Arithmetic determination of the coefficients $a_{ij}(p)$ of Hecke operators in the level-1 mod-2 Hecke algebra $A\simeq\mathbf{F}_2[[x,y]]$, equivalently the image of the universal mod-2 Galois representation unramified outside 2

Nicolas and Serre (2012) proved that the Hecke algebra $A$ acting on level-1 modular forms modulo 2 of all weights is isomorphic to the formal power series ring $\mathbf{F}_2[[x,y]]$, with $x=T_3$ and $y=T_5$. Every odd-prime Hecke operator therefore has a unique expansion $T_p=\sum_{i+j\ge1}a_{ij}(p)x^iy^j$, and for fixed $(i,j)$ the function $p\mapsto a_{ij}(p)$ is frobenian: its value depends only on the Frobenius substitution of $p$ in a finite Galois extension of $\mathbf{Q}$, unramified outside $\{2\}$, with Galois group a 2-group. Only the first cases are known — $a_{10},a_{01}$ detect $p\bmod 8$, $a_{11}$ detects $p\bmod 16$, and $a_{20},a_{02}$ detect representation of $p$ by $a^2+8b^2$ and $a^2+16b^2$ (in each case with $b$ odd) — with governing fields $\mathbf{Q}(\mu_8)$, $\mathbf{Q}(\mu_{16})$ and two order-8 dihedral extensions. The problem asks for a complete, systematic arithmetic description of $a_{ij}(p)$ for all $(i,j)$, identifying for each $(i,j)$ the finite 2-extension through which it factors; equivalently, via Bellaïche's universal continuous representation $r:G\to\mathrm{SL}_2(A)$ with $\mathrm{tr}\,r(\mathrm{Frob}_p)=T_p$ ($G$ the maximal pro-2 quotient of the Galois group of $\mathbf{Q}$ unramified outside 2 and $\infty$), a determination of the image of $r$. No known result determines the $a_{ij}(p)$ beyond $i+j\le2$ or describes $\mathrm{im}(r)$; the problem is open.

Background

Let $q=e^{2\pi i\tau}$ and let $\Delta(q)=q\prod_{n\ge1}(1-q^n)^{24}=\sum_{n\ge1}\tau(n)q^n$ be Ramanujan's $\Delta$-function, $\tau$ the Ramanujan tau-function. Reducing coefficients modulo 2 gives $\Delta\equiv\sum_{m\ge0}q^{(2m+1)^2}\in\mathbf{F}_2[[q]]$. A classical consequence of the congruences satisfied by $\tau(n)$ mod 2, established by Swinnerton-Dyer and Serre, is that every cuspidal modular form modulo 2 of level 1 (i.e. for the group $\mathrm{SL}_2(\mathbf{Z})$) is a polynomial $f(\Delta)$ over $\mathbf{F}_2$, and that the $q$-expansion of $\Delta^k$ is supported on exponents $n\equiv k\pmod 8$.

For an odd prime $p$, the Hecke operator $T_p$ acts on $f=\sum c_nq^n$ by $(T_pf)=\sum_n\gamma_nq^n$ with $\gamma_n=c_{pn}$ if $p\nmid n$ and $\gamma_n=c_{pn}+c_{n/p}$ if $p\mid n$. Modulo 2 each $T_p$ strictly lowers the degree in $\Delta$, so every $T_p$ is nilpotent on each finite-dimensional space of forms; Nicolas and Serre determined the exact order of nilpotence in a companion Note. Let $F\subset\mathbf{F}_2[[q]]$ be the span of the odd powers $\Delta,\Delta^3,\Delta^5,\dots$, let $F(n)$ be the span of $\Delta,\Delta^3,\dots,\Delta^{2n-1}$, and let $A(n)\subset\mathrm{End}(F(n))$ be the $\mathbf{F}_2$-algebra generated by all $T_p$ with $p$ odd. The projective limit $A=\varprojlim A(n)$ is the big level-1 mod-2 Hecke algebra (called shallow, because it is built from the spaces of forms of all weights, in contrast to the deep Hecke algebra acting on forms of a single fixed weight), a compact local ring acting on $F$.

