $\overline{V}_{k,a_{p}(f)}$ is regular by (ii). By Proposition
2.2, and the classification of $\overline{V}_{p+2,a_{p}}$ in equation (2.1),
the latter can only happen if $k=p+2$ and $v_{p}(a_{p}(f))=1$. This proves (1)
implies (2).

Now we will prove (2) implies (1). By [16, Theorem 3.4] there exists an
eigenform $f$ of weight $1\leq k\leq p+1$ where
$\overline{\rho}\cong\overline{\rho}_{f}\otimes\omega^{j}$, with the caveat
that $f$ may be an Eisenstein series or, in the case $k=1$, $f$ may be only a
mod $p$ modular form. Furthermore, the theory of $\theta$-cycles (see [16,
Proposition 3.3]) predicts the least positive weight in which
$\overline{\rho}_{f}\otimes\omega^{j}$ lifts to a modular eigenform.

Now consider (a), where $\overline{\rho}|_{D}$ is irreducible. Then, the
possibility of $k=1$ cannot actually occur. Indeed, if $f$ has weight one then
$\overline{\rho}_{f}|_{D}$ is unramified by theorems of Gross, Coleman–Voloch,
and Wiese. See [23, Corollary 1.3]. The theory of $\theta$-cycles ($a_{p}=0$
and $k\geq 2$ in [16, Proposition 3.3] cited above) then implies there exists
a non-ordinary eigenform $g$ of weight between $2$ and $\frac{p+3}{2}$ such
that $\overline{\rho}_{g}\cong\overline{\rho}_{f}\otimes\omega^{j}$ for some
$j$. This completes the proof of (a).

In case (b), we assume that $p\geq 3$ and after twisting that
$\overline{\rho}=\overline{\rho}_{f}$ with $\overline{\rho}_{f}|_{D}$
reducible, yet irregular and $f$ has weight $1\leq k\leq p+1$. We first claim
that either: $k=p$ and $f$ is ordinary, or $k=1$. Further, in both cases
$(\overline{\rho}_{f}|_{I})^{\operatorname{ss}}\cong 1\oplus 1$.

To see this, note first that if $k=1$ then, as above, $\overline{\rho}_{f}$ is
unramifed at $p$ and thus $(\overline{\rho}_{f}|_{I})^{\operatorname{ss}}\cong
1\oplus 1$. If $k\geq 2$, we first check that $f$ is ordinary. Indeed, if $f$
is Eisenstein, then it is ordinary because $p\nmid N$. If $f$ were cuspidal
non-ordinary, then (ii) and Proposition 2.2 would imply
$\overline{\rho}_{f}|_{D}$ is irreducible, which it is not. So, $f$ is
ordinary. Then, by (i) we see
$(\overline{\rho}_{f}|_{I})^{\operatorname{ss}}\cong 1\oplus\omega^{k-1}$. But
$\overline{\rho}$ is irregular, and so Proposition 2.2 implies
$\omega^{k-1}=1$, which implies $k=p$ since $2\leq k\leq p+1$.

Finally, regardless of whether $k=p$ and $f$ is ordinary, or $k=1$, the theory
of $\theta$-cycles guarantees there exists a modular eigenform $g$ of weight
$p+2$ such that $\overline{\rho}_{g}\cong\overline{\rho}_{f}\otimes\omega$.
For such $g$, we have
$(\overline{\rho}_{g}|_{I})^{\operatorname{ss}}\cong\omega\oplus\omega$ and
thus $g$ is non-ordinary by (i). Here we use $p\neq 2$ to know $\omega\neq 1$.
In particular we’ve shown that
$S_{p+2}(\overline{\rho}_{f}\otimes\omega)\neq\\{0\\}$. We also see from (ii)
that $\overline{V}_{p+2,a_{p}(g)}$ is reducible and irregular for any $g\in
S_{p+2}(\overline{\rho}_{f}\otimes\omega)$. Thus, by Proposition 2.2,
$v_{p}(a_{p}(g))>1$ for all $g\in S_{p+2}(\overline{\rho}_{f}\otimes\omega)$.
This completes the proof of (b). ∎

