WHAT IS A PERIOD ? Stefan M¨uller-Stach Mathematical constants like π, e and γ arise for integers si ≥ 1 with s1 ≥ 2 are very interesting frequently in number theory and other areas of periods. Even for the odd zeta-values ζ(3), ζ(5), mathematics and physics. Mathematicians have ... only a few results about their irrationality are long wondered whether such numbers are irra- known. By work of Ap´ery, ζ(3) is irrational. Like tional or perhaps even transcendental, that is, all multiple zeta values, it can be represented as not algebraic. Because they are solutions of an iterated integral ZZZ polynomial equations with rational coefficients, dxdydz algebraic numbers form a countable subset of ζ(3) = : (1 − x)yz the complex numbers. Therefore, most complex 0<x<y<z<1 numbers are transcendental, although, for any Using notions from algebraic geometry, one given number, it is usually difficult to figure out can also define periods in a different form as fol- whether it is transcendental. lows. Let X be a smooth algebraic variety over Q This essay is about the arithmetic notion of of dimension d. Take a regular algebraic d-form periods, a countable subalgebra P of the complex ! on X and a normal crossing divisor D in X; numbers defined around 1999 by Maxim Kontse- both ! and D are also defined over Q. Then let vich and Don Zagier [4]. Periods contain all alge- γ be a singular chain on the underlying topologi- braic numbers but also many other transcenden- cal manifold X(C) with boundary in D(C). The tal numbers important for number theory. This integral Z notion of periods generalizes in algebraic geome- p = ! try and yields the theory of periods and period γ domains for algebraic varieties, see [2]. is the period of the quadruple (X; D; !; γ). Kontsevich and Zagier define periods as those From an even higher viewpoint, periods are complex numbers whose real and imaginary parts matrix coefficients of the period isomorphism are values of absolutely convergent integrals ∼ Z ∗ ⊗ C−!= ∗ ⊗ C HdR(X; D) Q Hsing(X; D) Q f(x1; : : : ; xn) ··· p = dx1 dxn: between algebraic de Rham cohomology and sin- ∆ g(x1; : : : ; xn) gular cohomology after choosing Q-bases in both Here f and g are polynomials with coefficients in groups. In this case, X need not be smooth and Q, and the integration domain ∆ ⊂ Rn is given forms need not be of top degree. Sophisticated by polynomial inequalities with rational coeffi- arguments show that all three given definitions cients. of periods agree. Some initial examples of periods are Z Z In this setting, we find 2πi as the period of n dx H1(X) with X = P1 nf0; 1g, D = ;, ! = dx and log(n) = and π = dxdy: x 1 x x2+y2≤1 γ the unit circle. In a similar way, log(n) is one of the periods of H1(X; D), where X = P1 n f0; 1g, The representation of a period by an integral is D = f1; ng, ! = dx and γ = [1; n]. Certain not unique, in the sense that there are many dif- x special Γ-values occur in the Chowla-Selberg for- ferent integrals representing that period. P mula for periods of abelian varieties with com- The values ζ(s) = n−s of the Riemann n plex multiplication. The Beilinson conjectures, zeta function at positive integers s ≥ 2, and their which extend Dirchlet's class number formula in natural generalizations, the multiple zeta values X algebraic number theory, would imply that lead- − − − s1 s2 ··· sk ing terms in the Taylor series of L-functions of ζ(s1; :::; sk) = n1 n2 nk ; n1>···>nk>0 motives are periods in the extended period ring 1 2 Pb P 1 = [ π ], where π is inverted. A completely dif- assumptions [3]. The motivic Galois group is the ferent example comes from quantum field theory, pro-algebraic fundamental group G = Aut⊗(T ) where periods arise as values of regularized Feyn- of MM(Q) in the Tannaka sense. We call G a man amplitudes. Periods of homotopy groups are Galois group, as the viewpoint gives a far-ranging another source of examples. extension of the Galois theory of zero-dimensional In addition to the additivity in the integrand varieties. In MM(Q) cohomology groups of al- and the integration domain, periods inherit from gebraic pairs (X; D) are immediately mixed mo- calculus