Birch–Tate conjecture

The Birch–Tate conjecture is a conjecture in mathematics (more specifically in algebraic K-theory) proposed by both Bryan John Birch and John Tate.

Statement

In algebraic K-theory, the group K₂ is defined as the center of the Steinberg group of the ring of integers of a number field F. K₂ is also known as the tame kernel of F. The Birch–Tate conjecture relates the order of this group (its number of elements) to the value of the Dedekind zeta function $\zeta _{F}$. More specifically, let F be a totally real number field and let N be the largest natural number such that the extension of F by the Nth root of unity has an elementary abelian 2-group as its Galois group. Then the conjecture states that

$\#K_{2}=|N\zeta _{F}(-1)|.$

Status

Progress on this conjecture has been made as a consequence of work on Iwasawa theory, and in particular of the proofs given for the so-called "main conjecture of Iwasawa theory." The Birch-Tate conjecture is a deep and interesting conjecture in algebraic number theory, which relates to the order of the center of the Steinberg group associated with the ring of integers of a number field and the value of the Dedekind zeta function at 𝑠 = 1 s=1.

Let's break this down step by step:

1. Steinberg Group The Steinberg group 𝑆 ( 𝑅 ) S(R) of a commutative ring 𝑅 R is a group defined in the context of algebraic groups, typically with respect to a reductive group over the ring 𝑅 R. For number fields and their rings of integers, these groups come into play when studying integral points on algebraic groups, and they often reveal deep structural information about the ring 𝑅 R and its arithmetic properties.

The center of the Steinberg group 𝑍 ( 𝑆 ( 𝑅 ) ) Z(S(R)) can be of particular interest, especially in the case of the ring of integers of a number field 𝐾 K, as the order of this center provides insight into the arithmetic structure of the field.

2. Dedekind Zeta Function The Dedekind zeta function 𝜁 𝐾 ( 𝑠 ) ζ K ​

(s) of a number field 

𝐾 K is defined as:

𝜁 𝐾 ( 𝑠 ) = ∑ 𝑎 ⊆ 𝑂 𝐾 1 N ( 𝑎 ) 𝑠 ζ K ​

(s)= 

a⊆O K ​

∑ ​

N(a) s

1 ​

where the sum is over all non-zero ideals 𝑎 a of the ring of integers 𝑂 𝐾 O K ​

, and 

N ( 𝑎 ) N(a) denotes the norm of the ideal. This function generalizes the Riemann zeta function for number fields, and its analytic properties encode important arithmetic data about 𝐾 K.

3. Birch-Tate Conjecture The Birch-Tate conjecture conjectures a relationship between the order of the center of the Steinberg group 𝑍 ( 𝑆 ( 𝑂 𝐾 ) ) Z(S(O K ​

)) of the ring of integers 

𝑂 𝐾 O K ​

 of a number field 

𝐾 K and the value of the Dedekind zeta function 𝜁 𝐾 ( 𝑠 ) ζ K ​

(s) at 

𝑠 = 1 s=1. More specifically, it states:

∣ 𝑍 ( 𝑆 ( 𝑂 𝐾 ) ) ∣ = 𝜁 𝐾 ( 1 ) ∣Z(S(O K ​

))∣=ζ 

K ​

(1)

where 𝜁 𝐾 ( 1 ) ζ K ​

(1) is the value of the Dedekind zeta function at 

𝑠 = 1 s=1, which is known to be related to the class number of the number field 𝐾 K.

4. Interpretation The order of the center 𝑍 ( 𝑆 ( 𝑂 𝐾 ) ) Z(S(O K ​

)) of the Steinberg group can be interpreted as an arithmetic invariant of the field 

𝐾 K, encapsulating the relationship between the ideal class group and the structure of algebraic groups over 𝑂 𝐾 O K ​

.

The Dedekind zeta function 𝜁 𝐾 ( 1 ) ζ K ​

(1), when evaluated at 

𝑠 = 1 s=1, encodes important arithmetic properties of 𝐾 K, particularly the class number (i.e., the order of the ideal class group of 𝐾 K). For fields with a non-trivial class group, 𝜁 𝐾 ( 1 ) ζ K ​

(1) diverges, while for fields with a trivial class group (like 

𝑄 Q), it gives a finite value that is closely related to the order of certain associated groups.

The Birch-Tate conjecture suggests that there is a deep link between these two concepts: the center of the Steinberg group of the ring of integers is directly related to the class number (and hence the value of the Dedekind zeta function at 𝑠 = 1 s=1).

5. Applications and Implications The conjecture connects algebraic topology, group theory, and number theory by linking the geometry of algebraic groups and their arithmetic data (through the Steinberg group) to the analytic properties of the Dedekind zeta function.

A proof of the Birch-Tate conjecture would have significant implications for the study of number fields, particularly in understanding the arithmetic of number fields and their rings of integers through both algebraic and analytic lenses.

Conclusion The Birch-Tate conjecture provides a fascinating bridge between two important objects in algebraic number theory: the Steinberg group and the Dedekind zeta function. Its proof or disproof would deepen our understanding of the arithmetic of number fields, and it remains a central topic in the study of algebraic groups, zeta functions, and ideal class groups

References

• J. T. Tate, Symbols in Arithmetic, Actes, Congrès Intern. Math., Nice, 1970, Tome 1, Gauthier–Villars(1971), 201–211

External links

• Hurrelbrink, J. (2001) [1994], "Birch–Tate conjecture", Encyclopedia of Mathematics, EMS Press