On the Nature of the Special Values of Zeta Functions

Eki Ghate

November 14, 1996^{1}

1Updated April 19, 2001.

The Riemann zeta function

To say that the harmonic series

\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ⋯

diverges, is just to say that the Riemann zeta function

ζ(s) = ∞ ; ∑ ; n=1 \frac{1}{n^{s}} = _{p}^{∏} \frac{1}{1 − p^{−s}}

has a pole at s = 1.

In fact it is well known that ζ(s) converges absolutely in the half plane Re(s) > 1, can be analytically continued to the entire complex plane (except for the simple pole at s = 1), and satisfies the functional equation

ζ^{∗}(s) = ζ^{∗}(1 − s)

where

ζ^{∗}(s) = π^{−s/2}Γ(s/2)ζ(s).

Long ago (∼ 1740), Euler showed that one can explicitly compute ζ(2k) for k ≥ 1 :

\frac{1}{1^{2}} + \frac{1}{2^{2}} + \frac{1}{3^{2}} + \frac{1}{4^{2}} + ⋯ = \frac{π^{2}}{6}

\frac{1}{1^{4}} + \frac{1}{2^{4}} + \frac{1}{3^{4}} + \frac{1}{4^{4}} + ⋯ = \frac{π^{4}}{90}

\frac{1}{1^{6}} + \frac{1}{2^{6}} + \frac{1}{3^{6}} + \frac{1}{4^{6}} + ⋯ = \frac{π^{6}}{945}

⋮

\frac{1}{1^{2k}} + \frac{1}{2^{2k}} + \frac{1}{3^{2k}} + \frac{1}{4^{2k}} + ⋯ = \frac{(2π)^{2k}(−1)^{k+1}B_{2k}}{2(2k)!}

where B_{2k} is the 2k^{th} Bernoulli number defined by

\frac{z}{e^{z} − 1} = ∞ ; ∑ ; n=0 \frac{B_{n}}{n!}z^{n} |z| < 2π.

One way to show this: is to use Parseval’s Formula. This says that if f : [0, 2π] → R is Riemann integrable with Fourier coefficients

c_{n} = \frac{1}{2π} ∫_{0}^{2π} f(x)e^{−inx}dx

then

\frac{1}{2π} ∫_{0}^{2π} |f(x)|^{2}dx = ∞ ; ∑ ; n=−∞ |c_{n}|^{2}.

Example: f(x) = x, then

c_{0} = \frac{1}{2π} ∫_{0}^{2π} xdx = π and c_{n} = \frac{1}{2π} ∫_{0}^{2π} xe^{−inx}dx = i/n, for n ≠ 0

so that

π^{2} + 2 ∞ ; ∑ ; n=1 \frac{1}{n^{2}} = \frac{1}{2π} ∫_{0}^{2π} x^{2}dx = \frac{(2π)^{2}}{3} = \frac{4π^{2}}{3},

or

∞ ; ∑ ; n=1 \frac{1}{n^{2}} = \frac{π^{2}}{6}.

To get ζ(2k), just use F (x) = x^{k}.

What about the odd integers?

On the other hand, the special values ζ(2k + 1) are much more “mysterious” and have defied computation! In fact, it was only in 1978 that Apéry showed that

ζ(3) is irrational!!

He used the identity

∞ ; ∑ ; n=1 \frac{1}{n^{3}} = \frac{5}{2} ∞ ; ∑ ; n=1 \frac{(−1)^{n}}{n^{3}\binom{2n}{n}}.

• No such identity is known for ζ(5). But recently Rivoal has shown by other methods that infinitely many ζ(2k + 1) are irrational and that at least one ζ(2k + 1) with 5 ≤ 2k + 1 ≤ 169 is irrational.

• We will see later that ζ(2k+1) is connected to the K-groups of Z and to “higher logarithms ”...

Dirichlet L-functions

For each (primitive) character

χ : (Z/NZ)^{×} ⟶ C^{×},

Dirichlet defined the L-function

L(s, χ) = ∞ ; ∑ ; n=1 \frac{χ(n)}{n^{s}} = _{p}^{∏} \frac{1}{1 − χ(p)p^{−s}}.