some well-knownR relations:R a change of tives, i.e., finite-dimensional Q-representations of variables formula f ∗! = !, and Stokes' G, or equivalently comodules over the associ- R R γ f∗γ P ated Hopf algebra A. Both singular and de formula γ δ! = @γ !. Fubini's theorem gives a multiplication, hence it becomes a Q-algebra. Rham cohomology provide fiber functors Tsing, TdR from MM(Q) to Q-vector spaces. The At the time Kontsevich and Zagier formulated ⊗ their idea, not a single explicit non-period num- pro-algebraic torsor Isom (TdR;Tsing) is given b b ber was known. In 2008, Masahiko Yoshinaga by Spec(Pformal), where Pformal is the algebra (arXiv:0805.0349) wrote down a computable non- of formal periods, i.e., generated by quadruples period, using a variant of Cantor's diagonal argu- (X; D; !; γ), and subject only to the relations of ment. Moreover, he showed that all periods are linearity, change of variables and Stokes [3, 4]. elementary computable, i.e., they lie in a certain In this setting, Pb is the set of periods of all proper subset of all computable complex num- mixed motives over Q. Multiple zeta values form bers, so that there are computable non-periods. the subset of periods of mixed Tate motives over It is still unknown whether e or 1/π are pe- Z. The motivic Galois group restricted to mixed riods. Presumably they are not. The notion of Tate motives over Z gives much control over mul- exponential periods was invented to extend peri- tiple zeta values and implies relations among mul- ods to a larger set containing e [4]. tiple zeta values ζ(s1; : : : ; sk) of a fixed weight Let us now turn to deeper properties of peri- s1 + ··· + sk. The work of Francis Brown on ods, so that we find out more about the struc- multiple zeta values and the fundamental group ture of P. It turns out that the very abstract of P1 n f0; 1; 1g demonstrates again the value of viewpoint of mixed motives provides insights, and the philosophy of motives [1]. Also in other parts brings into the game a big symmetry group G, the of number theory, for example in the area of ra- motivic Galois group. tional points, motivic arguments can be applied Pure and mixed motives were envisioned by in finiteness proofs. Alexander Grothendieck in order to formal- Grothendieck formulated the famous and dif- ize properties of algebraic varieties. In the ficult period conjecture, stating that any relation 1990s, Madhav Nori defined an abelian category among periods is coming from algebraic geom- MM(Q) of mixed motives over Q. In Nori's etry, in particular through algebraic cycles on construction, one starts with a directed graph, products of varieties. In the setting of Nori, this where the vertices are pairs of algebraic vari- is equivalent to saying that the evaluation map Q b b eties (X; D) defined over , and the edges be- ev : Pformal ! P is injective. This conjecture tween them are deduced from morphisms of pairs would have strong consequences for the transcen- 0 0 (X; D) ! (X ;D ) ("change of variables") and dence degree of the space of all periods of a given chains of inclusions Z ⊂ D ⊂ X ("Stokes' for- algebraic variety X via the action of G. mula"). The edges thus immediately resemble relations among periods, and this is what makes Further Reading: the idea so helpful. These arrows are not closed [1] F. Brown: Mixed Tate motives over Z, Annals of under composition. However, if one fixes a repre- Math. 175(2), 949-976 (2012). sentation T with values in vector spaces over Q, [2] J. Carlson and Ph. Griffiths: What is a period do- e.g. singular or de Rham cohomology, then there main ? AMS Notices Vol. 55 Nr. 11, 1418-1419 C (2008). is a universal diagram category (T ), and an ex- [3] A. Huber and S. M¨uller-Stach: On the rela- tension of T as a functor. After formally invert- tion between Nori motives and Kontsevich periods, ing the Tate motive Z(−1) = (P1 n f0; 1g; f1g) arXiv:1105.0865v5 (2011). in C(T ), one obtains a Q-linear Tannakian, hence [4] M. Kontsevich and D. Zagier: Periods, in Engquist, abelian, category MM(Q) without any further Bj¨orn(ed.) et al., Mathematics unlimited - 2001 and beyond, Springer Verlag, 771-808 (2001)..