Small normalized discriminants for non-solvable Galois extensions of $\mathbb{Q}$ ramified at a single prime: extending the Dembélé–Serre analysis beyond $p = 2$

Dembélé (2009) constructed a non-solvable Galois extension of $\mathbb{Q}$ ramified only at $2$ whose root discriminant is at most $2^{1518251/262144} \approx 55.394$, a bound refined by Serre through a sharp conductor-filtration analysis of the character group of the local wild extension. Gross's conjecture — that every prime $p$ admits such an extension — was subsequently resolved ($p = 3$ and 5 by Dembélé–Greenberg–Voight in 2011, $p = 7$ by Dieulefait in 2012), and the quantitative side has not seen a comparable sharpening: for $p = 3$ the best proven root discriminant bound is $3^{22/9+3/2} \approx 76.21$, for $p = 5$ the best known bound is $5^{17/10+5/4} \approx 115.34$, and for $p = 7$ no estimate exists at all, all obtained by cruder Fontaine–Moon-type local estimates. The problem is to construct non-solvable Galois extensions of $\mathbb{Q}$ ramified only at a single prime $p \ge 3$ (or at $2$ with other local decomposition types) whose normalized discriminant is comparably small — improving the quoted bounds and supplying the missing $p = 7$ estimate — or dually to prove lower bounds for such discriminants beyond the Odlyzko bounds, which are blind to non-solvability.

Background

A number field $K$ of degree $n$ has a discriminant $d_K$ whose prime factors are exactly the rational primes that ramify in $K$, i.e. that divide at least one of the ramification indices. The root (or normalized) discriminant $\delta_K = |d_K|^{1/n}$ measures ramification per degree and is the quantity constrained by the Odlyzko discriminant bounds: analytic lower bounds on $\delta_K$ that grow with $n$, e.g. $\delta_K$ is eventually at least $8\pi e^{\gamma} \approx 44.76$ for totally complex fields (fields with no real embedding) and $8\pi e^{\gamma+\pi/2} \approx 215.33$ for totally real fields under the Generalized Riemann Hypothesis (GRH), where $\gamma$ is the Euler–Mascheroni constant. A Galois extension $K/\mathbb{Q}$ is called solvable or non-solvable according to its Galois group; non-solvable means the group has a non-abelian simple composition factor, which forces very large degree.

Restricting the ramification locus is a classical refinement of the inverse Galois problem. Gross's conjecture, recorded in Dembélé's paper and in Dembélé–Greenberg–Voight (attributed there to B. Gross), states that for every prime $p$ there is a finite non-solvable Galois extension of $\mathbb{Q}$ ramified only at $p$. For $p \ge 11$ this follows from residual (mod $p$) representations attached to level-1 classical modular forms; for $p \le 7$ the direct route is blocked because there are no irreducible odd 2-dimensional mod-$p$ representations of the absolute Galois group $G_{\mathbb{Q}}$ unramified outside $p$ (a consequence of the proof of the first case of Serre's modularity conjecture). The successful constructions come instead from automorphic forms over larger totally real fields or of higher dimension. Dembélé (2009) used mod-2 representations attached to Hilbert modular forms — automorphic forms for $\mathrm{GL}_2$ over a totally real field, computed via Hecke operators on quaternionic cohomology — of parallel weight 2 over the degree-8 field $\mathbb{Q}(\zeta_{32})^+$: the residual representation has image containing $8 \cdot \mathrm{SL}_2(\mathbb{F}_{2^8})^2$, and the field $K$ cut out by its kernel is Galois over $\mathbb{Q}$, non-solvable, ramified only at $2$, of degree $2^{19}(3 \cdot 5 \cdot 17 \cdot 257)^2 \approx 2.25 \times 10^{15}$, and totally complex. Dembélé–Greenberg–Voight (2011) did the same over $\mathbb{Q}(\zeta_{27})^+$ for $p = 3$ and over the degree-5 subfield of $\mathbb{Q}(\zeta_{25})$ for $p = 5$, and Dieulefait (2012) used a mod-7 representation attached to a genus-2 Siegel cusp form of level 1 and weight 28 with projective image $\mathrm{PGSp}(4,7)$ for $p = 7$. This resolves the existence part of Gross's conjecture for all primes. Throughout, the local decomposition type of a prime $p$ in such an extension $K$ is the isomorphism class of the local extension $K_{\mathfrak{p}}/\mathbb{Q}_p$ (for $\mathfrak{p}$ a prime of $K$ above $p$), equivalently of the decomposition group at $p$ together with its ramification filtration.

