On André periods of mixed Tate motives

Technical Summary: On André Periods of Mixed Tate Motives

Problem Statement
The paper addresses the relationship between two distinct notions of $p$-adic periods in the context of Mixed Tate Motives (MTM). On one hand, there is the "classical" notion of $p$-adic periods, developed by Yamashita, Chatzistamatiou, and Üner, which arises in the study of $p$-adic iterated integrals and Chabauty-Kim theory. This theory relies on comparing the Hodge filtration and the Frobenius automorphism on the $p$-adic de Rham realization. On the other hand, Ancona and Frătilă recently introduced a notion of "$p$-adic periods" (termed "André periods") for pure motives, inspired by Y. André's work. This approach compares the rational de Rham realization with the crystalline realization, emphasizing the factorization through the category of motives over the residue field.

While the classical theory is well-established for Mixed Tate motives, the Ancona-Frătilă framework was primarily developed for pure motives, with mixed motives treated as an afterthought. The central problem is to determine whether the Ancona-Frătilă definition, when applied to the specific setting of Mixed Tate motives, recovers the classical $p$-adic periods associated with $p$-adic multiple zeta values and polylogarithms.

Methodology
The author, Ishai Dan-Cohen, employs the language of Tannakian categories and fiber functors to bridge these two definitions. The methodology involves:

1. Translation of Frameworks: The Ancona-Frătilă construction is transported to the setting of Mixed Tate motives over an open integer scheme $Z \subset \text{Spec } \mathcal{O}_K$. This involves defining a functor $H(Z)$ representing the space of $\otimes$-compatible natural transformations between the specialization of the crystalline realization (via a functor $\zeta$) and the de Rham realization.
2. Construction of Period Maps:
   • The Ancona-Frătilă period map ($per_{AF}$) is defined via the "Berthelot point" ($c_{Ber}$), a specific $K_p$-point of $H(Z)$ constructed using the Berthelot comparison isomorphism between crystalline and de Rham realizations.
   • The classical period map ($per_{Cl}$) is defined via the "classical $p$-adic period loop" ($u$), which compares the arithmetic Hodge path and the arithmetic crystalline path within the unipotent radical $U_{dR}(Z)$ of the Tannakian Galois group.
3. Tannakian Comparison: The core technical work involves proving that the presheaf $H(Z)$ is a trivial torsor under the prounipotent group $U_{dR}(Z)$. This is achieved by:
   • Utilizing the equivalence between the category of semisimple Mixed Tate motives and representations of $\mathbb{G}_m$.
   • Applying a general Tannakian result (Proposition 4.2) regarding the isomorphism between quotient schemes and spaces of natural transformations.
   • Demonstrating that the orbit map induced by a specific de Rham point ($c_{dR}$) provides an isomorphism between $U_{dR}(Z)$ and $H(Z)$ that intertwines the two period loops.

Key Contributions and Results

• Theorem 1.1 (Equivalence of Period Maps): The primary result establishes an isomorphism of $K$-algebras between the ring of "formal André periods" ($\mathcal{O}_{H(Z)}$) and the ring of "formal classical periods" ($\mathcal{O}_{U_{dR}(Z)}$). This isomorphism commutes with the respective period maps $per_{AF}$ and $per_{Cl}$. Consequently, the two seemingly different definitions of $p$-adic periods are equivalent in the Mixed Tate setting.
• Independence of the Prime: A notable corollary is that the ring of formal André periods, while a priori defined using a specific prime $p$, is independent of $p$ up to canonical isomorphism, as it is isomorphic to the coordinate ring of the unipotent group $U_{dR}(Z)$, which does not depend on $p$.
• Motivic Nature of Frobenius-Fixed Paths: The paper resolves a conceptual ambiguity regarding the Frobenius-fixed de Rham paths identified by Besser and Vologodsky. By showing that these paths arise from motivic paths in characteristic $p$ (specifically, from the specialization of motivic path torsors), the author demonstrates that these paths are "motivic after all," provided one replaces motives over an arithmetic base with motives over a residue field.
• Realization of Polylogarithms: The author explicitly constructs functions on the principal homogeneous space $H(Z)$ corresponding to motivic iterated integrals. It is shown that the André periods of these functions yield the special values of $p$-adic multiple polylogarithms and $p$-adic multiple zeta values (Coleman functions).

Significance
The paper claims significance in unifying two strands of research: the classical theory of $p$-adic periods used in arithmetic geometry (Chabauty-Kim theory) and the newer, more general framework of André periods proposed by Ancona and Frătilă. By proving their equivalence in the Mixed Tate setting, the work validates the extension of the Ancona-Frătilă machinery to mixed motives without loss of the classical properties.

Furthermore, the paper provides a concrete realization of $p$-adic multiple polylogarithms as André periods. This connects the abstract Tannakian formalism of Ancona and Frătilă directly to the computable objects of $p$-adic integration. The author notes that this connection clarifies the "inner mechanisms" of mixed Tate Chabauty-Kim theory and offers a $p$-adic perspective on Goncharov's conjectures regarding motivic iterated integrals. The work also lays a foundation for potentially extending Chabauty-Kim theory beyond the Mixed Tate setting, although the current results are strictly confined to the Mixed Tate case.