######  Remark 2.6.

A key step in the proof of part (b) of Theorem 2.5 is that if $f$ is an
ordinary eigenform of weight $p$, then $\overline{\rho}_{f}\otimes\omega$
lifts to an eigenform of weight $p+2$. This statement also holds when $f$ is
non-ordinary, as long as $p\geq 5$. For instance, one could directly argue
using modular symbols modulo $p$ as in [3, Theorem 3.4(a,b)]. (That reference
assumes $p\geq 5$ throughout.)

For $p=3$, though, there exist weight 3 forms $f$ where
$\overline{\rho}_{f}\otimes\omega$ doesn’t lift to weight 5. For instance, if
$\chi$ is the quadratic character modulo 7, then $S_{3}(\Gamma_{1}(7),\chi)$
is spanned by a unique eigenform ([22, Newform 7.3.b.a]) whose $q$-expansion
begins $q-3q^{2}+\dotsb$, whereas $S_{5}(\Gamma_{1}(7),\chi)$ is spanned by a
unique eigenform ([22, Newform 7.5.b.a]) whose $q$-expansion is
$q+q^{2}+\dotsb$. These two eigenforms are not twists of each other modulo 3.

##  3\. The salvaged conjecture and evidence

We now state for the record our salvaged $\overline{\rho}$-ghost conjecture.
We place into the conjecture the hypotheses we need, as a reminder.

######  Conjecture 3.1 (The salvaged $\overline{\rho}$-ghost conjecture).

Let $p\geq 5$. If $\overline{\rho}$ is regular in the sense of Definition 2.4,
then, for each $\kappa\in\mathcal{W}_{b(\overline{\rho})}$, the Newton polygon
of $G_{\bar{\rho}}(w_{\kappa},t)$ is equal to the Newton polygon of the
characteristic series of the $U_{p}$-operator acting on
$S_{\kappa}^{\dagger}(\bar{\rho})$.

In the next subsections, we provide computational and theoretical evidence for
our salvaged conjecture.

######  Remark 3.2.

Notice that Conjecture 3.1 includes
$\overline{\rho}\cong\omega^{j}\oplus\omega^{j+1}$, in contrast with
Conjecture 1.1. See Section 3.1 below.

######  Remark 3.3.

Condition (2) in Theorem 2.5 appears in Buzzard’s slope conjectures when $p=2$
(see [11, Definition 1.3]). For odd primes, it was irrelevant in Buzzard’s
work, since restricting to level $\Gamma_{0}(N)$ removes concerns about odd
weights $p$ and $p+2$.

###  3.1. Galois multiplicities

In order to construct the series $G_{\overline{\rho}}(w,t)$, and thus test
Conjecture 3.1, one must understand the functions $d(k,\overline{\rho})$ and
$d_{p}(k,\overline{\rho})$. In fact, each of these functions are determined
after a finite computation. Specifically, either can be determined from the
values $d(k^{\prime},\overline{\rho}\otimes\omega^{i})$ with $2\leq
k^{\prime}\leq p+1$ and $0\leq i\leq p-2$ along with multiplicities of
Eisenstein series (if $\overline{\rho}$ is Eisenstein). This is explained in
[6, Section 6]. The technique involves modular symbols modulo $p$, which is
the source of the assumption $p\geq 5$. We do not currently know dimension
formulas when $p=2,3$. For an alternative method via the trace formula, which
is adaptable to $p=2,3$, see forthcoming work of Anni, Ghitza, and Medvedovsky
[1].

