André's motivic formulation of the Grothendieck period conjecture

Let XX be a pure or mixed motive defined over Q\overline{\mathbb{Q}}. Write periods(X)\mathrm{periods}(X) for its periods and Gmot(X)\mathcal{G}_{\mathrm{mot}}(X) for its motivic Galois group. Grothendieck period conjecture.

tr.degQQ(periods(X))=dimGmot(X).\operatorname{tr.deg}_{\mathbb{Q}}\,\overline{\mathbb{Q}}(\mathrm{periods}(X))=\dim\mathcal{G}_{\mathrm{mot}}(X).

Here Q(periods(X))\overline{\mathbb{Q}}(\mathrm{periods}(X)) is the field generated over Q\overline{\mathbb{Q}} by the periods of XX. The source notes that the full conjecture also includes connectedness of the period torsor, while this equality is the formulation used in transcendence theory.

Sources & referencesView supporting material

Primary source

Cristiana Bertolin, Patrice Philippon, Biswajyoti Saha and Ekata Saha, “Semi-abelian analogues of Schanuel Conjecture and applications”, arXiv:2010.15170 (2022).

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