[Paper Review] Galois Codescent For Motivic Tame Kernels

Jilali Assim, A. Movahhedi|arXiv (Cornell University)|Jan 22, 2019

Algebraic Geometry and Number Theory14 references4 citations

TL;DR

This paper provides an explicit description of the kernel of the Galois descent map on motivic tame K-groups for finite Galois extensions of number fields, establishing genus formulas for even K-groups via motivic cohomology. It derives arithmetic conditions—based on ramification and Frobenius actions—for the vanishing of étale cohomology and the 2-primary parts of K-groups, particularly in real and totally real 2-extensions of Q.

ABSTRACT

Let $L/F$ be a finite Galois extension of number fields with an arbitrary Galois group $G$. We give an explicit description of the kernel of the natural map on motivic tame kernels $H^2_{\mathcal{M}}(o_L, {\bf Z}(i))_{G} { ightarrow} H^2_{\mathcal{M}}(o_F, {\bf Z}(i))$. Using the link between motivic cohomology and $K$-theory, we deduce genus formulae for all even $K$-groups $K_{2i-2}(o_F)$ of the ring of integers. As a by-product, we also obtain lower bounds for the order of the kernel and cokernel of the functorial map $H^2_{\mathcal{M}}(F, {\bf Z}(i)) ightarrow H^2_{\mathcal{M}}( L, {\bf Z}(i) )^{G}$.

Motivation & Objective

• To describe the kernel of the natural map on motivic tame K-groups induced by a finite Galois extension $ L/F $ with group $ G $, using motivic cohomology.
• To deduce genus formulae for even $ K $-groups $ K_{2i-2}(o_F) $ via the Bloch-Kato isomorphism between $ K $-theory and motivic cohomology.
• To establish necessary and sufficient arithmetic conditions for the vanishing of positive étale cohomology groups $ H^2_+(o_F[1/2], \mathbb{Z}_2(i)) $ in $ 2 $-extensions of $ \mathbb{Q} $.
• To determine when the $ 2 $-primary part of $ K_{2i-2}(o_L) $ vanishes for Galois $ 2 $-extensions $ L/\mathbb{Q} $, depending on the congruence class of $ 2i-2 \mod 8 $ and ramification behavior.
• To resolve a question of B. Kahn on the image of the signature map in étale cohomology by constructing number fields with arbitrary $ 2 $-rank for the cokernel of the signature map.

Proposed method

• Use the motivic Bloch-Kato isomorphism to translate $ K $-theory questions into motivic cohomology, enabling the use of localization sequences.
• Apply Geisser’s localization sequence in motivic cohomology to relate $ H^2_{\mathcal{M}}(o_F, \mathbb{Z}(i)) $ to $ H^2_{\mathcal{M}}(F, \mathbb{Z}(i)) $, facilitating kernel analysis.
• Employ the étale Tate kernel $ D_F^{(i)} $ and the extension $ E(\sqrt[p]{D_F^{(i)}})/E $ to construct arithmetic obstructions to Galois descent.
• Use the Frobenius action on the Galois group of $ E(\sqrt[p]{D_F^{(i)}})/E $ to characterize when the transfer map is an isomorphism.
• Analyze the signature map $ \mathrm{sgn}_F: H^1(F, \mathbb{Z}_2(i))/2 \to \bigoplus_{v \text{ real}} \mathbb{Z}/2 $ to determine the structure of $ H^2_+(o_F[1/2], \mathbb{Z}_2(i)) $.
• Use exact sequences involving $ H^2_+ $ and $ H^2 $ to relate the vanishing of $ K $-groups to cohomological conditions, especially in $ 2 $-extensions of $ \mathbb{Q} $.

Experimental results

Research questions

• RQ1 What is the explicit structure of the kernel of the Galois descent map $ H^2_{\mathcal{M}}(o_L, \mathbb{Z}(i))_G \to H^2_{\mathcal{M}}(o_F, \mathbb{Z}(i)) $?
• RQ2 Under what arithmetic conditions does the transfer map $ \mathrm{tr}_i $ become an isomorphism for motivic cohomology of rings of integers?
• RQ3 When does the $ 2 $-primary part of $ K_{2i-2}(o_L) $ vanish for a Galois $ 2 $-extension $ L/\mathbb{Q} $, depending on $ i \mod 8 $?
• RQ4 What is the image of the signature map $ \mathrm{sgn}_F $, and can it be made to have arbitrary $ 2 $-rank $ \rho_i $?
• RQ5 How does the ramification behavior of $ L/F $, especially at primes dividing the ramification index, affect the kernel and cokernel of the motivic transfer map?

Key findings

• The kernel of the map $ H^2_{\mathcal{M}}(o_L, \mathbb{Z}(i))_G \to H^2_{\mathcal{M}}(o_F, \mathbb{Z}(i)) $ is explicitly described via the linear independence of Frobenius elements in $ \mathrm{Gal}(E(\sqrt[p]{D_F^{(i)}})/E) $ for primes $ p $ dividing ramification indices.
• For a cyclic $ p $-extension $ L/F $, the genus formula for $ K_{2i-2}(o_F) $ depends only on ramification data in $ L/F $, as shown in Corollaries 4.12 and 4.13.
• For $ i \geq 2 $ odd, there exist totally real number fields $ F $ such that the image of the signature map $ \mathrm{sgn}_F $ has $ 2 $-rank $ \rho_i = n $, for any $ n \geq 1 $, as proven in Theorem 4.15.
• The $ 2 $-primary part of $ K_{2i-2}(o_L) $ vanishes for a totally real Galois $ 2 $-extension $ L/\mathbb{Q} $ precisely when $ L $ is unramified outside $ \{2, \infty, \ell\} $ with $ \ell \equiv \pm 3 \pmod{8} $, as in Corollary 6.5.
• For $ 2i-2 \equiv 0 \pmod{8} $, the $ 2 $-primary part of $ K_{2i-2}(o_L) $ vanishes for totally imaginary $ L $ iff $ L $ is unramified outside $ \{2, \infty, \ell\} $ with $ \ell \equiv \pm 3 \pmod{8} $, as in Corollary 6.6.
• For totally real cyclic $ 2 $-extensions with $ 2i-2 \equiv 0 \pmod{8} $, the $ 2 $-primary part of $ K_{2i-2}(o_L) $ vanishes iff $ L $ is unramified outside $ \{2, \ell_1, \ell_2\} $ with $ \ell_1, \ell_2 \not\equiv 1 \pmod{8} $ and $ \ell_1 \not\equiv \ell_2 \pmod{8} $, as in Proposition 6.7.