Generalized hypergeometric arithmetic 𝒟-modules under a 𝓅-adic non-Liouvilleness condition

Miyatani, Kazuaki

doi:10.1090/proc/15814

Generalized hypergeometric arithmetic $\mathscr {D}$-modules under a $p$-adic non-Liouvilleness condition
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Proc. Amer. Math. Soc. 150 (2022), 3215-3229 Request permission

Abstract:

We prove that the arithmetic $\mathscr {D}$-modules associated with the $p$-adic generalized hypergeometric differential operators, under a $p$-adic non-Liouvilleness condition on parameters, are described as an iterative multiplicative convolution of (hypergeometric arithmetic) $\mathscr {D}$-modules of rank one. As a corollary, we prove the overholonomicity of hypergeometric arithmetic $\mathscr {D}$-modules under a $p$-adic non-Liouvilleness condition.

References

Tomoyuki Abe, Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic $\scr {D}$-modules, Rend. Semin. Mat. Univ. Padova 131 (2014), 89–149. MR 3217753, DOI 10.4171/RSMUP/131-7
Pierre Berthelot, ${\scr D}$-modules arithmétiques. I. Opérateurs différentiels de niveau fini, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 2, 185–272 (French, with English summary). MR 1373933, DOI 10.24033/asens.1739
Daniel Caro, Fonctions $L$ associées aux $\scr D$-modules arithmétiques. Cas des courbes, Compos. Math. 142 (2006), no. 1, 169–206 (French, with English summary). MR 2197408, DOI 10.1112/S0010437X05001880
Daniel Caro, Sur la stabilité par produit tensoriel de complexes de $\mathcal {D}$-modules arithmétiques, Manuscripta Math. 147 (2015), no. 1-2, 1–41 (French, with English summary). MR 3336937, DOI 10.1007/s00229-014-0716-4
Daniel Caro, Unipotent monodromy and arithmetic $\mathcal {D}$-modules, Manuscripta Math. 156 (2018), no. 1-2, 81–115. MR 3783567, DOI 10.1007/s00229-017-0959-y
Christine Huyghe, $\scr D^\dagger (\infty )$-affinité des schémas projectifs, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 4, 913–956 (French, with English and French summaries). MR 1656002, DOI 10.5802/aif.1643
Nicholas M. Katz, Exponential sums and differential equations, Annals of Mathematics Studies, vol. 124, Princeton University Press, Princeton, NJ, 1990. MR 1081536, DOI 10.1515/9781400882434
Kiran S. Kedlaya, $p$-adic differential equations, Cambridge Studies in Advanced Mathematics, vol. 125, Cambridge University Press, Cambridge, 2010. MR 2663480, DOI 10.1017/CBO9780511750922
Kazuaki Miyatani, $p$-adic generalized hypergeometric equations from the viewpoint of arithmetic $\mathcal {D}$-modules, Amer. J. Math. 142 (2020), no. 4, 1017–1050. MR 4124114, DOI 10.1353/ajm.2020.0030
Christine Noot-Huyghe, Transformation de Fourier des $\scr D$-modules arithmétiques. I, Geometric aspects of Dwork theory. Vol. I, II, Walter de Gruyter, Berlin, 2004, pp. 857–907 (French). MR 2099091