Around 1880 Hurwitz showed that

\begin{cases}L(2k, χ) = \frac{G(χ)(2π/N)^{2k}(−1)^{k+1}B_{2k,χ̄}}{2(2k)!} & if χ(−1) = 1 \\ L(2k + 1, χ) = \frac{G(χ)(2π/N)^{2k+1}(−1)^{k+1}B_{2k+1,χ̄}}{2i(2k+1)!} & if χ(−1) = −1 \end{cases}

where

•

B_{n,χ} = N^{n−1} N ; ∑ ; a=1 χ(a)B_{n}(a/N)

is the n^{th} generalized Bernoulli number, defined in terms of the n^{th} Bernoulli polynomial B_{n}(x):

\frac{ze^{xz}}{e^{z} − 1} = ∞ ; ∑ ; n=0 \frac{B_{n}(x)}{n!}z^{n}

• G(χ) is the Gauss sum

G(χ) = N ; ∑ ; a=1 χ(a)e^{2πai/N}.

Example: Let χ : (Z/7Z)^{×} → C^{×} be given by χ(3) = ω = e^{2πi/3}. Then

L(2, χ) = \frac{1}{1^{2}} + \frac{ω^{2}}{2^{2}} + \frac{ω}{3^{2}} + \frac{ω}{4^{2}} + ⋯ = \frac{G(χ)4π^{2}(2 − ω^{2})}{343}

• There is also a formula for L(1, χ), χ ≠ 1

• Most of the other special values of ‘opposite parity’ are more mysterious

Dedekind zeta functions

One can attach a zeta function to any number field F, namely

ζ_{F} (s) = _{𝔄}^{∑} \frac{1}{N(𝔄)^{s}} = _{℘}^{∏} \frac{1}{1 − N(℘)^{−s}},

where N(𝔄) = [O_{F} : 𝔄], and O_{F} is the ring of integers of F.

Again, ζ_{F} (s) can be analytically continued to all of C, except for a simple pole at s = 1, and satisfies a functional equation:

ζ_{F}^{∗} (s) = ζ_{F}^{∗} (1 − s).

Here

ζ_{F}^{∗} (s) = \sqrt{|D|}^{s/2}(π^{−s/2}Γ(s/2))^{n_{+}}(π^{−s/2}Γ(s + 1/2))^{n_{−}}ζ_{F} (s).

where

• D is the discriminant of F,

• n = n_{+} + n_{−} is the degree of F over Q, with

n_{+} = r_{1} + r_{2}, n_{−} = r_{2},

where r_{1} = number of imbeddings F into R and r_{2} = number of imbeddings of F into C (modulo complex conjugation).

Example: Let F = Q(\sqrt{5}). Then n = 2, n_{+} = 2 and n_{−} = 0. Moreover every (integral) ideal of O_{F} is principal, generated by an element of the form

m + n(\frac{1 + \sqrt{5}}{2}) m, n ∈ Z,

so that

ζ_{F} (s) = ∞ ; ∑ ; m=1 m−1 ; ∑ ; n=1 \frac{1}{(m^{2} + mn − n^{2})^{s}}.

What can we say about the special values of ζ_{F} (s)? We have the Theorem 1 (Klingen-Siegel, 1962-1969) If F it totally real, i.e. n_{−} = 0, then for all k ≥ 1,

ζ_{F} (2k) = rational number × \frac{π^{2kn_{+}}}{\sqrt{D}} .

Moreover this rational number is explicitly computable.

Example: If F = Q(\sqrt{5}), then

ζ(12) = ∞ ; ∑ ; m=1 m−1 ; ∑ ; n=1 \frac{1}{(m^{2} + mn − n^{2})^{12}} = \frac{2^{4} · 691 · 1150921}{3^{10} · 5^{16} · 7^{3} · 13} × \frac{π^{24}}{\sqrt{5}}.

This example is really “cheating”, since if F/Q is abelian then

ζ_{F} (s) = _{χ}^{∏} L(s, χ)

where the product is over all χ belonging to the subgroup of the characters of

Gal(Q(ω_{N})/Q) = (Z/NZ)^{×}

which are trivial on Gal(Q(ω_{N})/F). Here N is the smallest integer such that F ⊂ Q(ω_{N}). The point of the theorem is that it gives information in the NON-ABELIAN totally real case!

