Hilbert's twelfth problem for totally real fields: complex-analytic generation of the maximal abelian extension

For the rational numbers, the Kronecker–Weber theorem realizes every abelian extension inside cyclotomic fields, and for imaginary quadratic fields the theory of complex multiplication (values of modular functions) plays the same role; these are the classical solutions of Hilbert's twelfth problem. Dasgupta and Kakde (2021/2024) proved the Brumer–Stark conjecture away from 2 and the $p$ -part of the integral Gross–Stark conjecture, and deduced the first unconditional construction of the maximal abelian extension $F^{ab}$ of an arbitrary totally real field $F$ — but via $p$ -adic integration performed separately for infinitely many rational primes $p$ . They emphasize that this does not answer Hilbert's question in its original complex-analytic sense: no single collection of complex-analytic functions is known whose special values or transformation properties generate $F^{ab}$ for general totally real $F$ . The problem stated here is this surviving open core.

Background

Hilbert's twelfth problem (from his 1900 Paris address) asks to extend the Kronecker–Weber theorem — every finite abelian extension of $\mathbb{Q}$ is contained in a cyclotomic field generated by roots of unity, i.e. by special values of the exponential function — to other base number fields: for a number field $F$ , one wants explicit analytic functions whose special values or transformation properties produce the maximal abelian extension $F^{ab}$ of $F$ . For imaginary quadratic $F$ this is solved by the theory of complex multiplication: special values of modular functions such as the j-invariant at CM points, together with torsion coordinates (Weber functions), generate $F^{ab}$ , with the Galois action governed by Shimura's reciprocity law, which explicitly describes how Galois automorphisms act on these special values. Beyond $\mathbb{Q}$ and imaginary quadratic fields, no general solution is known. A number field is called totally real if every embedding into $\mathbb{C}$ has real image; Stark (1976) identified the totally real case as the natural next target and conjectured that certain exponentials of derivatives at $s=0$ of Artin $L$ -functions (the complex-analytic $L$ -functions attached to representations of finite Galois groups) or of partial zeta functions attached to each element of the Galois group of an abelian extension are algebraic units — 'Stark units' — whose Galois orbit generates the extension. These conjectures remain unproved in general and are considered the analytic heart of Hilbert's twelfth problem for totally real fields. A related arithmetic statement, the Brumer–Stark conjecture of Tate, predicts for a finite abelian CM extension $H/F$ (CM meaning $H$ is totally imaginary with a totally real subfield of index 2) and a prime $\mathfrak{p}$ of $F$ splitting completely in $H$ , the existence of a $\mathfrak{p}$ -unit $u \in H$ — an element of absolute value 1 at every place not above $\mathfrak{p}$ — whose valuations at the primes above $\mathfrak{p}$ are the partial zeta values $\zeta(\sigma,0)$ (the Stickelberger element). Such $u$ are the Brumer–Stark units. Dasgupta and Kakde proved the Brumer–Stark conjecture over $\mathbb{Z}[1/2]$ (Annals 2023) and, in the source paper, the $p$ -part of the integral Gross–Stark conjecture for odd $p$ (a statement about the Fitting ideal — the invariant measuring the relations among generators in a module presentation — of a Galois module $\nabla_{\mathscr{L}}$ built from Ritter–Weiss class modules, which encode class field theory in a cohomological complex), deducing that the Brumer–Stark units — computed by an explicit $p$ -adic analytic formula involving Shintani zeta functions (holomorphic functions parametrizing partial zeta values on a cone decomposition of a fundamental domain for the units) and $p$ -adic integration — together with $n-1$ explicitly described square roots of elements of $F$ (where $n=[F:\mathbb{Q}]$ ) generate $F^{ab}$ as one ranges over all primes $p$ and conductors. This is the first unconditional explicit class field theory for arbitrary totally real fields, but the construction uses $p$ -adic integration for infinitely many primes $p$ , and Hilbert had explicitly asked for complex-analytic functions. The source paper states plainly that its constructions 'do not fit neatly into his framework'; all subsequent analytic progress toward real-multiplication analogues of complex multiplication (e.g. the rigid-analytic theta cocycles of Darmon–Pozzi–Vonk surveyed in Dasgupta's ICM 2022 report) is likewise $p$ -adic rather than complex-analytic.

