• 沒有找到結果。

# On Algebraic Independence of Special Zeta Values in Characteristic p

N/A
N/A
Protected

Share "On Algebraic Independence of Special Zeta Values in Characteristic p"

Copied!
20
0
0

(1)

### On Algebraic Independence of Special Zeta Values in Characteristic p

Jing Yu

National Taiwan University and TIMS, Taipei

July, 2010, Workshop on Arithmetic Geometry

(2)

### t-motives

Let t, σ be variables independent of θ.

Let ¯k(t)[σ, σ−1] be noncommutative ring of Laurent polynomials in σ with coefficients in ¯k(t), subject to the relation

σf := f(−1)σ for all f ∈ ¯k(t).

Here f(−1) is the rational function obtained from f ∈ ¯k(t) by twisting all its coefficients a ∈ ¯k to a1q.

A pre-t-motive M over Fq is a left ¯k(t)[σ, σ−1]-module which is finite-dimensional over ¯k(t).

Let m ∈ Matr×1(M ) be a ¯k(t)-basis of M .

Multiplying by σ on M is represented by σ(m) = Φm for some matrix Φ ∈ GLr(¯k(t)).

The category of pre-t-motives over Fq forms a rigid abelian Fq(t)-linear tensor category.

(3)

### t-motives

Let t, σ be variables independent of θ.

Let ¯k(t)[σ, σ−1] be noncommutative ring of Laurent polynomials in σ with coefficients in ¯k(t), subject to the relation

σf := f(−1)σ for all f ∈ ¯k(t).

Here f(−1) is the rational function obtained from f ∈ ¯k(t) by twisting all its coefficients a ∈ ¯k to a1q.

A pre-t-motive M over Fq is a left ¯k(t)[σ, σ−1]-module which is finite-dimensional over ¯k(t).

Let m ∈ Matr×1(M ) be a ¯k(t)-basis of M .

Multiplying by σ on M is represented by σ(m) = Φm for some matrix Φ ∈ GLr(¯k(t)).

The category of pre-t-motives over Fq forms a rigid abelian Fq(t)-linear tensor category.

(4)

### Frobenius difference equations

From a pre-t-motive, one associates a “system of Frobenius difference equation”which has solutions in series of t.

Consider the operator on C by x 7→ x1q. Then extend this operator to C((t)) as follows, for f =P

iaiti∈ C((t)) define f(−1) :=P

iaqi−1ti. More generally, for matrix B with entries in C((t)) define twisting B(−1) by the rule B(−1)ij = Bij(−1).

Let m ∈ Matr×1(M ) be a ¯k(t)-basis of M .

Multiplying by σ on M is represented by σ(m) = Φm for some matrix Φ ∈ GLr(¯k(t)).

Theequationto be solved in Ψ ∈ Matr(C((t)) is : Ψ(−1)= ΦΨ.

(5)

### Frobenius difference equations

From a pre-t-motive, one associates a “system of Frobenius difference equation”which has solutions in series of t.

Consider the operator on C by x 7→ x1q. Then extend this operator to C((t)) as follows, for f =P

iaiti∈ C((t)) define f(−1) :=P

iaqi−1ti. More generally, for matrix B with entries in C((t)) define twisting B(−1) by the rule B(−1)ij = Bij(−1).

Let m ∈ Matr×1(M ) be a ¯k(t)-basis of M .

Multiplying by σ on M is represented by σ(m) = Φm for some matrix Φ ∈ GLr(¯k(t)).

Theequationto be solved in Ψ ∈ Matr(C((t)) is : Ψ(−1)= ΦΨ.

(6)

### Frobenius difference equations

From a pre-t-motive, one associates a “system of Frobenius difference equation”which has solutions in series of t.

Consider the operator on C by x 7→ x1q. Then extend this operator to C((t)) as follows, for f =P

iaiti∈ C((t)) define f(−1) :=P

iaqi−1ti. More generally, for matrix B with entries in C((t)) define twisting B(−1) by the rule B(−1)ij = Bij(−1).