Idea of Proof: Uses the theory of modular forms (of even weight 2k), i.e. the theory of functions on the upper-half complex plane H

f : H → C

which are holomorphic on H and the cusp ∞ and which satisfy the ‘automorphy condition’

f(γz) = (cz + d)^{2k}f(z)

for all γ = ({a \atop c} {b \atop d}) ∈ SL_{2}(Z) and z ∈ H.

Indeed, for each ideal 𝔄 ⊂ O_{F}

E_{k}(z, 𝔄) = \frac{1}{N(𝔄)^{2k}} _{(m,n)∈𝔄×𝔄/O_{F}^{×}}^{∑} _{σ}^{∏} \frac{1}{(m^{σ}z + n^{σ})^{2k}}

is such a modular form of weight 2kn. Set

E_{k}(z) = _{[𝔄]∈Cl_{F}}^{∑} E_{k}(z, 𝔄).

Then on the one hand E_{k}(z) has the Fourier expansion

E_{k}(z) = ζ_{F} (2k) + \frac{(2πi)^{2kn_{+}} \sqrt{D}^{2k−1/2}}{(k − 1)!} ∞ ; ∑ ; l=1 c_{l}q^{l}.

On the other hand, Siegel shows that any f = ∑ a_{l}q^{l} satisfies the relation

c_{k,0}a_{0} + c_{k,1}a_{1} + ⋯ + c_{k_{r}}a_{r} = 0

for some r, and for some explicitly computable rational constants c_{k,r}, with c_{k,0} ≠ 0. □

Deligne’s philosophy

What makes the special values considered so far really ‘special’???

Answer: These are the so called ‘critical values’ of the zeta functions we have been looking at, that is, these are exactly the values at which none of the Gamma factors appearing in (both sides of) the functional equation have poles!

Deligne (1979) “It is possible to describe the critical values of any MOTIVIC zeta functions up to multiplication by a rational number”

Definition: A motive M over Q, of weight w, is a collection of cohomology theories with certain structure:

• M_{B} = Q-vector space with an action of an involution F_{∞}, having ‘Hodge decomposition’

M_{B} ⊗ C = _{p+q=w}^{⨁} H^{p,q}.

Here F_{∞} ⊗ 1 : H^{p,q} ↦ H^{q,p}. Set h^{p,q} = dim H^{p,q}.

• M_{dR} = Q-vector space with a decreasing filtration F^{p}(M_{dR})

• M_{ℓ} = Q_{ℓ}-vector space with an action of Gal(Q̄/Q)

that are comparable via certain ‘comparison isomorphisms’:

• I : M_{B} ⊗_{Q} C ⟶ M_{dR} ⊗_{Q} C with ⨁_{p′≥p} H^{p′,q} ↦ F^{p}(M_{dR}) ⊗_{Q} C

• I_{ℓ} : M_{B} ⊗_{Q} Q_{ℓ} ⟶ M_{ℓ}.

The common dimension

dim_{Q}M_{B} = dim_{Q}M_{dR} = dim_{_{Q}_{ℓ}}M_{ℓ} = d

is call the rank of M.

To such a motive M we may associate an L-function L(s, M) defined as follows:

L(s, M) = _{p}^{∏} L_{p}(s, M)

where

L_{p}(s, M) = det(1 − Frob(p)|_{M_{ℓ}_{Ip}}p^{−s})^{−1}.

Set

Γ_{R}(s) = π^{−s/2}Γ(s/2)
Γ_{C}(s) = 2 · (2π)^{−s}Γ(s)

and let

L_{∞}(s, M) = ∏_{{(p,q) \atop p<q}} Γ_{C}(s − p)^{h^{p,q}} · _{(p,p)}^{∏} 1 ; ∏ ; ϵ=0 Γ_{R}(s − p + ϵ)^{h_{ϵ}^{p,p}}

where h_{ϵ}^{p,p} is the dimension of the (−1)^{p+ϵ} eigenspace of F_{∞} ⊗ 1 acting on H^{p,p}.

