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special values of L-functions

Idea

At certain integers $s = n$ the values $L(s)$ of L-functions $L$ are typically of special interest, these are known as the “special values of L-functions”.

For instance some of special values of the Riemann zeta function found (for the non-trivial region of non-positive integers) by Leonhard Euler in 1734 and 1749 are

n	-5	-3	-1	0	2	4	6	8
$\zeta(n)$	$-\frac{1}{252}$	$\frac{1}{120}$	$-\frac{1}{12}$	$-\frac{1}{2}$	$\frac{\pi^2}{6}$	$\frac{\pi^4}{90}$	$\frac{\pi^6}{945}$	$\frac{\pi^8}{9450}$

where for instance the value $\zeta(-1) = -\frac{1}{12}$ turns out to be the Euler characteristic of the moduli stack of complex elliptic curves and as such controls much of string theory (notably the critical dimension).

Periods and Deligne’s conjecture

More generally, it is known that for $K$ a number field and $\zeta_K$ its Dedekind zeta function, then all values $\zeta_K(n)$ for integer $n$ happen to be periods (MO comment).

Based on this (Deligne 79) identified critical values of $L$-functions (at certain integers) and conjectured that these all are algebraic multiples of determinants of matrices whose entries are periods. Review of this relation to periods in in (Konsevich Zagier, section 3).

Notice that under the function field analogy these arithmetic $L$-functions are supposed to be analogous to eta functions and zeta functions of elliptic differential operators, which, when regarding these operators as Dirac operators/Hamiltonians encode partition functions and path integrals of quantum mechanical systems, and that periods naturally appear as expectation values in quantum field theory (highlighted in Kontsevich 99).

Regulators and Beilinson’s conjectures

Another kind of special value is given by the classical class number formula which says that the residue of the Dedekind zeta function $\zeta_K$ of a number field $K$ at $n = 1$ is proportional to the regulator of the number field.

Also the Birch and Swinnerton-Dyer conjecture expresses the first non-vanishing derivative of the $L$-series of an elliptic curve over $\mathbb{Q}$ at $s= 1$ in terms of the regulator.

Motivated by this (Beilinson 85) generalized both Deligne’s conjecture and the statements about regulators by defining higher regulators (“Beilinson regulators”) and conjecturing a general statement that expresses special values of $L$-function in terms of these, now known as Beilinson's conjectures.

Periods and Scholl’s conjecture

However (Scholl 91) observed that also these Beilinson regulators have an expression in terms of periods and (Konsevich Zagier, section 3.2 and 3.5) advertises the

Deligne-Beilinson-Scholl conjecture: For $r$ the order of vanishing of a motivic $L$-function at some integer $n$, then the $r$th derivative of the function at that point is a period.