Nicolas and Serre proved that $A$ is isomorphic to the two-variable formal power series ring $\mathbf{F}_2[[x,y]]$, the isomorphism sending $x\mapsto T_3$ and $y\mapsto T_5$; in particular $A$ is a regular local ring of dimension 2 and an integral domain, and it is topologically generated by any pair $(T_p,T_{p'})$ with $p\equiv3$, $p'\equiv5\pmod 8$. Consequently every Hecke operator has a unique expansion $T_p=\sum_{i+j\ge1}a_{ij}(p)x^iy^j$ with $a_{ij}(p)\in\mathbf{F}_2$. The same paper establishes the main structural facts about the modules: $F^\ast$ is a free $A$-module of rank one, $F$ is the module of continuous linear forms on $A$ and the injective envelope of the residue field $\mathbf{F}_2$ (the maximal injective module containing the residue field, in the sense of module theory), and there is a canonical basis $m(a,b)$ of $F$ adapted to $T_3$ and $T_5$.

A function from the primes to a finite set is called frobenian (in the terminology of Serre's book Lectures on $N_X(p)$) if its value at $p$ depends only on the Frobenius conjugacy class $\mathrm{Frob}_p$ in some finite Galois extension $K/\mathbf{Q}$, for every $p$ unramified in $K$; equivalently, it extends to a locally constant class function on the absolute Galois group. For fixed $(i,j)$ the function $p\mapsto a_{ij}(p)$ is frobenian; more precisely, its value depends only on $\mathrm{Frob}_p$ in a finite Galois extension of $\mathbf{Q}$, unramified outside $\{2\}$, whose Galois group is a 2-group (a finite group of order a power of 2). Only the initial cases are explicit, with their governing fields: $a_{10}(p)=1\iff p\equiv3\pmod8$ and $a_{01}(p)=1\iff p\equiv5\pmod8$ (field $\mathbf{Q}(\mu_8)$, the cyclotomic field of 8th roots of unity); $a_{11}(p)=1\iff p\equiv7\pmod{16}$ (field $\mathbf{Q}(\mu_{16})$); and $a_{20}(p)=1$, resp. $a_{02}(p)=1$, if and only if $p$ is of the form $a^2+8b^2$, resp. $a^2+16b^2$, with $a,b\in\mathbf{Z}$ and $b$ odd, the governing fields being order-8 dihedral extensions obtained from $\mathbf{Q}(\mu_8)$ by adjoining square roots of $1+i$ and of $\sqrt2$. One also knows that $T_p$ lies in $\mathbf{F}_2[[x^2,y^2]]$, $x\cdot\mathbf{F}_2[[x^2,y^2]]$, $y\cdot\mathbf{F}_2[[x^2,y^2]]$ or $xy\cdot\mathbf{F}_2[[x^2,y^2]]$ according as $p\equiv1,3,5,7\pmod8$.

Bellaïche, answering a question of Serre, constructed the Galois-theoretic object governing all these coefficients at once. Let $\mathbf{Q}_{\{2,\infty\}}$ be the maximal extension of $\mathbf{Q}$ unramified outside 2 and $\infty$ (where $\infty$ denotes the archimedean place, so only the prime 2 and infinity may ramify), and let $G$ be the maximal pro-2 quotient (inverse limit of finite 2-groups) of $\mathrm{Gal}(\mathbf{Q}_{\{2,\infty\}}/\mathbf{Q})$. Bellaïche proved that there exists a unique continuous representation $r:G\to\mathrm{SL}_2(A)$ with $\mathrm{tr}\,r(\mathrm{Frob}_p)=T_p$ for every odd prime $p$; that $A$ represents the functor of continuous 2-dimensional determinants (pseudo-representations, i.e. pairs $(t,d)$ satisfying the algebraic identities of trace and determinant of a 2-dimensional representation) of $G$ with $d=1$; and that the $T_p$ are algebraically independent over $\mathbf{F}_2$, which answers negatively the question whether $T_p$ for $p>5$ could be a polynomial in $x,y$. The residual representation modulo the maximal ideal $(x,y)$ is the nontrivial extension of the trivial character by itself attached to $\mathbf{Q}(\sqrt2)$; at the height-one primes $(x)$ and $(y)$ (the minimal nonzero prime ideals of the two-dimensional ring $A$) the image of $r$ is dihedral, and there is exactly one exceptional height-one prime $\mathfrak{p}_0=y+\tau_{\log 5/\log(-3)}(x)$ (a formal series defined through a 2-adic exponent) at which reducibility after a ramified quadratic extension of the coefficient field occurs.

