Colored multizeta values in positive characteristic

Let V l be a finite k ̄ -linearly dependent subset of CZ w . Then V l is a finite k ′ -linearly dependent subset of CZ w .

Proof

We set V l = { Z 1 , … , Z m } . Without loss of generality, we can assume that m ≥ 2 by Theorem 3.1 and

Again, we can assume that Z 1 ∈ Span k ̄ ⁡ { Z 2 , … , Z m } ; then, by assumption (A.1), { Z 2 , … , Z m } is a linearly independent set over k ̄ . By Proposition 5.5, we take the matrix Φ j ∈ Mat d j ⁡ ( k ̄ ⁢ [ t ] ) and the column vector ψ j ∈ Mat d j × 1 ⁡ ( E ) ( 1 ≤ j ≤ m ) so that the 4-tuple ( Φ j , ψ j , Z j , r j ) satisfies Lemma 5.4 (i)–(iv).

We define the direct sums of any matrices M 1 , … , M m and any column vectors v 1 , … , v m whose entries belong to C ∞ ⁢ ( ( t ) ) by

respectively.

By using this notation, we define the block diagonal matrix Φ and column vector 𝜓 as follows:

By Lemma 5.4, we have

for some a j ∈ F ̄ q × , b j ∈ k × and

for some c j ∈ F q × , N ∈ N . Using Theorem 5.3, there exist row vectors f j = ( f j ⁢ 1 , … , f j ⁢ d j ) ∈ Mat 1 × d j ⁢ ( k ̄ ⁢ [ t ] ) ( j = 1 , … , m ) so that if we put F = ( f 1 , … , f m ) , then we have

Since f 1 , d 1 | t = θ is the coefficient of a 1 ⁢ b 1 ⁢ Z 1 / ( π ̃ w ) in ( F ⁢ ψ ) | t = θ = 0 and by assumption (A.1), Z 1 is expressed by nontrivial k ̄ -linear combinations of Z 2 , … , Z m . We write F ′ : = ( 1 / f 1 ⁢ d 1 ) F and d : = ∑ j = 1 m d j . Note that the vector F ′ is of the form

where f 1 ⁢ d 1 ′ = 1 . We have the following by F ⁢ ψ = 0 and f j ⁢ i | t = θ = 0 :

By using Lemma 5.4, for r = lcm ⁡ ( r 1 , … , r m ) , we obtain F ′ ⁣ ( − r ) ⁢ Φ ⁢ ψ = ( F ′ ⁢ ψ ) ( − r ) = 0 and thus

The last column of the matrix Φ j is ( 0 , … , 0 , 1 ) tr for each 𝑗, and consequently, the d 1 -th entry of row vector F ′ − F ′ ⁣ ( − r ) ⁢ Φ is zero since the d 1 -th entries of the row vectors F ′ and F ′ ⁣ ( − r ) ⁢ Φ are 1. The ∑ i = 1 j d i -th column of Φ is

Then the ∑ i = 1 j d i -th entry of the row vector F ′ ⁣ ( − r ) ⁢ Φ is f j ⁢ d j ′ ⁣ ( − r ) . Thus, the ∑ i = 1 j d i -th entry of F ′ − F ′ ⁣ ( − r ) ⁢ Φ is written as follows:

We claim that f j ⁢ d j ′ − f j ⁢ d j ′ ⁣ ( − r ) = 0 for j = 2 , … , m . Indeed, if there exist some 2 ≤ j ≤ m such that f j ⁢ d j ′ − f j ⁢ d j ′ ⁣ ( − r ) ≠ 0 , we can derive the contradiction in the following way.

Let us take a sufficiently large N ∈ N so that ( f j ⁢ d j ′ − f j ⁢ d j ′ ⁣ ( − r ) ) | t = θ q N ≠ 0 and all entries of ( F ′ − F ′ ⁣ ( − r ) ⁢ Φ ) are regular at t = θ q N . By using (A.3) and substituting t = θ q N in (A.5), we obtain

Thus, combined with f 1 ⁢ d 1 ′ − f 1 ⁢ d 1 ′ ⁣ ( − r ) = 1 − 1 = 0 , we obtain the nontrivial k ̄ -linear relations among Z 2 q N , … , Z m q N as follows:

Then, by taking the q N -th root of the relation, we obtain the following nontrivial k ̄ -linear relation among Z 2 , … , Z m :

This contradicts our assumption that { Z 2 , … , Z m } is a k ̄ -linearly independent set.

