The Charollois–Darmon conjecture: Stark unit arguments as periods of Hilbert Eisenstein series for ATR fields (Conjecture 4.1, 2008)

For a quadratic extension $K/F$ of a totally real field $F$ of degree $n\geq 2$ in which $K$ is almost totally real (ATR: exactly one complex place), Charollois and Darmon (2008) defined analytic invariants $J_\tau\in\mathbb{C}/\Lambda'_{\mathrm{Eis}}$ by integrating weight-two Eisenstein series on $\mathrm{GL}_2(F)$ over homologically trivial cycles on the Hilbert modular variety of $F$ , and conjectured (Conjecture 4.1) that $J_\tau$ — regarded modulo the larger lattice $\Lambda''_{\mathrm{Eis}}\supseteq\Lambda'_{\mathrm{Eis}}$ — pins the argument of a Stark unit $u_\tau$ in the class field $H_I$ of $K$ , with prescribed Galois equivariance. This conjecture remains unproven. The period lattice $\Lambda''_{\mathrm{Eis}}$ is fully defined in the source; the source's numerical sections additionally leave a period-lattice parameter $m_F$ only empirically determined ( $15$ for $\mathbb{Q}(\sqrt{5})$ and $6$ for $\mathbb{Q}(\sqrt{2})$ ) and give an algorithm computing $J_\tau$ only for $n=2$ , but neither auxiliary matter is part of the conjecture. The problem is a concrete analytic attack on Hilbert's twelfth problem for fields with exactly one complex place, where no explicit class field theory is known in any form.

Background

Hilbert's twelfth problem asks for explicit analytic constructions of the abelian extensions of a number field, in the way the exponential function generates $\mathbb{Q}^{\mathrm{ab}}$ (Kronecker–Weber) and elliptic functions with complex multiplication (CM) generate abelian extensions of imaginary quadratic fields. Beyond CM and the recently settled $p$ -adic theory for totally real fields, the next natural test case consists of ATR fields ("almost totally real"), i.e. fields $K$ with exactly one pair of complex embeddings and all other embeddings real; for these no explicit class field theory is known.

Stark's conjectures predict, in the rank-one situation where the relevant $L$ -function — an Artin $L$ -function (the Dirichlet- $L$ -series analogue attached to a Galois character), here a partial zeta function of $K$ attached to a class of the relevant class group — has a simple zero at $s=0$ , the existence of a "Stark unit" whose logarithms at distinguished places are the leading terms $L'(\chi,0)$ . Let $F$ be a totally real field of degree $n\geq 2$ with narrow class number one (the narrow class group identifies ideals differing by a principal ideal with a totally positive generator, so it takes the signs at the real places into account), and let $K/F$ be a quadratic ATR extension: the first archimedean place $v_1$ of $F$ becomes complex in $K$ , while $v_2,\dots,v_n$ stay real, so $K$ is ATR. For an ideal $I$ of $\mathcal{O}_F$ , set $\mathcal{O}_I=\mathcal{O}_F+I\mathcal{O}_K$ , an order of conductor $I$ in $K$ , and let $H_I$ be the class field of $K$ attached to $\mathcal{O}_I$ , i.e. the abelian extension that class field theory canonically associates with the generalized ideal class group of the order $\mathcal{O}_I$ (for $I=\mathcal{O}_F$ , the Hilbert class field in the restricted sense). Stark's conjecture in this setting predicts, for each ideal class of the restricted class group $G_I=\mathrm{Gal}(H_I/K)$ (the class group of $\mathcal{O}_I$ in which ideals differing by a principal fractional ideal with totally positive generator are identified), a unit $u$ whose image under a place $\tilde v_1$ of $H_I$ above $v_1$ has a prescribed complex logarithm $\log\tilde v_1(u)$ , and whose image under every complex embedding above $v_2,\dots,v_n$ has absolute value $1$ ; but it determines only the real part of $\log\tilde v_1(u)$ , hence only the absolute value of the unit, leaving its argument (phase) undetermined. Numerical work on the rank-one Stark conjecture over complex cubic fields (Dummit–Tangedal–van Wamelen, 2004) and Dasgupta's senior thesis (1999) made this phase ambiguity explicit: there was no candidate formula for the argument of the expected Stark unit.

Charollois and Darmon (2008) proposed such a formula. On the Hilbert modular variety $X$ attached to $\mathrm{GL}_2(F)$ (the quotient of $\mathbb{H}^n$ , $n$ copies of the complex upper half-plane, by the Hilbert modular group $\Gamma=\mathrm{SL}_2(\mathcal{O}_F)$ ), the weight-two Eisenstein series — the explicitly constructible non-cuspidal Hilbert modular forms, obtained by averaging translates of a boundary term — define a closed $\Gamma$ -invariant $n$ -form $\omega_{\mathrm{Eis}}$ . To a projective $\mathcal{O}_I$ -module $M\subset K$ one associates a point $\tau=\omega_2/\omega_1\in K$ (for a positive $\mathcal{O}_F$ -basis $\omega_1,\omega_2$ of $M$ ) and a closed $(n-1)$ -cycle $\Delta_\tau$ on $X$ , homologically trivial, built from the geodesic joining the two real embeddings of $\tau$ at the places $v_2,\dots,v_n$ that survive in $K$ . Integrating $\omega_{\mathrm{Eis}}$ over an $n$ -chain with boundary $n_F\Delta_\tau$ (where $n_F$ is an explicit integer) defines a formal analogue of the Abel–Jacobi map on algebraic cycles (classically: the map sending a homologically trivial cycle to the integrals of holomorphic forms over it, modulo periods),