Problem Statement

Solve Hilbert's twelfth problem for totally real fields in Hilbert's original complex-analytic sense: for an arbitrary totally real number field $F$ of degree $n \geq 2$ , construct explicitly a single collection of complex-analytic functions (defined on complex domains with explicitly described transformation properties, in the way the exponential function serves $\mathbb{Q}$ via Kronecker–Weber and modular functions serve imaginary quadratic fields via complex multiplication) such that their special values at explicitly specifiable arguments are algebraic and generate the maximal abelian extension $F^{ab}$ , with the Galois action on these values described explicitly. Equivalently, the special values sought would play the role that the Brumer–Stark units (computed by $p$ -adic integration at varying primes) and the $n-1$ square roots of elements of $F$ play in the Dasgupta–Kakde $p$ -adic construction of $F^{ab}$ . Within such a complex-analytic framework, the special values would in particular discharge the underlying Stark-type predictions (the Galois-equivariant algebraicity of $\exp$ of partial zeta or Artin $L$ -function derivatives at $s=0$ ) for totally real fields.

References

Samit Dasgupta, Mahesh Kakde, Brumer–Stark units and explicit class field theory, Duke Mathematical Journal 173 (2024), no. 8, 1477–1555, DOI 10.1215/00127094-2023-0039, https://doi.org/10.1215/00127094-2023-0039; arXiv:2103.02516, https://arxiv.org/abs/2103.02516
Samit Dasgupta, Mahesh Kakde, On the Brumer–Stark conjecture, Annals of Mathematics (2) 197 (2023), no. 1, 289–388, DOI 10.4007/annals.2023.197.1.5, https://doi.org/10.4007/annals.2023.197.1.5
Samit Dasgupta, Mahesh Kakde, Jesse Silliman, Jiuya Wang, The Brumer–Stark Conjecture over Z, arXiv:2310.16399 [math.NT] (2023), https://arxiv.org/abs/2310.16399
Samit Dasgupta, Matthew H. L. Honnor, Michael Spieß, On the equality of three formulas for Brumer–Stark units, Journal of the London Mathematical Society (2) 112 (2025), no. 4, Article ID e70296, 44 pp., DOI 10.1112/jlms.70296, https://doi.org/10.1112/jlms.70296; arXiv:2211.01715, https://arxiv.org/abs/2211.01715
Matthew H. L. Honnor, On the root of unity ambiguity in a formula for the Brumer–Stark units, Canadian Mathematical Bulletin 67 (2024), no. 3, 611–623, DOI 10.4153/S0008439523001005, https://doi.org/10.4153/S0008439523001005; arXiv:2302.12332, https://arxiv.org/abs/2302.12332
Samit Dasgupta, Mahesh Kakde, On the Brumer–Stark conjecture and refinements, in: International Congress of Mathematicians, ICM 2022 (Helsinki), Vol. 3, European Mathematical Society, 2023, pp. 1768–1799, DOI 10.4171/ICM2022/124, https://doi.org/10.4171/ICM2022/124; arXiv:2204.09037, https://arxiv.org/abs/2204.09037
David Hilbert, Mathematical problems, Bulletin of the American Mathematical Society 8 (1902), no. 10, 437–479, DOI 10.1090/S0002-9904-1902-00923-3, https://doi.org/10.1090/S0002-9904-1902-00923-3
H. M. Stark, L-functions at s=1. III: Totally real fields and Hilbert's twelfth problem, Advances in Mathematics 22 (1976), no. 1, 64–84, DOI 10.1016/0001-8708(76)90138-9, https://doi.org/10.1016/0001-8708(76)90138-9
Samit Dasgupta, Michael Spieß, Partial zeta values, Gross's tower of fields conjecture, and Gross–Stark units, Journal of the European Mathematical Society 20 (2018), no. 11, 2643–2683, DOI 10.4171/JEMS/821, https://doi.org/10.4171/jems/821