Math 2.0

Number theory Conjecture

Birch–Tate conjecture

Builds on K-groups of rings of integersDedekind zeta functionAlgebraic number fieldCyclotomic field

Statement

Let FF be a totally real number field with r1r_1 real embeddings and ring of integers 𝒪F\mathcal{O}_F, and let ζF\zeta_F be its Dedekind zeta function. Let w2(F)w_2(F) be the largest integer NN such that the Galois group of the cyclotomic extension F(μN)/FF(\mu_N)/F has exponent dividing 22. Then ζF(−1)=(−1)r1 ∣K2(𝒪F)∣w2(F),\zeta_F(-1) = (-1)^{r_1} \, \frac{\abs{K_2(\mathcal{O}_F)}}{w_2(F)}, where K2(𝒪F)K_2(\mathcal{O}_F) is the second K-group of 𝒪F\mathcal{O}_F, a finite group.

Remarks

Example: for F=ℚF = \Q, r1=1r_1 = 1, K2(ℤ)≅ℤ/2K_2(\Z) \cong \Z/2 and w2(ℚ)=24w_2(\Q) = 24, since (ℤ/24ℤ)×(\Z/24\Z)^\times has exponent 22 and 2424 is the largest such modulus. The formula gives −2/24=−1/12=ζ(−1)-2/24 = -1/12 = \zeta(-1), which is correct.

The finiteness of K2(𝒪F)K_2(\mathcal{O}_F) is a theorem of Garland. The odd part of the conjecture follows from the main conjecture of Iwasawa theory for totally real fields, proved by Wiles. The deduction also uses work of Tate relating K2K_2 to Galois cohomology. The conjecture is the case n=2n = 2 of Lichtenbaum's conjecture on special values.

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Algebraic number field, concept, no evidence levelAlgebraic number fieldconceptCyclotomic field, concept, no evidence levelCyclotomic fieldconceptDedekind zeta function, concept, no evidence levelDedekind zeta functionconceptK-groups of rings of integers, concept, no evidence levelK-groups of rings of …conceptBirch–Tate conjecture, conjecture, no evidence levelBirch–Tate conjectureconjecture
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  1. Algebraic number field concept
  2. Cyclotomic field concept
  3. Dedekind zeta function concept
  4. K-groups of rings of integers concept
  5. Birch–Tate conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Birch–Tate_conjecture
About this page
TagS2CHWAZX4V
URIm2://S2CHWAZX4V
Kindconjecture
WritingW0 stub
MSC11R70, 11R42, 19F27
StatusThis copy of the site has no build data. Run m2 status S2CHWAZX4V.
AboutK-groups of rings of integers, Dedekind zeta function
UsesK-groups of rings of integers, Dedekind zeta function, Algebraic number field, Cyclotomic field
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Changed3 November 2026

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