Periods in Arithmetic and Motivic Geometry – PERGAMO

Periods are a class of complex numbers obtained by integrating algebraic differential forms over algebraically defined domains which only involve rational coefficients. Examples include logarithms of integers, multiple zeta values and certain amplitudes in string and quantum field theory. From the modern point of view, periods appear as entries of a matrix of the comparison isomorphism between algebraic de Rham and Betti cohomology of varieties over number fields. Thanks to this interpretation, the theory of motives becomes a powerful tool to predict all algebraic relations among these numbers and, in some favourable cases, to prove them. It should be thought of as a higher analogue of the Galois theory of algebraic numbers. Indeed, all recent breakthroughs in the study of periods, namely Ayoub's theorem (a relative version of the Kontsevich-Zagier conjecture) and Brown's theorem (every multiple zeta value can be written as a linear combination of those having only 2 and 3 as exponents), were the reflection of the emergence of new ideas and techniques in motivic Galois theory. This JCJC project aims at gathering together young researchers working on the theory of periods and motives form different points of view, the interaction of which seems particularly promising. The researchers were selected according to pre-existing collaborations and the perspective of initiating new ones. We plan to tackle, among others, the following questions: 1) Mixed Tate motives – Give geometric constructions of the extensions of Q(0) by Q(n) and relate them to irrationality proofs of zeta values. Find generators of the category of mixed Tate motives over the ring of integers of cyclotomic fields. 2) Motivic Feynman amplitudes – Study the motives associated with Feynman graphs and the coaction conjecture by Brown, Panzer, and Schnetz, according to which motivic Feynman amplitudes are stable under the Galois action. 3) Exponential motives – Establish a Newton above Hodge theorem for the irregular Hodge filtration and the eigenvalues of Frobenius on the de Rham cohomology associated with a smooth variety together with a regular function. 4) Operads, motives, and the Grothendieck-Teichmüller group – Explain the role of periods in the proofs of the formality of the little disks operad. Study the action of the tannakian group of the category of mixed Tate motives over Z on certain operads coming from algebraic topology. Compute the Galois action on the multiple zeta values appearing as "Kontsevich weights" in deformation quantisation. 5) Motivic Galois group in positive characteristic – Pursue the study of de Rham-like realisation functors on motives in positive characteristic. Understand the structure of the derived Hopf algebras obtained by applying the weak tannakian formalism in order to define a motivic Galois group and tackle the variant of the Grothendieck period conjecture in this setting. 6) p-adic periods – Use the recent work of Bhatt-Morrow-Scholze to study integral aspects of p-adic periods (e.g. multiple zeta values). Compare the rational structures on the l-adic cohomology of varieties over number fields arising from algebraic cycles on their reductions modulo p. 7) BCOV invariants of arithmetic Calabi-Yau varieties –Compute the unknown constant in the proof of the BCOV conjecture for the Dwork pencil in terms of special values of the gamma function and explain the result in the spirit of the Gross-Deligne conjecture by exhibiting a motive with complex multiplication. Undertake a similar study in other arithmetic situations.