==== TABLE OF CONTENTS ====
The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level//main.tex
The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level//references.bib
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==== BEGIN The-Dimension-of-the-Two-Adic-Hecke-Algebra-at-Odd-Level//main.tex ====
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\hypersetup{pdftitle={The Dimension of the Two-Adic Hecke Algebra at Odd Level},pdfauthor={OpenAI}}
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\title{The Dimension of the Two-Adic Hecke Algebra\\at Odd Level}
\author{OpenAI}
\date{October 5, 2026}
\begin{document}
\maketitle
\begin{abstract}
For every odd positive integer $N$, every irreducible component of the
full two-adic Hecke algebra of level $\Gamma_1(N)$ has Krull dimension
four. This proves the $p=2$ case of Emerton's dimension conjecture,
including all residual components.
\end{abstract}
\section{Introduction}
\label{sec:introduction}
The congruences between classical modular forms of different weights
assemble their Hecke eigenvalues into a single $p$-adic algebra. Its
spectrum is larger than an individual family of eigenforms: many such
families can pass through the same classical point. The dimension of
this spectrum measures how much variation is present when no slope or
residual irreducibility condition is imposed.
Fix an odd positive integer $N$. For $i\geq1$, let $M_i(N)$ be the
complex vector space of modular forms of weight $i$ and level
$\Gamma_1(N)$, including Eisenstein series. For every prime $\ell\nmid N$,
write $T_\ell$ for the usual Hecke operator and let $S_\ell$ act on
$M_i(N)$ as $\ell^{i-2}\langle\ell\rangle$, where
$\langle\ell\rangle$ is the diamond operator. Let
\[
T_{\leq k}^{(2)}(N)
\subseteq \End_{\C}\!\left(\bigoplus_{i=1}^k M_i(N)\right)
\]
be the $\Z$-algebra generated by $T_\ell$ and $\ell S_\ell$ for
$\ell\nmid2N$. Set
\begin{equation}\label{eq:algebra}
A_k=\Z_2\otimes_{\Z}T_{\leq k}^{(2)}(N),
\qquad A=T_2(N)=\varprojlim_k A_k,
\end{equation}
with transition maps given by restriction. The topology on $A$ is the
inverse-limit topology, with the $2$-adic topology on each $A_k$.
All dimensions below are Krull dimensions.
\begin{theorem}\label{thm:main}
For every odd positive integer $N$, every irreducible component of
$\Spec T_2(N)$ has dimension exactly $4$.
\end{theorem}
This is the $p=2$ case of Emerton's Conjecture~2.9
\cite{Emerton11}. The algebra in \eqref{eq:algebra} retains all weights,
all Eisenstein systems, and all residual systems of eigenvalues. In
particular, the theorem does not require a residual Galois representation
to be irreducible, nonscalar, or distinguished at $2$.
\subsection*{History and the main input}
Hida's ordinary families \cite{Hida86} and the eigencurve of Coleman and
Mazur \cite{ColemanMazur98} give fundamental constructions of $p$-adic
families of modular eigenforms. Gouv\^ea and Mazur's infinite fern
\cite{GouveaMazur98} explains how families intersect and produce
higher-dimensional Zariski closures. Emerton's account
\cite{Emerton11} formulates the dimension question for the full
prime-to-$Np$ Hecke algebra, without restricting to one residual
deformation problem. His Corollary~2.28 gives the lower bound needed
here: every irreducible component has dimension at least $4$.
The complementary upper bound is a deformation-theoretic question.
Mazur's deformation theory \cite{Mazur89} relates infinitesimal
deformations of a Galois representation to its adjoint cohomology.
For automorphic representations, the finite local conditions of
Bloch and Kato \cite{BlochKato90} isolate a Selmer group within this
cohomology. Under suitable hypotheses, Kisin \cite{Kisin04} proved
geometric adjoint Selmer vanishing and applied it to local deformation
and eigencurve geometry. Allen \cite{Allen16} established adjoint Selmer vanishing
under residual hypotheses and related local genericity to unobstructed
deformations. Newton and Thorne \cite{NewtonThorne23} subsequently
proved the vanishing theorem used here without the residual restrictions
that would obstruct its application to all components at $2$.
We use their Theorem~5.4 for classical cuspidal forms over $\Q$.
The theorem includes $p=2$ and also includes a CM form provided its
CM field is not contained in the cyclotomic $2$-power extension.
Odd level guarantees this last condition. The result concerns the
\emph{full} adjoint representation, of dimension four, which is the
one required when the determinant is allowed to vary.
\subsection*{Method and organization}
The proof works at characteristic-zero points, even on components
whose residual systems are reducible or scalar. There are three steps.
First, the integral algebra is a finite product of complete Noetherian
local rings, and its classical points are Zariski dense. At $2$ we
justify Noetherianity using Chenevier's determinant laws
\cite{Chenevier14}, avoiding division by $2$ in the integral
pseudorepresentation. Second, the Eisenstein locus has dimension at
most $2$. Together with Emerton's lower bound, this supplies a classical
cuspidal point of weight at least $3$ on every component. Third, the
tangent space at such a point has dimension at most $3$: the
Newton--Thorne theorem annihilates the global finite Selmer group,
and the only nonzero local quotient has dimension $3$ at $2$.
A dimension formula then adds the one-dimensional arithmetic quotient.