Let m ∈ Matr×1(M ) be a ¯k(t)-basis of M .

Multiplying by σ on M is represented by σ(m) = Φm for some matrix Φ ∈ GLr(¯k(t)).

Theequationto be solved in Ψ ∈ Matr(C((t)) is : Ψ(−1)= ΦΨ.

(7)

### Let t = θ

We view Ψ as giving a “fundamental”solution of the system of Frobeniusdifference equations described by the algebraic matrix Φ coming from M .

Note that if Ψ0∈ Matr(C((t))) is also a solution of the Frobenius system from Φ, then Ψ0−1Ψ ∈ GLr(Fq(t)).

A power series f =P

i=0aiti ∈ C[[t]] that converges everywhere and satisfies

[k(a0, a1, a2, . . . ) : k] < ∞

is called an entire power series. As a function of t it takes values in k, when restricted to k. The ring of the entire power series is denoted by E.

If all entries of a solution Ψ of the Frobenius system in question are in E, one canspecializ Ψ to Ψ(θ).

(8)

### Let t = θ

We view Ψ as giving a “fundamental”solution of the system of Frobeniusdifference equations described by the algebraic matrix Φ coming from M .

Note that if Ψ0∈ Matr(C((t))) is also a solution of the Frobenius system from Φ, then Ψ0−1Ψ ∈ GLr(Fq(t)).

A power series f =P

i=0aiti ∈ C[[t]] that converges everywhere and satisfies

[k(a0, a1, a2, . . . ) : k] < ∞

is called an entire power series. As a function of t it takes values in k, when restricted to k. The ring of the entire power series is denoted by E.

If all entries of a solution Ψ of the Frobenius system in question are in E, one canspecializ Ψ to Ψ(θ).

(9)

### Rigid analytic trivialization

| · |:= a fixed absolute value for the completed field C. T := {f ∈ C[[t]] | f converges on |t|≤ 1}.

L := the fraction field of T.

Pre t-motive M is called rigid analytically trivial if there exists Ψ ∈ GLr(L) such that

Ψ(−1)= ΦΨ.

Such matrix Ψ is called a rigid analytic trivialization of the pre t-motive in question.

The category R of rigid analytically trivial pre-t-motives over Fq

forms a neutral Tannakian category over Fq(t).

(10)

### Rigid analytic trivialization

| · |:= a fixed absolute value for the completed field C. T := {f ∈ C[[t]] | f converges on |t|≤ 1}.

L := the fraction field of T.

Pre t-motive M is called rigid analytically trivial if there exists Ψ ∈ GLr(L) such that

Ψ(−1)= ΦΨ.

Such matrix Ψ is called a rigid analytic trivialization of the pre t-motive in question.

The category R of rigid analytically trivial pre-t-motives over Fq

forms a neutral Tannakian category over Fq(t).

(11)

### Tannakian duality

Given object M in R and let TM be the strictly full Tannakian subcategory of R generated by M . That is, TM consists of all objects of R isomorphic to subquotients of finite direct sums of

M⊗u⊗ (M)⊗v for various u, v,

where M is the dual of M . By Tannakianduality, TM is representable by an affine algebraic group scheme ΓM over Fq(t).

Such ΓM is called the motivic Galois groupof M .

Given rigid analytically trivial pre-t-motive M , the motivic Galois group ΓM is isomorphic over Fq(t) to the linear algebraic Galois group ΓΨ of the associated Frobenius difference equation.

(12)

### Tannakian duality

Given object M in R and let TM be the strictly full Tannakian subcategory of R generated by M . That is, TM consists of all objects of R isomorphic to subquotients of finite direct sums of

M⊗u⊗ (M)⊗v for various u, v,

where M is the dual of M . By Tannakianduality, TM is representable by an affine algebraic group scheme ΓM over Fq(t).