There is one caveat: the exposition in [6] ignores the case
$\overline{\rho}\cong\omega^{j}\oplus\omega^{j+1}$ (“twists of $E_{2}$”) for
convenience. These are precisely the Galois representations modulo $p$ that
contribute to torsion in $H^{2}_{c}$’s, which creates complications. However,
the complications are luckily limited to exposition. The actual result, that
low weight calculations for $\overline{\rho}$ and all its twists will
determine $\overline{\rho}$-multiplicities in all weights, are still valid for
twists of $E_{2}$.

So, while omitting more specific details, we used this principle in order to
carry out the numerical testing explained in the next section, regardless of
whether $\overline{\rho}$ is a twist of $E_{2}$ or not. (But still, $p\geq
5$.) For the interested reader, we also created computer code that implements
the natural process to calculate $d(k,\overline{\rho})$ from low weight
values. See [4].

###  3.2. Numerical evidence

In [6, Section 7.2], we gave significant numerical evidence for Conjecture
3.1. Here we report more numerical evidence, especially highlighting the
following two cases:

  1. (1)

Representations
$\overline{\rho}:G_{{\mathbb{Q}}}\rightarrow\operatorname{GL}_{2}(\overline{{\mathbb{F}}}_{p})$
with $(\overline{\rho}|_{D})^{\operatorname{ss}}$ a twist of unramified with
non-zero Frobenius trace (Examples 3.4-3.5).

  2. (2)

Twists of $1\oplus\omega$ (Example 3.6).

The latter was a case not numerically tested in [6], and the former is meant
to ensure the issue with the counter-examples in Section 1.2 has been
accurately diagnosed.

In each example, we describe a single $\overline{\rho}$ and explain why it is
regular. Then, when we write that we “verified Conjecture 3.1 in this case for
weights $k$…” we mean: for each
$\overline{\rho}^{\prime}=\overline{\rho}\otimes\omega^{i}$ we calculated the
$T_{p}$-slopes on $S_{k}(\overline{\rho}^{\prime})$ and checked that they
matched the first $d(k,\overline{\rho}^{\prime})$-many slopes on
$G_{\overline{\rho}^{\prime}}(w_{k},t)$. That is, we checked (using the code
referenced in Remark 1.2) the ghost series accurately predicts the classical
slopes for $\overline{\rho}$ and each of its twists (for a range of weights).

######  Example 3.4.

Let $\chi$ denote the quadratic character of conductor $N$ thought of as
taking values in ${\mathbb{F}}_{p}^{\times}$ and let
$\overline{\rho}=1\oplus\chi$. If $p\nmid N$ then $\overline{\rho}$ is
unramified at $p$ and it is regular if and only if $\chi(p)=1$. For each of
$N=3,7,11$ and for all primes $5\leq p\leq 97$ for which $\overline{\rho}$ is
regular, we verified Conjecture 3.1 in this case for weights $k\leq
385,173,191$ respectively.

######  Example 3.5.

Let $p=5$, $N=23$, and $\chi$ be the quadratic character modulo 23. Then,
$S_{5}(\Gamma_{1}(23),\chi)$ supports a unique $\overline{\rho}$ whose
Frobenius trace at $\ell=2$ is equal to 2. Specifically, if $f$ is a member of
the newform orbit [22, Newform 23.5.b.b] and $\mathfrak{p}$ is the unique
prime above 5 with inertia degree 2 for the number field generated by $f$’s
Hecke eigenvalues, then $\overline{\rho}$ is the Galois representation
associated with $f$ modulo $\mathfrak{p}$. The eigenform $f$ is
$\mathfrak{p}$-ordinary and $\overline{\rho}\otimes\omega$ arises from a
weight 7 form with slope 1. By Theorem 2.5, $\overline{\rho}$ is regular.
Unlike the previous example, this $\overline{\rho}$ is globally irreducible.
We verified Conjecture 3.1 in this case for weights $k\leq 155$.

######  Example 3.6.

Let $\overline{\rho}=1\oplus\omega$ which is regular. For both of $N=1$ and
$11$ and for all primes $p$ with $5\leq p\leq 97$, we verified Conjecture 3.1
in this case for weights $k\leq 750,172$ respectively.