Two passages in this argument are useful beyond the immediate
application. Section~\ref{sec:eisenstein} constructs the Eisenstein
interpolation maps directly from the inverse-limit Hecke algebra by a
compact graph argument. Section~\ref{sec:tangent} gives a
characteristic-zero tangent bound for a compact Noetherian algebra
carrying a two-dimensional determinant. In particular, it proves the
continuity of the derivations that occur in its algebraic tangent
space. Neither passage assumes residual irreducibility.
We supply these arguments in detail. The deep external inputs are
the classical attachment and local compatibility of Galois
representations, Emerton's lower bound, Chenevier's determinant
theorems, and Newton--Thorne's Selmer vanishing. Their relevant
forms and hypotheses are stated where they are used.
\section{The integral algebra and its Galois determinant}
\label{sec:integral}
Our first goal is to put the inverse limit \eqref{eq:algebra} in the
category of Noetherian rings while retaining all of its classical
systems. Write $S$ for the set of finite primes dividing $2N$, and
let $G=G_{\Q,S}$ be the Galois group of the maximal extension of $\Q$
unramified at finite primes outside $S$. Ramification at infinity is
allowed. All Galois cohomology will be continuous cohomology.
\subsection{Finite stages and classical systems}
We use the classical rationality, integrality, and simultaneous
semisimplicity of the prime-to-level Hecke action; a convenient
statement including weight one and Eisenstein forms is
\cite[Proposition~1.12 and Remark~1.13]{Emerton11}. For cusp forms,
old copies have the same good-prime eigenvalues as their associated
newforms. On the Eisenstein subspace the eigenvalues are sums and
products of Dirichlet character values and powers of primes.
Consequently, for every nonzero finite stage there is an embedding
\begin{equation}\label{eq:finite-embedding}
A_k\lhook\joinrel\longrightarrow
\prod_{j=1}^{m_k}\OO_{E_{k,j}},
\end{equation}
where each $E_{k,j}/\Q_2$ is finite and the coordinate maps are
classical eigenvalue systems, with all necessary $2$-adic embeddings
included. Indeed, before tensoring with $\Z_2$ the algebra embeds
into a finite product of rings of integers of number fields. It is
therefore finite and torsion-free over $\Z$; flatness of $\Z_2$
preserves the embedding. Thus $A_k$ is finite free over $\Z_2$, and
its image in \eqref{eq:finite-embedding} is closed. Zero initial
stages, if present, can be discarded.
A \emph{classical point} of $\Spec A$ means the kernel of a
classical eigenvalue map $A\to\OO_E$ obtained from some finite
stage. The following consequences will be used repeatedly.
\begin{lemma}\label{lem:density}
The ring $A$ is compact, Hausdorff, and reduced. Its classical points
are Zariski dense, and the operators $T_\ell,\ell S_\ell$,
$\ell\nmid2N$, topologically generate it over $\Z_2$.
\end{lemma}
\begin{proof}
The restriction maps between finite-stage algebras are surjective:
the generators at the smaller stage are restrictions of the same
generators at the larger stage. Compactness and Hausdorffness follow
from \eqref{eq:algebra}. Reducedness follows from the injections
\eqref{eq:finite-embedding}. Those injections also show that the
intersection of all classical kernels in $A$ is zero, which is
equivalent to the asserted density. Finally, the $\Z_2$-algebra of
polynomials in the indicated generators surjects onto each $A_k$,
and is therefore dense in the inverse limit.
\end{proof}
For every classical system, the theorems of Deligne and
Deligne--Serre \cite{Deligne71,DeligneSerre74} give a continuous
two-dimensional representation of $G$ over a finite extension of
$\Q_2$; in the Eisenstein case one uses a direct sum of characters.
Choose the Frobenius convention so that its characteristic polynomial
at $\ell\nmid2N$ is
\begin{equation}\label{eq:frobenius}
Z^2-T_\ell Z+\ell S_\ell
\end{equation}
after specialization. A stable lattice ensures integral traces and
determinants at every element of $G$.
\subsection{An integral determinant law}
A two-dimensional \emph{determinant} over a commutative ring $R$ is
a unital multiplicative homogeneous polynomial law of degree two
$R[G]\to R$, compatible with extension of scalars; see
\cite[Section~1]{Chenevier14}. For a determinant $D$, its trace
and determinant on a group element $g$ are defined by
$D(Z-g)=Z^2-t(g)Z+d(g)$. This notion works over rings
in which $2$ is not invertible.
\begin{lemma}\label{lem:determinant}
There is a continuous two-dimensional determinant over $A$ whose
trace and determinant satisfy
\[
t(\Frob_\ell)=T_\ell,\qquad
d(\Frob_\ell)=\ell S_\ell\qquad(\ell\nmid2N).
\]
\end{lemma}
\begin{proof}
At a fixed finite stage, take the tuple of traces and determinants
of its classical Galois representations. The tuple belongs to the
image of \eqref{eq:finite-embedding} on Frobenius conjugacy classes,
by \eqref{eq:frobenius}. The union of these classes is dense in $G$
by Chebotarev. The tuples are continuous class functions, and the
image of \eqref{eq:finite-embedding} is closed. Hence the tuples
belong to $A_k$ at every $g\in G$. Uniqueness on the dense union of
Frobenius classes makes these functions compatible as $k$ varies.
They give continuous functions $t,d$ with values in $A$.
For completeness, the determinant law itself can be constructed
without recovering it by division by $2$. On a finite formal sum
$\sum_g x_g g$, prescribe the quadratic polynomial
\begin{equation}\label{eq:quadratic-law}
\sum_g d(g)x_g^2+
\sum_{g