Such ΓM is called the motivic Galois groupof M .

Given rigid analytically trivial pre-t-motive M , the motivic Galois group ΓM is isomorphic over Fq(t) to the linear algebraic Galois group ΓΨ of the associated Frobenius difference equation.

(13)

### Tannakian duality

Given object M in R and let TM be the strictly full Tannakian subcategory of R generated by M . That is, TM consists of all objects of R isomorphic to subquotients of finite direct sums of

M⊗u⊗ (M)⊗v for various u, v,

where M is the dual of M . By Tannakianduality, TM is representable by an affine algebraic group scheme ΓM over Fq(t).

Such ΓM is called the motivic Galois groupof M .

Given rigid analytically trivial pre-t-motive M , the motivic Galois group ΓM is isomorphic over Fq(t) to the linear algebraic Galois group ΓΨ of the associated Frobenius difference equation.

(14)

### Papanikolas theory 2008

This algebraic Galois group ΓΨ from solution Ψ has the key property

dim ΓΨ = tr.degk(t) k(t)(Ψ).

If furthermore Ψ ∈ Matr(E) and satisfies

tr.deg¯k(t)¯k(t)(Ψ) = tr.deg¯k¯k(Ψ(θ)), then we say that M has the GP property. It follows that

dim ΓM = tr.deg¯k ¯k(Ψ(θ)).

Pre-t-motives having the GP property first come from

Anderson-Brownawell-Papanikolas 2004, through reformulating the submodule theorem of Yu 1997 which plays the role of W¨ustholz subgroup theorem (1989).

(15)

### Papanikolas theory 2008

This algebraic Galois group ΓΨ from solution Ψ has the key property

dim ΓΨ = tr.degk(t) k(t)(Ψ).

If furthermore Ψ ∈ Matr(E) and satisfies

tr.deg¯k(t)¯k(t)(Ψ) = tr.deg¯k¯k(Ψ(θ)), then we say that M has the GP property. It follows that

dim ΓM = tr.deg¯k ¯k(Ψ(θ)).

Pre-t-motives having the GP property first come from

Anderson-Brownawell-Papanikolas 2004, through reformulating the submodule theorem of Yu 1997 which plays the role of W¨ustholz subgroup theorem (1989).

(16)

### The Carlitz motive

The Carlitz motive C. Let C = ¯k(t) with σ-action:

σf = (t − θ)f(−1), f ∈ C.

Here Φ = (t − θ). Analytic solution Ψ of the equation Ψ(−1) = (t − θ)Ψ is given by

ΨC(t) = (−θ)−q/(q−1)

Y

i=1

(1 − t/θqi).

Note Galois group here is ΓC = Gm which has dimension 1.

Therefore ΨC(θ) = −1π˜ is transcendental over ¯k,

(17)

### Motivic Galois theory

We are interested in finitely generated extension of ¯k = Fq(θ) generated by a set S of special values, denoted by KS. In particular we want to determine all algebraic relations among elements of S.

From known algebraic relations, we can guess the transcendence degree of KS over ¯k, and the goal is to prove that is indeed the specific degree in question.

To proceed, we construct a t-motive MS for this purpose, so that it has the GP property and its “periods”ΨS(θ) from rigid analytic trivialization generate also the field KS, then computing the dimension of this motivic Galois group ΓMS.

(18)

### Motivic Galois theory

We are interested in finitely generated extension of ¯k = Fq(θ) generated by a set S of special values, denoted by KS. In particular we want to determine all algebraic relations among elements of S.

From known algebraic relations, we can guess the transcendence degree of KS over ¯k, and the goal is to prove that is indeed the specific degree in question.

To proceed, we construct a t-motive MS for this purpose, so that it has the GP property and its “periods”ΨS(θ) from rigid analytic trivialization generate also the field KS, then computing the dimension of this motivic Galois group ΓMS.

