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Periods. (English) Zbl 1039.11002
Engquist, Björn (ed.) et al., Mathematics unlimited – 2001 and beyond. Berlin: Springer (ISBN 3-540-66913-2/hbk). 771-808 (2001).
The authors define a period to be a complex number whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coefficients integrated over domains in \(\mathbb R^n\) described by polynomial inequalities with rational coefficients. For example the numbers \(\sqrt 2\), \(\pi\), \(\zeta(3)\) and \(\Gamma(5/4)^4\) are periods but the authors conjecture that \(e\) and Euler’s constant \(\gamma\) are not.
The first section of the paper explores the conjecture (the authors call it a “widely held belief”) that if a period has two integral representations then one can prove the identity of the two representations by means of the rules of elementary calculus. They illustrate this with some well-chosen examples.
Section 2 of the paper explores periods that appear as special values of the solutions to linear differential equations evaluated at algebraic arguments and exhibits some striking examples from the theory of elliptic curves, hypergeometric functions and modular forms.
In Section 3, the authors discuss conjectures expressing special values of \(L\)-functions as periods, in particular the far reaching conjecture of Deligne-Beilinson-Scholl concerning motivic \(L\)-functions and the Birch-Swinnerton-Dyer conjecture. In particular it is shown how the latter conjecture might be formulated as proving the equality of two periods and hence should be capable of an elementary proof for any particular elliptic curve by the meta conjecture of the first section.
The final section 4, due to the first author, is a sketch of an elementary approach to motives via periods.
The paper is well worth reading and should appeal to a wide variety of tastes.
For the entire collection see [Zbl 0955.00011].

11-02 Research exposition (monographs, survey articles) pertaining to number theory
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G55 Polylogarithms and relations with \(K\)-theory