Then the full L-function associated to M

Λ(s, M) = L_{∞}(s, M)L(s, M)

(conjecturally) satisfies the functional equation

Λ(s, M) = ε(s, M)Λ(M^{∨}, 1 − s)

where

• ϵ(s, M) = a · b^{s} is an ε-factor

• M^{∨} denotes the ‘dual motive’.

Definition: An integer n is called critical if neither

L_{∞}(s, M) nor L_{∞}(M^{∨}, 1 − s)

has a pole at s = n.

We want to describe Deligne’s conjecture on the critical values of L(s, M). If M(n) denotes the n-fold Tate twist of M, then we have the identity

L(M(n), s) = L(s + n, M).

In view of this, it is only necessary to consider the case when 0 is a critical value of L(s, M). In this case we have

Conjecture 1 (Deligne, 1979)

L(0, M) = rational number × c^{+}(M)

where c^{+}(M) = det(I^{+}) is the determinant of the map

I^{+} : M_{B}^{+} ⊗_{Q} C → M_{B} ⊗_{Q} C ⟶^{I} M_{dR} ⊗_{Q} C → M_{dR}^{+} ⊗_{Q} C.

Example: Let M = Res_{F /Q}(Q(0)_{/F}). This is a motive of weight 0 and rank n. There is only one (p, p) = (0, 0) space in the Hodge decomposition, on which F_{∞} acts by

(\begin{matrix}1 \\ & 1 \\ & & ⋯ \\ & & & ⋯ \\ & & & & −1 \\ & & & & & −1 \end{matrix} )

It turns out that L(s, M) = ζ_{F} (s) is the Dedekind zeta function and

L_{∞}(s, M) = Γ_{R}(s)^{n_{+}}Γ_{R}(s + 1)^{n_{−}}

are exactly the Gamma factors that appeared in the functional equation!

What are the critical integers of ζ_{F} (s)? Answer: those integers for which none of the Gamma factors

Γ(s/2)^{n_{+}}, Γ((s + 1)/2)^{n_{−}}, Γ((1 − s)/2)^{n_{+}}, Γ((2 − s)/2)^{n_{−}}

have a pole. But there are no such integers UNLESS n^{−} = 0, that is, F is totally real. In this case the even positive integers are critical:

Furthermore, one may compute

c^{+}(M(2k)) = \frac{(2πi)^{2kn}}{\sqrt{D}}

up to multiplication by a rational number!

What about the non-critical values of ζ_{F}(s)?

We start by looking at the residue of the pole at s = 1.

Theorem 2 (Dirichlet)

lim_{s→1} (s − 1)ζ_{F} (s) = \frac{2^{r_{1}}(2π)^{r_{2}}hR}{w\sqrt{|D|}}

where

• h is the class number of F

• w is the number of roots of unity in F • R is the REGULATOR = the co-volume of the lattice

Im(r), where r : O_{F}^{×} → R^{r_{1}+r_{2}−1} is the map given by

ϵ ↦ (ln|ϵ^{σ_{1}}|, ⋯, ln|ϵ^{σ_{r_{1}}}|, 2ln|ϵ^{σ_{r_{1}+1}}|, ⋯, 2ln|ϵ^{σ_{r_{1}+r_{2}−1}}|)

Hence R is the absolute value of det[e_{j}ln|ϵ_{i}^{σ_{j}}|], where

ϵ_{1}, ⋯, ϵ_{r_{1}+r_{2}−1}

generate the torsion free part of O_{F}^{×}, and e_{j} = 1 or 2.

It is enlightening to rewrite this as

lim_{s→0} \frac{ζ_{F} (s)}{s^{r_{1}+r_{2}−1}} = −\frac{hR}{w} = −\frac{#K_{0}(O_{F})_{tors}}{#K_{1}(O_{F})_{tors}} R.