$\Psi_{\mathrm{Eis}}:\tilde C^0_{n-1}(X)\longrightarrow\mathbb{C}/\Lambda'_{\mathrm{Eis}},$

well defined modulo the lattice $\Lambda'_{\mathrm{Eis}}$ of Eisenstein periods, and

$J_\tau:=e_I(-2)^{n-1}\Psi_{\mathrm{Eis}}(\Delta_\tau)\in\mathbb{C}/\Lambda'_{\mathrm{Eis}},$

an element that the main conjecture regards modulo the larger lattice $\Lambda''_{\mathrm{Eis}}\supseteq\Lambda'_{\mathrm{Eis}}$ , where $e_I,\delta_I$ are explicit normalizing constants attached to the order $\mathcal{O}_I$ and $\Lambda''_{\mathrm{Eis}}$ is the lattice in the real line $(2i\pi)^n\mathbb{R}$ generated by $\Lambda'_{\mathrm{Eis}}$ and $(2i\pi)^n\mathbb{Z}$ . The authors prove that their construction is compatible with Stark's conjecture — it recovers the predicted absolute values — and that it is a genuine refinement, since the congruence class of $J_\tau$ also encodes the argument of the expected unit. In the case $n=2$ they give an algorithm computing $J_\tau$ via an explicit Sczech-type group $2$ -cocycle (a cocycle evaluating Eisenstein periods by finite continued-fraction-style sums), and they verify numerically, for $F=\mathbb{Q}(\sqrt{5})$ and $F=\mathbb{Q}(\sqrt{2})$ and several ATR quartic $K/F$ with restricted class groups $C_4$ , $C_2\times C_4$ , and $C_8$ , that exponentials of the computed invariants coincide to high precision with roots of the minimal polynomials of units in $H_I$ , under the empirically suggested optimal period-lattice constants $m_{\mathbb{Q}(\sqrt{5})}=15$ and $m_{\mathbb{Q}(\sqrt{2})}=6$ , where $m_F$ is a period-lattice parameter introduced in the source's numerical sections under the working assumption $\Lambda''_{\mathrm{Eis}}=\Lambda'_{\mathrm{Eis}}$ (its optimal values are only conjectured there, not proven).

Problem Statement

Prove Conjecture 4.1 of Charollois–Darmon (2008) in the source's own generality. Let $F$ be a totally real field of degree $n\geq 2$ with narrow class number one, let $K/F$ be a quadratic ATR extension (the place $v_1$ of $F$ becomes complex in $K$ , the places $v_2,\dots,v_n$ stay real), let $\mathcal{O}_I=\mathcal{O}_F+I\mathcal{O}_K$ be the order of conductor $I$ with class field $H_I$ of $K$ , and let $\Lambda''_{\mathrm{Eis}}\subset(2i\pi)^n\mathbb{R}$ be the Eisenstein period lattice generated by $\Lambda'_{\mathrm{Eis}}$ and $(2i\pi)^n\mathbb{Z}$ , attached to the weight-two Eisenstein $n$ -form $\omega_{\mathrm{Eis}}$ on the Hilbert modular variety of $F$ , all as defined in the source.

Show that for every projective $\mathcal{O}_I$ -module $M$ attached to $\tau\in K$ there exists a unit $u_\tau\in\mathcal{O}_{H_I}^{\times}$ such that

$J_\tau\equiv 2\delta_I(2i\pi)^{n-1}\log(\tilde v_1(u_\tau))\pmod{\Lambda''_{\mathrm{Eis}}},$

that the image of $u_\tau$ under every complex embedding of $H_I$ above $v_j$ has absolute value $1$ for each $2\le j\le n$ , and that for every $\alpha\in G_I=\mathrm{Gal}(H_I/K)$ the equivariance $u_{\tau^\alpha}=\mathrm{rec}(\alpha)^{-1}u_\tau$ holds, where $\mathrm{rec}$ is the Artin reciprocity map (the canonical isomorphism identifying the restricted class group $G_I$ with $\mathrm{Gal}(H_I/K)$ ) and $\tau^\alpha$ is the point attached to the conjugate module $M^\alpha$ .

The scientific objective is to turn the conjectural analytic construction of Stark units — arguments included, not just absolute values — in abelian extensions of ATR fields into a theorem, in the spirit of Hilbert's twelfth problem.

References

Pierre Charollois, Henri Darmon, Arguments des unités de Stark et périodes de séries d'Eisenstein, Algebra & Number Theory 2 (2008), no. 6, 655–688. DOI: 10.2140/ant.2008.2.655, https://doi.org/10.2140/ant.2008.2.655
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