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Cyclotomic p-adic multi-zeta values in depth two

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Abstract

In this paper we compute the values of the p-adic multiple polylogarithms of depth two at roots of unity. Our method is to solve the fundamental differential equation satisfied by the crystalline frobenius morphism using rigid analytic methods. The main result could be thought of as a computation in the p-adic theory of higher cyclotomy. We expect the result to be useful in proving non-vanishing results since it gives quite explicit formulas.

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References

  1. Brown F.: Mixed Tate motives over \({\mathbb{Z}}\). Ann. Math. (2) 175, 2–949976 (2012)

    Article  Google Scholar 

  2. Deligne, P.: Equations différentielles à points singuliers réguliers. In: Lecture Notes in Mathematics, vol. 163. Springer, Berlin, pp. iii+133 (1970)

  3. Deligne, P.: Le groupe fondamental de la droite projective moins trois points. In: Galois Groups Over \({\mathbb{Q}}\). Mathematical Sciences Research Institute Publications, vol. 16, Springer, New York, pp. 79–297 (1989)

  4. Deligne, P.: Catégories tannakiennes. In: The Grothendieck Festschrift, Vol. II, Progress in Mathematics, vol. 87. Birkhäuser Boston, Boston, MA, pp. 111–195 (1990)

  5. Deligne P.: Le groupe fondamental unipotent motivique de \({\mathbb{G}_{m}-\mu_{N}}\), pour N = 2, 3, 4, 6 ou 8. Publ. Math. Inst. Hautes Études Sci. 112, 101–141 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deligne P., Goncharov A.: Groupes fondamentaux motiviques de Tate mixte. Ann. Sci. École Norm. Sup. (4) 38(1), 1–56 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Fresnel, J., van der Put, M.: Rigid analytic geometry and its applications. In: Progress in Mathematics, vol. 218, Birkhäuser Boston, Boston, MA, pp. xii+296 (2004)

  8. Furusho H.: p-adic multiple zeta values II. Tannakian interpretations. Am. J. Math. 129(4), 1105–1144 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Goncharov, A.: Multiple \({\zeta}\) -values, Galois groups, and geometry of modular varieties. In: European Congress of Mathematics, vol. I (Barcelona, 2000), Progress in Mathematics, vol. 201, Birkhäuser, Basel, pp. 361–392 (2001)

  10. Shiho A.: Crystalline fundamental groups II. J. Math. Sci. Univ. Tokyo 9, 1–163 (2004)

    MathSciNet  Google Scholar 

  11. Ünver S.: p-adic multi-zeta values. J. Number. Theory 108, 111–156 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ünver S.: A note on the algebra of p-adic multi-zeta values. arXiv:1410.8648

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  1. Mathematics Department, Koç University, Rumelifeneri Yolu, 34450, Istanbul, Turkey

    Sinan Ünver

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  1. Sinan Ünver
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Correspondence to Sinan Ünver.

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Ünver, S. Cyclotomic p-adic multi-zeta values in depth two. manuscripta math. 149, 405–441 (2016). https://doi.org/10.1007/s00229-015-0789-8

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  • DOI: https://doi.org/10.1007/s00229-015-0789-8

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