Here K_{i}(O_{F})_{tors} is the torsion part of the (finitely generated abelian) group K_{i}(O_{F}). Recall that:

K_{0}(O_{F}) = Z ⊕ Cl_{F}
K_{1}(O_{F}) = O_{F}^{×} = μ_{f} ⊕ Z^{r_{1}+r_{2}−1}
K_{2m}(O_{F}) has rank = 0, m > 0
K_{2m−1}(O_{F}) has rank = \begin{cases}n_{−} = r_{2} & if (−1)^{m} = 1 \\ n_{+} = r_{1} + r_{2} & if (−1)^{m} = −1, m > 1 \end{cases}

Coincidence? The rank of K_{2m−1}(O_{F}) = n_{∓} = order of vanishing of ζ_{F} (s) at s = 1 − m for m ≥ 1.

Conjecture 2 (Lichtenbaum, 1973) Let m > 1. Then

lim_{s→1−m} \frac{ζ_{F} (s)}{(s − (1 − m))^{n^{∓}}} = ±\frac{#K_{2m−2}(O_{F})}{#K_{2m−1}(O_{F})_{tors}} R_{m}

for a certain regulator, R_{m}, defined via K_{2m−1}(O_{F}).

Borel showed that the above conjecture is true up to multiplication by a rational number:

Theorem 3 (Borel, 1977)

lim_{s→1−m} \frac{ζ_{F} (s)}{(s − (1 − m))^{n^{∓}}} = rational number × R_{m}.

Idea of Proof: Let G = Res_{F/Q}SL_{n}_{/F} for n a sufficiently large odd integer. Let K = SO_{n}^{r_{1}} × SU_{n}^{r_{2}} be the standard maximal compact subgroup of G(R), let G_{u} = SU_{n}^{r_{1}+2r_{2}} be the standard maximal subgroup of G(C), and let Γ be an arithmetic subgroup of G(Q) Then there exists a map

j_{Γ} : H^{∗}(K\G_{u}, R) ⟶ H^{∗}(Γ, R)

There is a canonical rational structure on both these spaces, and comparing the indecomposable parts of degree 2m − 1, which both have dimension n_{∓}, in rational bases, one can show that they differ by the regulator

R_{m} = \frac{\sqrt{|D|}ζ_{F} (m)}{π^{mn^{∓}}}. □

Can we compute this regulator explicitly?

Zagier realized that we may be able to express these “higher regulators” in terms of the “higher logarithms” or “polylogarithms”. The following theorem computes the value of the Dedekind zeta function at s = 2.

Theorem 4 (Zagier, 1986)

ζ_{F} (2) = \frac{π^{2n_{−}}}{\sqrt{|D|}} × _{v}^{∑} c_{v}A(x_{v,1})A(x_{v,2}) ⋯ A(x_{v,n_{−}})

where A(x) is the real valued function

A(x) = ∫_{0}^{x} \frac{1}{1 + t^{2}}ln\frac{4}{1 + t^{2}}dt

and the x_{v,j} are real algebraic numbers of degree at most 8 over F.

Example: Let F = Q(\sqrt{−7}), so n_{−} = 1. Then

ζ_{F} (2) = \frac{2π^{2}}{7\sqrt{7}}(2A(\sqrt{7}) + A(\sqrt{7} + 2\sqrt{3}) + A(\sqrt{7} − 2\sqrt{3}))

Remark: We are ‘cheating’ again! For if F/Q is imaginary quadratic (more generally abelian) then the theorem is well known: Say F = Q(\sqrt{D}) has discriminant D < 0. Recall

ζ_{F} (s) = ζ(s)L(s, χ)

where χ(n) = (\frac{D}{n} ) : Z/|D|Z → ±1 is the quadratic character associated to F. Since χ is odd and periodic it has a Fourier sine expansion:

χ(n) = \frac{1}{\sqrt{|D|}} _{0<k<|D|}^{∑} χ(k) sin\frac{2πkn}{|D|}

Let

Π(θ) = \frac{1}{2} ∞ ; ∑ ; n=1 sin\frac{2nθ}{n^{2}} = − ∫_{0}^{θ} log |2sint|dt.

be the Lobachevsky function. Since

\frac{d}{dx}Π(arccot x) = \frac{1}{2} \frac{1}{1 + x^{2}}log\frac{4}{1 + x^{2}},

we see that

A(x) = 2Π(arccot x).