(19)

### Motivic Galois games

Suppose we have pre-t-motive M1 (M2) with GP property for set of values S1 (S2 respectively), and we are able to determine the Galois group ΓM1M2 respectively). To handle the set S1∪ S2, we form the direct sum of pre-t-motive M = M1⊕ M2. Then the dimension of the Galois group ΓM equals to the transcendence degree over ¯k of the compositum of the field KS1 and KS2 which is KS1∪S2. We have surjective morphisms from ΓM onto both ΓM1 and ΓM2. On many occasions this makes it possible to deduce the dimension of ΓM from the algebraic group structures of ΓM1 and ΓM2. As an example

(20)

### The End. Thank You.

P s ( dBm )=( P t ) dBm +( G t ) dB +( G r ) dB ( PL ( d )) dB (12) where P r ( d ) is the received power in dBm, which is a function of the T-R separation distance in meters, P t

In particular, if s = f(t) is the position function of a particle that moves along a straight line, then f ′(a) is the rate of change of the displacement s with respect to the

Then we can draw a right triangle with angle θ as in Figure 3 and deduce from the Pythagorean Theorem that the third side has length.. This enables us to read from the

T transforms S into a region R in the xy-plane called the image of S, consisting of the images of all points in S.... So we begin by finding the images of the sides

An n×n square is called an m–binary latin square if each row and column of it filled with exactly m “1”s and (n–m) “0”s. We are going to study the following question: Find

6 《中論·觀因緣品》，《佛藏要籍選刊》第 9 冊，上海古籍出版社 1994 年版，第 1

Chang-Yu 2005 proves that the Euler-Carlitz relations and the Frobenius relations generate all the algebraic relations among special Carlitz zeta values over the field ¯ k.. Jing

GCG method is developed to minimize the residual of the linear equation under some special functional.. Therefore, the minimality property does not hold... , k) in order to construct

Then, it is easy to see that there are 9 problems for which the iterative numbers of the algorithm using ψ α,θ,p in the case of θ = 1 and p = 3 are less than the one of the

● Using canonical formalism, we showed how to construct free energy (or partition function) in higher spin theory and verified the black holes and conical surpluses are S-dual.

Hence, we have shown the S-duality at the Poisson level for a D3-brane in R-R and NS-NS backgrounds.... Hence, we have shown the S-duality at the Poisson level for a D3-brane in R-R

3: Calculated ratio of dynamic structure factor S(k, ω) to static structure factor S(k) for &#34;-Ge at T = 1250K for several values of k, plotted as a function of ω, calculated

The case where all the ρ s are equal to identity shows that this is not true in general (in this case the irreducible representations are lines, and we have an inﬁnity of ways

• elearning pilot scheme (Four True Light Schools): WIFI construction, iPad procurement, elearning school visit and teacher training, English starts the elearning lesson.. 2012 •

– The futures price at time 0 is (p. 275), the expected value of S at time ∆t in a risk-neutral economy is..

– S+U can also preserve annotations of synthetic images – Refined images really help improving the testing result – Generate &gt; 1 images for each synthetic

For your reference, the following shows an alternative proof that is based on a combinatorial method... For each x ∈ S, we show that x contributes the same count to each side of

In the table boldface line was the S curve means students and dotted line was the P curve means problem.

(5)The Direction-Giving Language and the Empathetic Language of the principal have reach to the outstanding level of anticipa t i on f or t he t e a c he r ’ s j ob

It also has great influence on their leisure lives, or even health.The purpose of this study is to explore the extent to which leisure constraint and leisure activity demand to

At the 3000 m deep submarine trench near Lanyu, the Kz in the bottom layer is also very high (~ 1 m 2 s -1 ), indicating that effective turbulent mixing in the bottom layer is

CPI is a characteristic of the time series, so the purpose of this research is to use both the ARIMA models and the classical models in the time series analysis and to seek out

It would be a significant contribution in this academic field on Guei-Fu’ s philology after this research accomplished by showing a complete picture of his fabulous works as well as