The cohomological Lichtenbaum conjecture for abelian extensions of imaginary quadratic fields at the excluded primes

For a finite abelian extension $F$ of an imaginary quadratic field $K$ and an integer $m\geq 2$, the cohomological Lichtenbaum conjecture predicts (up to sign and powers of $2$) that the special value $\zeta_F(1-m)$ of the Dedekind zeta function, the Beilinson regulator $R_m(F)$, and the orders of the étale cohomology groups $H^i(\mathcal{O}_F[1/p],\mathbb{Z}_p(m))$ satisfy an explicit identity for every rational prime $p$. This identity is known unconditionally when $F$ is abelian over $\mathbb{Q}$ (Huber–Kings, with the $2$-part via Flach), and a 2021 preprint of Chaochao Sun proves it for ray class fields of $K$ only outside a finite exceptional set of primes (containing $2$ and the bad-reduction primes of an auxiliary CM elliptic curve, and in its key descent theorem also requiring $p\nmid[F:K]$), in an unfinished draft with proof steps marked '???' and a step conditional on a standard finiteness conjecture. The open problem is to establish the identity for all finite abelian extensions $F/K$, all $m\geq 2$, and all primes $p$ — in particular $p=2$, primes dividing $[F:K]$, and inert or ramified primes of bad reduction — unconditionally and with complete proofs.

Background

The analytic class number formula relates the residue of the Dedekind zeta function $\zeta_F(s)$ of a number field $F$ at $s=1$ to its classical invariants. S. Lichtenbaum (1973) conjectured an analogue at the negative integers $s=1-m$, $m\geq 2$: the leading term $\zeta_F(1-m)$ should be expressible through algebraic $K$-theory, or equivalently through étale cohomology. In the cohomological formulation, one writes $H^i(\mathcal{O}_F[1/p],\mathbb{Z}_p(m))$ for the étale cohomology of the ring of $S$-integers with all primes above $p$ inverted, with coefficients in the $m$-fold Tate twist $\mathbb{Z}_p(m)$ (the $p$-adic Galois module built from the $m$-th tensor power of the group of $p$-power roots of unity); $R_m(F)$ denotes the Beilinson (equivalently Borel) regulator, the covolume of the image of the odd $K$-group $K_{2m-1}(F)$ in a suitable real vector space. The conjecture then predicts, up to a sign and a power of $2$, that $\zeta_F(1-m)^{*}=R_m(F)\prod_p\#H^2(\mathcal{O}_F[1/p],\mathbb{Z}_p(m))/\#H^{1}(\mathcal{O}_F[1/p],\mathbb{Z}_p(m))_{\mathrm{tors}}$, where $\zeta_F(1-m)^{*}$ is the modified zeta value with Euler factors at the primes above $2$ removed (the convention of Kolster–Nguyen Quang Do–Fleckinger); the value $\zeta_F(1-m)$ itself is a rational number. This statement is equivalent to the Bloch–Kato (Tamagawa number) conjecture for the Tate motive $\mathbb{Q}(m)$ over $F$, which is open for general number fields $F$.

The conjecture is a theorem when $F$ is abelian over $\mathbb{Q}$: Kolster, Nguyen Quang Do and Fleckinger (1996) proved it (modulo a standard finiteness conjecture, with errors corrected in the appendix of Benois–Nguyen Quang Do, 2002) via descent on cyclotomic units, Huber and Kings (2003) proved the Bloch–Kato conjecture for all Dirichlet characters, and Flach (2011) established the cyclotomic main conjecture for the prime $2$, completing the $2$-part. The natural next class consists of finite abelian extensions $F$ of an imaginary quadratic field $K$ (a CM setting): such $F$ need not be abelian over $\mathbb{Q}$, and $\zeta_F$ factors into Hecke $L$-functions (the $L$-series attached to characters of the ray class groups of $K$, i.e. of the abelian extensions of $K$). Here the role of cyclotomic units is played by elliptic units (special values of Siegel functions at torsion points of a CM elliptic curve, i.e. an elliptic curve whose endomorphism ring is the full ring of integers of $K$), and the descent goes through Iwasawa theory (the study of arithmetic invariants over infinite towers of number fields) over the $\mathbb{Z}_p^2$-extension of $K$ (the compositum of all $\mathbb{Z}_p$-extensions of $K$, an infinite Galois extension with Galois group $\mathbb{Z}_p^2$), following the method Kings (2001) used for the Tamagawa number conjecture for CM elliptic curves and using Poitou–Tate duality (the standard nine-term exact sequence relating global and local Galois cohomology).

A separate and active line of work addresses the complementary boundary case $s=0$ (the equivariant analytic class number formula side): Burns–Greither (2003) for fields abelian over $\mathbb{Q}$, Bley (2006) for abelian extensions of an imaginary quadratic field at primes $p>2$ splitting in $K$ with $p$ not dividing the class number of $K$, and Bullach–Hofer (2023) and Dasgupta–Kakde–Silliman (2023) more recently. These $s=0$ results concern the leading term at $0$ and do not touch the negative-integer values at issue here.