######  Remark 3.7.

Gouvêa and Mazur once conjectured that if $h$ is a rational number and
$k,k^{\prime}\geq 2h+2$, then the multiplicity of the ($T_{p}$-)slope $h$ in
weight $k$ will be the same as the multiplicity in weight $k^{\prime}$
provided $k\equiv k^{\prime}\bmod(p-1)p^{\lceil h\rceil}$ (see [17, Conjecture
1]). Buzzard and Calegari found many counter-examples, all occuring in
situations where $\overline{\rho}$ is locally irreducible at $p$ ([12]). Those
$\overline{\rho}$ are, in particular, irregular. In the final paragraphs of
loc. cit., however, it is mentioned that Clay noticed issues in a globally
reducible, and in fact regular, situation.

Specifically, let $\overline{\rho}=\omega+\omega\chi$ where $\chi$ is the
quadratic character modulo 4, which is regular for $p=5$ because
$\chi(5)=\chi(1)=1$ (compare with Example 3.4). The violation of Gouvêa–Mazur,
which Clay noticed, occurs because the $5$-adic $\overline{\rho}$-slopes in
weight $k=7$ at level $\Gamma_{1}(4)$ are $\\{1,1\\}$ (two times), whereas in
weight $k=27$ they are $\\{1,1,1,1\\}$ (four times). We highlight this because
this violation of the Gouvêa–Mazur conjecture is consistent with Conjecture
3.1. That is, the ghost series predicts exactly the same slope pattern.

The data collected in Example 3.4 gives rather systematic counter-examples to
the Gouvêa–Mazur conjecture (on $\overline{\rho}$-subspaces), all of which are
consistent with Conjecture 3.1. The issue is also a local one, as opposed to a
global one, it seems. We found the same phenomenon in the setting of Example
3.5, which is globally irreducible. Specifically, in that example, in weight
$7$ the slope sequence is $\\{1\\}$, in weight 27 it is $\\{1,1\\}$.
Conjecture 3.1 predicts these slopes.

###  3.3. Theoretical evidence

For theoretical evidence, we show that $\overline{\rho}$ must be regular if
the ghost series correctly calculates slopes of modular forms.

######  Proposition 3.8.

Let $p\geq 5$. If $\overline{\rho}$ is irregular, then the ghost series of
some twist of $\overline{\rho}$ does not correctly compute slopes of modular
forms. That is, there is some $j$ and some weight
$\kappa\in\mathcal{W}_{b(\overline{\rho}\otimes\omega^{j})}$ such the slopes
of the Newton polygon of $G_{\overline{\rho}\otimes\omega^{j}}(w_{\kappa},t)$
do not match the slopes of $U_{p}$ acting on
$S_{\kappa}^{\dagger}(\overline{\rho}\otimes\omega^{j})$.

###### Proof.

First assume that $\overline{\rho}|_{D}$ is irreducible. By Theorem 2.5, after
replacing $\overline{\rho}$ by a twist, we may assume that $\overline{\rho}$
arises from a non-ordinary cuspform of weight $k$ between $2$ and
$\frac{p+3}{2}$. We will show that the ghost series for $\overline{\rho}$ does
not correctly compute the slopes of $U_{p}$ in weight $k=b(\overline{\rho})$.