Therefore, we obtain f j ⁢ d j ′ − f j ⁢ d j ′ ⁣ ( − r ) = 0 for j = 2 , … , m , and this equation shows the following:

By substituting t = θ in equation F ′ ⁢ ψ = 0 , equations (A.2) and (A.4) enable us to obtain the following equalities:

By f 1 ⁢ d 1 ′ = 1 and (A.6), we have the following nontrivial k ′ -linear relation among Z 1 , … , Z m :

Therefore, we obtain the desired claim. ∎

Proof

We can assume that w l > ⋯ > w 1 without loss of generality. By the definition, CZ w i is a finite set for each i = 1 , … , l , and thus, its subset V i is also finite. Let each V i consist of { Z i ⁢ 1 , … , Z i ⁢ m i } , where Z i ⁢ j ∈ CZ w i ( j = 1 , … , m i ) are the same total weight w i . The proof is by induction on 𝑙.

We require on the contrary that { 1 } ∪ ⋃ i = 1 l V i is a k ̄ -linearly dependent set, and then we proceed to our proof by assuming the existence of nontrivial k ̄ -linear relations involving V l .

For 1 ≤ i ≤ l and 1 ≤ j ≤ m l , by combining Theorem 4.4 with Proposition 5.5, we can take r i ⁢ j > 0 , the matrix and the column vector

so that d i ⁢ j ≥ 2 and that each ( Φ i ⁢ j , ψ i ⁢ j , Z i ⁢ j , r i ⁢ j ) satisfies Lemma 5.4.

For Φ i ⁢ j and ψ i ⁢ j ( 1 ≤ i ≤ l , 1 ≤ j ≤ m l ), we define the following block diagonal matrix and column vector:

Here we used ⨁ in the same manner of the definitions of Φ and 𝜓 in the proof of Lemma A.1.

By the requirement in the beginning of proof, { 1 } ∪ ⋃ i = 1 l V i is a linearly dependent set over k ̄ . Thus, there exists a nonzero vector ρ = ( v 11 , … , v 1 ⁢ m 1 , … , v l ⁢ 1 , … , v l ⁢ m l ) such that

where v i ⁢ j ∈ Mat 1 × d i ⁢ j ⁢ ( k ̄ ) for 1 ≤ i ≤ l and 1 ≤ j ≤ m i . Then we have the following nontrivial k ̄ -linear relation:

In the beginning of this proof, we assumed that there exist nontrivial k ̄ -linear relations between V l and { 1 } ∪ ⋃ i = 1 l − 1 V i , and thus, for some 1 ≤ s ≤ m l , the last entry of v l ⁢ s is nonzero. By using Theorem 5.3, we have F : = ( f 11 , … , f 1 ⁢ m 1 , … , f l ⁢ 1 , … , f l ⁢ m l ) , where f i ⁢ j = ( f i ⁢ 1 , … , f i ⁢ d i ⁢ j ) ∈ Mat 1 × d i ⁢ j ⁢ ( k ̄ ⁢ [ t ] ) for 1 ≤ i ≤ l , 1 ≤ j ≤ m i and it satisfies F ⁢ ψ ̃ = 0 and F | t = θ = ρ . The last entry of f l ⁢ s is a nontrivial polynomial because the last entry of v l ⁢ s is not zero. We choose a sufficiently large N ∈ Z so that f l ⁢ s | t = θ q N ≠ 0 and ξ q N = ξ for all factors ζ A ⁢ ( s ; ξ ) which appear in all Z i ⁢ j ( 1 ≤ i ≤ l , 1 ≤ j ≤ m i ). We rewrite the equation ( F ⁢ ψ ̃ ) | t = θ q N = 0 as follows by using Ω | t = θ q N = 0 , Lemma 5.4 (iv) and the definition of ψ ̃ :

when i ≠ l , Ω w l − w i | t = θ q N = 0 and then

Thus, we obtain the following nontrivial k ̄ -linear relation with some f l ⁢ d l ⁢ s ≠ 0 :

Therefore, by taking the q N -th root of the above relation, we obtain the following nontrivial k ̄ -linear relation for { Z l ⁢ 1 , … , Z l ⁢ m l } :

This shows that V l is a k ̄ -linearly dependent set. Then, from Lemma A.1, it follows that V l is a k ′ -linearly dependent subset. However, this contradicts the condition that V l is the k ′ -linearly independent set. Therefore, Theorem 5.7 holds. ∎