In this landscape, Chaochao Sun's preprint (arXiv:2112.12314, December 2021, version 1, unpublished) is the only direct attack on the $m\geq 2$ cohomological Lichtenbaum conjecture for these fields. Its key comparison theorem (Theorem 6.7) computes the index of the twisted elliptic units inside $H^{1}(\mathcal{O}_F[1/p],\mathbb{Z}_p(m))$ for each nontrivial character of $\operatorname{Gal}(F/K)$, under the hypotheses: $p$ odd, $p\nmid[F:K]$, and $p$ either split in $K$ or a prime of good reduction for the auxiliary CM elliptic curve; the final statement (Theorem 7.2) derives the cohomological Lichtenbaum identity for the ray class field $F=K(\mathfrak{f})$ (the maximal abelian extension of $K$ with conductor dividing the finite modulus $\mathfrak{f}$; every finite abelian extension of $K$ lies in one) 'up to the primes in $S$', where $S$ contains $2$ and the bad-reduction primes of the chosen CM elliptic curve. The manuscript is an unfinished draft: several load-bearing steps are marked with literal '???' placeholders (inside a generalized Tate lemma, spectral-sequence and Poitou–Tate comparisons, and local class field theory arguments), one descent input is quoted from Kolster–Nguyen Quang Do–Fleckinger 'modulo the standard conjecture' (a finiteness conjecture on the relevant Tate–Shafarevich groups), and the exclusion set of the headline theorem is smaller than what its own engine theorem (Theorem 6.7, which also requires $p\nmid[F:K]$) actually covers.

Problem Statement

Let $K$ be an imaginary quadratic field and let $F/K$ be a finite abelian extension. For an integer $m\geq 2$, let $R_m(F)$ be the Beilinson (Borel) regulator on $K_{2m-1}(F)$ and, for each rational prime $p$, let $H^{i}(\mathcal{O}_F[1/p],\mathbb{Z}_p(m))$ denote the étale cohomology of $\mathcal{O}_F$ with all primes above $p$ inverted, with coefficients in the $m$-fold Tate twist $\mathbb{Z}_p(m)$. Establish, unconditionally and with complete proofs, the cohomological Lichtenbaum conjecture for these fields in full: an explicit identity relating the modified special value $\zeta_F(1-m)^{*}$ (the Dedekind zeta value with Euler factors at the primes above $2$ removed; the identity is up to a sign and a power of $2$, as in the Kolster–Nguyen Quang Do–Fleckinger formulation), the regulator $R_m(F)$, and the product, over all rational primes $p$ without exception, of the ratio of orders $\#H^{2}(\mathcal{O}_F[1/p],\mathbb{Z}_p(m))/\#H^{1}(\mathcal{O}_F[1/p],\mathbb{Z}_p(m))_{\mathrm{tors}}$. At the prime $p=2$ the phrase 'up to a sign and a power of $2$' records a convention of the general statement rather than a licence to leave the $2$-adic term undetermined: the identity determines the sign and the power of $2$, that is, it pins down the factor contributed at $p=2$ in the product just displayed, so that the convention for $\zeta_F(1-m)^{*}$ does not leave the $2$-adic term free. Because the product ranges over all rational primes without exception, its scope includes the primes excluded by Sun's arXiv:2112.12314 — the prime $p=2$, the primes of bad reduction of the auxiliary CM elliptic curve used to construct elliptic units, the primes $p\mid[F:K]$, and the inert or ramified primes of $K$ at which that curve has bad reduction — and 'unconditionally and with complete proofs' means in particular that neither the standard finiteness conjecture quoted in the descent step of that preprint nor its incomplete '???' steps (in the descent, the Poitou–Tate comparison, or the index computation) may be taken as established inputs. The statement is uniform over all imaginary quadratic $K$, all finite abelian extensions $F/K$, and all $m\geq 2$; the problem is resolved either by such a proof or by an explicit counterexample (a specific $K$, $F/K$, $m\geq 2$, and prime $p$ for which the identity with the stated conventions fails).

References

1. Chaochao Sun, The Lichtenbaum conjecture for abelian extension of imaginary quadratic fields, preprint (2021), arXiv:2112.12314, DOI 10.48550/arXiv.2112.12314, https://arxiv.org/abs/2112.12314
2. Stephen Lichtenbaum, Values of zeta-functions, étale cohomology, and algebraic K-theory, Lecture Notes in Mathematics 342 (1973), 487-501, DOI 10.1007/bfb0073737, https://doi.org/10.1007/bfb0073737
3. Manfred Kolster, Thong Nguyen Quang Do, Vincent Fleckinger, Twisted S-units, p-adic class number formulas, and the Lichtenbaum conjectures, Duke Mathematical Journal 84 (1996), no. 3, 679-717, DOI 10.1215/s0012-7094-96-08421-5, https://doi.org/10.1215/s0012-7094-96-08421-5
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8. Matthias Flach, On the cyclotomic main conjecture for the prime 2, Journal für die reine und angewandte Mathematik 661 (2011), 1-36, DOI 10.1515/crelle.2011.082, https://doi.org/10.1515/crelle.2011.082
9. Werner Bley, On the equivariant Tamagawa number conjecture for Abelian extensions of a quadratic imaginary field, Documenta Mathematica 11 (2006), 73-118, DOI 10.4171/dm/205, https://doi.org/10.4171/dm/205
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