To this end, write
$G_{\overline{\rho}}(w,t)=1+g_{1}(w)t+g_{2}(w)t^{2}+\dotsb$. The smallest $k$
such that $w_{k}$ is possibly a zero of any $g_{i}(w)$ is
$k=b(\overline{\rho})$. From the definition of the ghost series, if
$w_{b(\overline{\rho})}$ is a zero is for $g_{i}(w)$ then

$i>d=d(b(\overline{\rho}),\overline{\rho})=\dim
S_{b(\overline{\rho})}(\overline{\rho}).$

Since $p\geq 5$, we have that $n\mapsto
d(b(\overline{\rho})+n(p-1),\overline{\rho})$ is increasing by [6, Proposition
6.12]. So, $G_{\overline{\rho}}(w,t)=1+\dotsb+t^{d}+\dotsb$ and thus the first
$d$ slopes of the Newton polygon of this ghost series (at any weight!) are all
0. In particular, if the slopes attached to this series equaled the true
slopes of $U_{p}$, then the ($T_{p}$-)slopes in
$S_{b(\overline{\rho})}(\overline{\rho})$ would be all be 0. But this
contradicts the fact that $\overline{\rho}$ arises from a non-ordinary form of
weight $b(\overline{\rho})$.

Now we consider the case where $\overline{\rho}|_{D}$ is reducible (but still
irregular). By part (b) of Theorem 2.5, after replacing $\overline{\rho}$ by a
twist, we may assume that $S_{p+2}(\overline{\rho})\neq\\{0\\}$ and
$v_{p}(a_{p}(g))>1$ for all $g\in S_{p+2}(\overline{\rho})$. We will show that
the ghost series for $\overline{\rho}$ does not correctly compute slopes in
weight $p+2$.

To this end, again write
$G_{\overline{\rho}}(w,t)=1+g_{1}(w)t+g_{2}(w)t^{2}+\dotsb$. Since
$S_{p+2}(\overline{\rho})$ is non-zero, we have that $g_{1}(w_{p+2})\neq 0$.
Thus, as argued above, $g_{1}(w)=(w-w_{3})^{m}$, and $m\leq 1$ because the
multiplicity pattern described on pg. 2 begins $1,2,\dotsc$. Thus, the lowest
slope of the Newton polygon of $G_{\overline{\rho}}(w_{p+2},t)$ is at most 1.
But we’ve also assumed every slope in $S_{p+2}(\overline{\rho})$ is larger
than one. This is a contradiction. ∎

##  4\. Conclusion

We have presented a new, salvaged, ghost conjecture for Galois representations
modulo $p$, along with theoretical and computational evidence. We end by
briefly describing two questions raised by our work (separate from whether or
not the conjecture is true).

###  4.1. Small primes

Let $p=2$. According to the definitions given here, the trivial representation
$\overline{\rho}=1\oplus 1$ is irregular. This is rather confouding. The
$2$-adic slopes at level $\operatorname{SL}_{2}({\mathbb{Z}})$, where the only
Galois representation modulo 2 is the trivial one, are the most carefully
studied of all examples. It is a major motivating example for both Buzzard’s
work and the work of the authors. The ghost formalism is believed to predict
the $2$-adic slopes at level $\operatorname{SL}_{2}({\mathbb{Z}})$, yet it
does not (nor does Buzzard’s work) correctly predict the $2$-adic
$\overline{\rho}$-slopes at level $\Gamma_{0}(5)$ where $\overline{\rho}$ is
still the trivial representation.

Of course, the second condition of Theorem 2.5 holds for the trivial
representation when $p=2$ and $N=1$, so perhaps that condition is the correct
one for which the ghost series will predict slopes (regardless of $p$ being
small). This is in-line with Buzzard’s ad hoc definition of
$\Gamma_{0}(N)$-regular when $p=2$ in [11]. Note however that even in the
$\Gamma_{0}(N)$-regular situation, the $2$-adic ghost conjecture requires a
modification as in [5, Section 5].

The authors have not seen a reason to exclude $p=3$ from Conjecture 3.1. Yet,
we want to be careful. We have not done extensive testing for $p=3$ because
the derivations of Galois multiplicities in Section 3.1 relies on [3], which
assumes that $p\geq 5$. The same methods work immediately when $p=2,3$ as long
as $\Gamma_{1}(N)$ is a group without $p$-torsion, but stating a conjecture
for $p=3$ under such a restriction would be rather prosaic.