Related concepts

• Beilinson conjectures

context/function field analogy	theta function $\theta$	zeta function $\zeta$ (= Mellin transform of $\theta(0,-)$)	L-function $L_{\mathbf{z}}$ (= Mellin transform of $\theta(\mathbf{z},-)$)	eta function $\eta$	special values of L-functions
physics/2d CFT	partition function $\theta(\mathbf{z},\mathbf{\tau}) = Tr(\exp(-\mathbf{\tau} \cdot (D_\mathbf{z})^2))$ as function of complex structure $\mathbf{\tau}$ of worldsheet $\Sigma$ (hence polarization of phase space) and background gauge field/source $\mathbf{z}$	analytically continued trace of Feynman propagator $\zeta(s) = Tr_{reg}\left(\frac{1}{(D_{0})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(0,\tau)\, d\tau$	analytically continued trace of Feynman propagator in background gauge field $\mathbf{z}$: $L_{\mathbf{z}}(s) \coloneqq Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)^s = \int_{0}^\infty \tau^{s-1} \,\theta(\mathbf{z},\tau)\, d\tau$	analytically continued trace of Dirac propagator in background gauge field $\mathbf{z}$ $\eta_{\mathbf{z}}(s) = Tr_{reg} \left(\frac{sgn(D_{\mathbf{z}})}{ { \vert D_{\mathbf{z}} } \vert }\right)^s$	regularized 1-loop vacuum amplitude $pv\, L_{\mathbf{z}}(1) = Tr_{reg}\left(\frac{1}{(D_{\mathbf{z}})^2}\right)$ / regularized fermionic 1-loop vacuum amplitude $pv\, \eta_{\mathbf{z}}(1)= Tr_{reg} \left( \frac{D_{\mathbf{z}}}{(D_{\mathbf{z}})^2} \right)$ / vacuum energy $-\frac{1}{2}L_{\mathbf{z}}^\prime(0) = Z_H = \frac{1}{2}\ln\;det_{reg}(D_{\mathbf{z}}^2)$
Riemannian geometry (analysis)		zeta function of an elliptic differential operator	zeta function of an elliptic differential operator	eta function of a self-adjoint operator	functional determinant, analytic torsion
complex analytic geometry	section $\theta(\mathbf{z},\mathbf{\tau})$ of line bundle over Jacobian variety $J(\Sigma_{\mathbf{\tau}})$ in terms of covering coordinates $\mathbf{z}$ on $\mathbb{C}^g \to J(\Sigma_{\mathbf{\tau}})$	zeta function of a Riemann surface	Selberg zeta function		Dedekind eta function
arithmetic geometry for a function field		Goss zeta function (for arithmetic curves) and Weil zeta function (in higher dimensional arithmetic geometry)
arithmetic geometry for a number field	Hecke theta function, automorphic form	Dedekind zeta function (being the Artin L-function $L_{\mathbf{z}}$ for $\mathbf{z} = 0$ the trivial Galois representation)	Artin L-function $L_{\mathbf{z}}$ of a Galois representation $\mathbf{z}$, expressible “in coordinates” (by Artin reciprocity) as a finite-order Hecke L-function (for 1-dimensional representations) and generally (via Langlands correspondence) by an automorphic L-function (for higher dimensional reps)		class number $\cdot$ regulator
arithmetic geometry for $\mathbb{Q}$	Jacobi theta function ($\mathbf{z} = 0$)/ Dirichlet theta function ($\mathbf{z} = \chi$ a Dirichlet character)	Riemann zeta function (being the Dirichlet L-function $L_{\mathbf{z}}$ for Dirichlet character $\mathbf{z} = 0$)	Artin L-function of a Galois representation $\mathbf{z}$ , expressible “in coordinates” (via Artin reciprocity) as a Dirichlet L-function (for 1-dimensional Galois representations) and generally (via Langlands correspondence) as an automorphic L-function

References

Original article on the main conjectures are

• Pierre Deligne, Valeurs de fonctions $L$ et périodes d’intégrales, in Automorphic forms, Representations, and $L$-functions, Proc. Symp. Pure Math. 33, AMS (1979) pages 313-346

• Alexander Beilinson, Higher regulators and values of L-functions, Journal of Soviet Mathematics 30 (1985), 2036-2070 (web (Russian original))

• Anthony Scholl, Remarks on special values of L-functions, in J. Coates, M. Taylor (eds.) $L$-Functins and Arithmetic, London Math. Soc. Lecture Notes 153, Cambridge University Press 1991, pages 373-392 (pdf)

Review and further developments:

• Michael Rapoport, Norbert Schappacher, Peter Schneider (eds.), Beilinson's Conjectures on Special Values of L-Functions Perspectives in Mathematics, Volume 4, Academic Press, Inc. 1988 (ISBN:978-0-12-581120-0)

• Maxim Kontsevich, Don Zagier, section 3 of Periods (pdf)

• Matthias Flach, The equivariant Tamagawa Number Conjecture: A survey (pdf)

• Bruno Kahn, Fonctions zêta et $L$ de variétés et de motifs, arXiv:1512.09250.