The quantitative refinement asks how small the ramification can be. By the conductor–discriminant formula, the discriminant of an abelian extension equals the product of the conductors (Artin conductors) of its characters, and discriminants of general Galois extensions decompose along the filtration of the Galois group; controlling the conductors of the characters of the wildly ramified local extension at $p$ controls $\delta_K$. In his complement to Dembélé's note, Serre carried this out sharply for the $p = 2$ example: the character group $X$ of the local multiquadratic extension sits in $Y^- = L^{\times}/L^{\times 2}$ ($L$ an abelian subextension of the local closure), which is an 8-dimensional vector space over $\mathbb{F}_4$ carrying a complete conductor filtration whose successive quotients have conductors $16 - 2i$; maximizing the resulting expression gives $\delta_K \le 2^{1518251/262144} \approx 55.394388$, remarkably close to the GRH lower bound 44.76. For $p = 3$ and 5, Dembélé–Greenberg–Voight instead used cruder local tools — Fontaine's corollary of his theorem that there is no abelian variety over $\mathbb{Z}$, applicable when the residual representation is peu ramifiée (mildly ramified) at $p$, and Moon's lemma for the très ramifiée (strongly ramified) case — yielding $\delta_K \le 3^{22/9+3/2} \approx 76.21$ at $p = 3$ and $\delta_K \le 5^{17/10+5/4} \approx 115.34$ at $p = 5$; Roberts computed the exact root discriminant $5^{3-1/12500} \approx 124.984$ of an explicit degree-25 polynomial generating one of the $p = 5$ closures. For Dieulefait's $p = 7$ field no root discriminant estimate is known.

This problem sits inside the Odlyzko–Martinet program on how small root discriminants can be. Its companion question — how small the root discriminant of an infinite tower of number fields can be — has records 78.4269 (totally complex) and 857.5662 (totally real) due to Hajir–Maire–Ramakrishna, against the GRH lower bounds 44.76 and 215.33; Dembélé asked whether infinite totally complex towers with root discriminant near 55 exist. On the complementary side, exclusion results (Jones, Hoelscher, Pollak, Ghitza–Yamauchi) restrict which groups and which low-dimensional mod-$p$ representations can occur unramified outside a single small prime, delimiting the possible routes to better constructions.

Problem Statement

Let $\delta_{\min}(p)$ denote the infimum of the root discriminants $\delta_K = |d_K|^{1/[K:\mathbb{Q}]}$ over finite non-solvable Galois extensions $K/\mathbb{Q}$ ramified only at the prime $p$; for $p = 2$ this infimum is at most $2^{1518251/262144} \approx 55.394388$ (Serre's bound for Dembélé's field), and all known examples are totally complex.

(1) Construct, for the remaining primes — concretely $p = 3, 5, 7$, and more generally for other local decomposition types (including at $p = 2$) — non-solvable Galois extensions of $\mathbb{Q}$ ramified only at $p$ whose root discriminant is comparably small, i.e. improve the best published bounds: $\delta_K \le 3^{22/9+3/2} \approx 76.21$ at $p = 3$ (the level-1 field of Dembélé–Greenberg–Voight), $\delta_K \le 5^{17/10+5/4} \approx 115.34$ at $p = 5$ (the degree-10 family of Dembélé–Greenberg–Voight; Roberts' explicit degree-25 field attains the exact value $5^{3-1/12500} \approx 124.984$), and supply the first proven estimate at $p = 7$ (Dieulefait's $\mathrm{PGSp}(4,7)$-extension) of the same order of magnitude as the $p = 3$ and $p = 5$ bounds rather than a crude Fontaine–Moon-type estimate; the same magnitude requirement applies to the general branch (other primes $p$ and other local decomposition types, including at $p = 2$).

(2) Dually, prove lower bounds for $\delta_{\min}(p)$ that use non-solvability — the Odlyzko bounds apply to all fields and are blind to the Galois group — or show that the $p = 2$ magnitude cannot be achieved at other primes.

Any single proved result of the form described in (1) or (2) — one improved bound in (1), the missing $p = 7$ estimate, or one lower-bound statement in (2) — resolves the task. The items listed in (1) are independent sub-goals rather than a joint requirement.

Two adjacent questions recorded in the same sources remain open and delimit this cluster: whether a totally real non-solvable extension of $\mathbb{Q}$ ramified only at one prime exists (Dembélé–Greenberg–Voight), and whether infinite towers of totally complex fields with root discriminant near 55 exist (Dembélé; current record 78.4269).

References

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