Hence

L(2, χ) = \frac{2}{\sqrt{|D|}} _{0<k<|D|}^{∑} χ(k)Π(\frac{πk}{|D|})
= \frac{1}{\sqrt{|D|}} _{0<k<|D|}^{∑} χ(k)A(cot\frac{πk}{|D|}).

Example revisited: Thus when F = Q(\sqrt{−7}), we get

ζ_{F} (2) = \frac{π^{2}}{3\sqrt{7}}(A(cot\frac{π}{7} )+A(cot\frac{2π}{7} )+A(cot\frac{4π}{7} )).

Thus the point of Zagier’s theorem is that it computes ζ_{F} (2) for NON-ABELIAN extensions F/Q as well!

Idea of Proof: The proof is based on computing the volume of certain arithmetic hyperbolic 3-manifolds. Consider the case n_{−} = 1 for simplicity. Let B ≠ M_{2}(F) be any quaternion algebra ramified at all real places σ:

B ⊗_{F,σ} R = H.

Let Γ be a subgroup of finite index in O^{×}, where O is an order of B. Picking one complex imbedding τ, we have

B ⊗_{F,τ} C = M_{2}(C)

which allows us to identify Γ as a discrete subgroup of SL_{2}(C).

Let ℋ denote hyperbolic 3-space. Then Γ acts on ℋ freely and properly discontinuously, and the manifold Γ\ℋ is smooth and compact. On the one hand it is known that

vol(Γ\ℋ) = \frac{ζ_{F} (2)}{π^{2n_{+}} \sqrt{|D|}}.

On the other hand, by choosing a suitable triangulation of this manifold, one can show that vol(Γ\ℋ) is given by rational linear combination of values of A(x) with algebraic arguments. □

Zagier realized that A(x) is closely related to the dilogarithm function Li_{2}(x), where for k ≥ 1

Li_{k}(x) = ∞ ; ∑ ; n=1 \frac{x^{k}}{n^{k}} |x| ≤ 1.

In fact, let

D_{2}(x) = \begin{cases}Imag(Li_{2}(x) + log|x|log(1 − x) & |x| ≤ 1 \\ −D_{2}(1/x) & |x| ≥ 1 \end{cases}

be the Bloch-Wigner function. Then D_{2} satisfies a number of ‘functional equations’ all of which have a geometric interpretation. In particular, if a, b, c, d ∈ P^{1}(C) = ∂ℋ, then

D_{2}(\frac{a − c}{a − d} \frac{b − d}{b − c})

is the volume of the ideal hyperbolic tetrahedron with vertices a, b, c, d. Thus, for instance (in the case n_{−} = 1), Zagier’s theorem can be rephrased as

ζ_{F} (2) = \frac{π^{2n_{+}}}{\sqrt{|D|}} _{v}^{∑} D_{2}(x_{v}), x_{v} ∈ F.

Example revisited again: Let F = Q(\sqrt{−7}). Then

ζ_{F} (2) = \frac{2^{2}π^{2}}{3\sqrt{7}^{3}} (2D_{2}(\frac{1 + \sqrt{−7}}{2}) + D_{2}(\frac{−1 + \sqrt{−7}}{4}))

This, along with extensive numerical work, led to the following Conjecture 3 (Zagier, 1989)

ζ_{F} (m) = rational number · \frac{π^{mn_{±}}}{\sqrt{|D|}} · R

where R is a n_{∓} × n_{∓}-determinant whose entries are Z linear combinations of D_{m}(x) with x ∈ F, where D_{m} : C → R is a linear combination of of the polylogarithm functions Li_{k}(x) for k ≤ m.

Let us look at some examples for m = 3. In this case

D_{3}(x) = Li_{3}(x) − log|x|Li_{2}(x) − \frac{1}{2}log^{2}|x|log(1 − x) + \frac{1}{12}log^{3}|x|.

Example: F = Q. Then

ζ_{F} (3) = ζ(3) = Li_{3}(1) = D_{3}(1),
ζ(3) = 5(D_{3}(\frac{1 + \sqrt{5}}{2}) + D_{3}(\frac{1 − \sqrt{5}}{2})).