###  4.2. Integral slopes and local Galois representations

It is conjectured that if $p$ is odd, $k$ is even, and
$v_{p}(a_{p})\not\in\mathbb{N}$ then $\overline{V}_{k,a_{p}}$ is irreducible.
See [14, Conjecture 4.1.1]. The possibility of such a result was discussed “in
emails between Breuil, Buzzard, and Emerton” in the early 2000’s according to
loc. cit. To give it a neutral name we’ll call it the integral slope
conjecture. We’ve been told by Breuil and Buzzard that the inspiration for
discussing the integral slope conjecture was numerical calculations of global
slopes and explicit local calculations in small weights. In light of this
article’s emphasis on reducibility being a faulty lens through which to
predict global slopes, we offer the following comments.

When $p$ is odd and $k$ is even, the irregular $\overline{V}_{k,a_{p}}$ in the
sense of Definition 2.1 are all irreducible. So, the integral slope conjecture
might as well say if $k$ is even and $v_{p}(a_{p})\not\in\mathbb{N}$ then
$\overline{V}_{k,a_{p}}$ is irregular. This re-phrasing opens up the
possibility of stating a more full conjecture. For instance, $p=2$ is not
included in the original integral slope conjecture because when $p=2$ and
$k\equiv 4\bmod 6$, then $\overline{V}_{k,0}$ is reducible and hence so is
$\overline{V}_{k,a_{p}}$ for any $v_{p}(a_{p})\gg 0$ (see [14, Remark 4.1.6]).
Yet, these examples are in fact all irregular. Separate objections are raised
when $k$ is odd (see [14, Remark 4.1.5]), each of which is rendered moot if
instead the conjecture begins with the assumption that
$v_{p}(a_{p})\not\in\frac{1}{2}\mathbb{N}$ and ends with the conclusion that
$\overline{V}_{k,a_{p}}$ is irregular. (One might also note that in
Proposition 2.2 we could additionally add the condition (d): $\overline{r}$ is
a twist of $\overline{V}_{k,0}$ for some $k$.)

So, we know no counter-examples or counter-arguments for the statement:

($\star$) “For any prime $p$, if $k$ is even and
$v_{p}(a_{p})\not\in{\mathbb{N}}$ or if $k$ is odd and
$v_{p}(a_{p})\not\in\frac{1}{2}{\mathbb{N}}$, then $\overline{V}_{k,a_{p}}$ is
irregular.”

In fact, ($\star$ ‣ 4.2) is supported by theorems of Buzzard and Gee [13,
Theorem 1.6] and Bhattacharya and Ghate [8, Theorem 1.1]. An example theorem
derived from the latter work is that
$\overline{V}_{k,a_{p}}\otimes\omega^{-1}\cong\operatorname{ind}(1)$, as long
as $p\geq 5$ and $1<v_{p}(a_{p})<2$ and $k\equiv p+2\bmod p^{2}(p-1)$. This
nicely complements the discussion in the previous paragraph, further
illustrating why “irregular” replacing “irreducible” is necessary for any
generalization of the integral slope conjecture. We note, finally, that
($\star$ ‣ 4.2) is also consistent with a theorem of Nagel and Pande [20,
Theorem 0.1] focused on $2<v_{p}(a_{p})<3$, and work of Arsovski [2, Theorem
1], which covers a more sporadic range of slopes.

This article began by correcting a false conjecture. So, here, it is best to
wait for a more conceptual explanation before declaring ($\star$ ‣ 4.2) as an
outright conjecture. Even the integral slope conjecture, limited to even
weights and odd $p$, is based on a combination of computational evidence and
theoretical evidence restricted to low weights and slopes. It has not, for
instance, been contextualized within a wider context of all local Galois
representations modulo $p$ nor, except in the above paragraphs, have issues
with odd $k$ or $p=2$ been really confronted. Does replacing reducibility by
regularity offer any conceptual clarity? There’s certainly a fascinating
opportunity here for further explanation.

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