What is not known is the arithmetic content of the coefficients. Beyond the five cases listed above there is no explicit description of $a_{ij}(p)$, no identification of the finite extensions through which $p\mapsto a_{ij}(p)$ factors, and no determination of the image $r(G)\subseteq\mathrm{SL}_2(A)$, which is only constrained, not described. The difficulty is that the governing fields plausibly range through ever deeper layers of the maximal pro-2 extension of $\mathbf{Q}$ unramified outside 2 — the known cases already require $\mathbf{Q}(\mu_{16})$ and the splitting fields of $X^4-2$ — while the algebraic independence of the $T_p$ rules out any finite polynomial parametrization of the coefficients. On the combinatorial side, the coefficients are effectively computable: Nicolas and Serre give a linear recursion (via symmetric polynomials $F_p$ computed by solving a linear system over $\mathbf{F}_2[t]$) that determines $T_p|\Delta^k$, from which the $a_{ij}(p)$ with $i+j$ bounded can be read off by duality; extensive tables exist for $p\le257$.

Problem Statement

Let $A\simeq\mathbf{F}_2[[x,y]]$ (with $x=T_3$, $y=T_5$) be the level-1 mod-2 Hecke algebra of Nicolas-Serre, and for each odd prime $p$ write the unique expansion $T_p=\sum_{i+j\ge1}a_{ij}(p)x^iy^j$. Determine, for every pair $(i,j)$ with $i+j\ge1$, the frobenian function $p\mapsto a_{ij}(p)$ explicitly and arithmetically. A complete solution must provide, for each $(i,j)$ — or via an effective procedure that takes $(i,j)$ as input and terminates — (1) a finite Galois extension $E_{ij}/\mathbf{Q}$, unramified outside $\{2\}$, whose Galois group is a 2-group, through which the function factors; and (2) the class function $\varphi_{ij}$ on $\mathrm{Gal}(E_{ij}/\mathbf{Q})$ such that $a_{ij}(p)=\varphi_{ij}(\mathrm{Frob}_p)$ for every odd prime $p$ unramified in $E_{ij}$; together with a proof of these identities for all $(i,j)$ and all odd $p$. Equivalently, by Bellaïche's reformulation, a complete solution may instead determine the image of the universal continuous representation $r:G\to\mathrm{SL}_2(A)$, $\mathrm{tr}\,r(\mathrm{Frob}_p)=T_p$, as an explicit closed subgroup of $\mathrm{SL}_2(A)$ from which the finite quotient of $G$ through which each coordinate of $\mathrm{tr}\,r(\mathrm{Frob}_p)$ factors can be read off. Any answer must reproduce the known initial cases: $a_{10}(p)=1\iff p\equiv3\pmod8$; $a_{01}(p)=1\iff p\equiv5\pmod8$; $a_{11}(p)=1\iff p\equiv7\pmod{16}$; $a_{20}(p)=1\iff p=a^2+8b^2$ with $a,b\in\mathbf{Z}$, $b$ odd; $a_{02}(p)=1\iff p=a^2+16b^2$ with $a,b\in\mathbf{Z}$, $b$ odd; together with their governing fields $\mathbf{Q}(\mu_8)$, $\mathbf{Q}(\mu_{16})$ and the two order-8 dihedral extensions of $\mathbf{Q}(\mu_8)$; and it must be compatible with the fact that $T_p$ belongs to $\mathbf{F}_2[[x^2,y^2]]$, $x\cdot\mathbf{F}_2[[x^2,y^2]]$, $y\cdot\mathbf{F}_2[[x^2,y^2]]$ or $xy\cdot\mathbf{F}_2[[x^2,y^2]]$ according as $p\equiv1,3,5,7\pmod8$.

References

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