Example: F = Q(\sqrt{5}) and χ = (^{\underline{5}}). Then

L(3, χ) = \frac{32}{75\sqrt{5}}·

(D_{3}(2 − \sqrt{5}) − D_{3}(2 + \sqrt{5}) + 3D_{3}(\frac{1 + \sqrt{5}}{2}) − 3D_{3}(\frac{1 − \sqrt{5}}{2})).

Assume now that F is totally real. Let F_{F} denote the free abelian group on F^{×}:

F_{F} = { _{i}^{∑} n_{i} x_{i} | n_{i} ∈ Z, x_{i} ∈ F^{×}},

and let A_{3}(F) = ker(β_{3}), where

β_{3} : F_{F} ⟶ F^{×} ⊗ F^{×} ⊗ F^{×} ⊗ Q

x ↦ x ⊗ x ⊗ (\frac{x}{(1 − x)^{2}} ), 1 ↦ 0.

Let B_{3}(F) = A_{3}(F)/C_{3}(F) denote the third Bloch group of F, where C_{3} ⊂ A_{3} is a suitably defined subgroup. For each (real) imbedding σ of F, define

D_{3}^{σ} : B_{3}(F) → R
D_{3}^{σ}(_{i}^{∑} n_{i} x_{i}) = _{i}^{∑} n_{i} D_{3}(x_{i}^{σ})

Finally let D_{3}^{F} : B_{3}(F) → R^{n_{+}} denote ∏_{σ} D_{3}^{σ}. Then in the example above we see that if

u = (\frac{1+\sqrt{5}}{2}) + (\frac{1−\sqrt{5}}{2})
v = (2 − \sqrt{5}) − (2 + \sqrt{5}) + 3(\frac{1+\sqrt{5}}{2}) − 3(\frac{1−\sqrt{5}}{2}),

then up to rational multiples,

D_{3}^{F} (u) = (ζ(3), ζ(3))
D_{3}^{F} (v) = (\sqrt{5}L(s, χ), −\sqrt{5}L(s, χ))

and so span a lattice with co-volume a rational multiple of \sqrt{5}ζ_{F} (3)!

Theorem 5 (Goncharov, 1991) There is an isomorphism

K_{5}(O_{F}) ⊗ Q → B_{3}(F) ⊗ Q

such that the composition with D_{3}^{F} is the Borel regulator map.

Remark: Thus Zagier’s conjecture is true for m = 3. For general m, Beilinson has constructed a map

B_{m}(F) ⊗ Q → K_{2m−1}(O_{F}) ⊗ Q

whose composition with the Borel regulator coincides with D_{m}^{F}, the analogue of D_{3}^{F} defined above. This shows that the image of D_{m}^{F} is CONTAINED in the Borel lattice, but may not be all of it!

So where does my thesis fit into all this?

We investigate the CRITICAL VALUES of the twisted tensor L-function:

Abelian L-functions Non-Abelian L-functions

Associated to GL_{1}_{/F} Associated to GL_{n}_{/F}, n ≥ 2 E.g. ζ(s), L(s, χ), ζ_{F} (s) E.g. L(s, Δ) = ∑_{n} \frac{τ(n)}{n^{s}} where

Δ = ∏_{n} q(1 − q^{n})^{24} = ∑_{n} τ(n)q^{n}

Degree 1 Euler products Degree n ≥ 2 Euler Products

Motivic Conjectured to be so!

Critical values investigated Critical values investigated by ..., Shimura, Deligne, by ..., Shimura, Hida, ... Blasius, ...

My Thesis: Let f be a cusp form over an imaginary quadratic field K, and let f^{c} denote the conjugate form. Then the Lfunction L(s, f × f^{c}) splits into two twisted tensor L-functions:

G(s, f) and G(s, f ⊗ χ) where χ is the quadratic character associated to F.

Theorem 6 Modulo the existence of motives, G(s, f) behaves as predicted by Deligne’s conjecture: for every even critical integer j,

G(j, f) = Ω(f) · (2π)^{3j−w}

up to multiplication by an algebraic number. Here Ω(f) is a constant, and w depends only on the weight of f.