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Bessel F-isocrystals for reductive groups


We construct the Frobenius structure on a rigid connection \({{\,\mathrm{Be}\,}}_{\check{G}}\) on \({\mathbb {G}}_{m}\) for a split reductive group \(\check{G}\) introduced by Frenkel–Gross. These data form a \(\check{G}\)-valued overconvergent F-isocrystal \({{\,\mathrm{Be}\,}}_{\check{G}}^{\dagger }\) on \({\mathbb {G}}_{m,{\mathbb {F}}_p}\), which is the p-adic companion of the Kloosterman \(\check{G}\)-local system \({{\,\mathrm{Kl}\,}}_{\check{G}}\) constructed by Heinloth–Ngô–Yun. By studying the structure of the underlying differential equation, we calculate the monodromy group of \({{\,\mathrm{Be}\,}}_{\check{G}}^{\dagger }\) when \(\check{G}\) is almost simple (which recovers the calculation of monodromy group of \({{\,\mathrm{Kl}\,}}_{\check{G}}\) due to Katz and Heinloth–Ngô–Yun), and prove a conjecture of Heinloth–Ngô–Yun on the functoriality between different Kloosterman \(\check{G}\)-local systems. We show that the Frobenius Newton polygons of \({{\,\mathrm{Kl}\,}}_{\check{G}}\) are generically ordinary for every \(\check{G}\) and are everywhere ordinary on \(|{\mathbb {G}}_{m,{\mathbb {F}}_p}|\) when \(\check{G}\) is classical or \(G_2\).

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  1. The sum ( is slightly different from the standard definition by a factor \((-\frac{1}{\sqrt{q}})^{n-1}\).

  2. We adopt the definition of [4], which is different from that of [5] by a Tate twist.


  1. Abe, T.: Explicit calculation of Frobenius isomorphisms and Poincaré duality in the theory of arithmetic \(\mathscr {D}\)-modules. Rend. Semin. Mat. Univ. Padova 131, 89–149 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abe, T.: Langlands program for \(p\)-adic coefficients and the petits camarades conjecture. J. Reine Angew. Math. 734, 59–69 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abe, T.: Langlands correspondence for isocrystals and existence of crystalline companion for curves. J. Am. Math. Soc. 31, 921–1057 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abe, T.: Around the nearby cycle functor for arithmetic \(\mathscr {D}\)-modules. Nagoya Math. J. 236, 1–28 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Abe, T., Caro, D.: On Beilinson’s equivalence for \(p\)-adic cohomology. Sel. Math. New Ser. 24, 591–608 (2018)

  6. Abe, T., Caro, D.: Theory of weights in \(p\)-adic cohomology. Am. J. Math. 140(4), 879–975 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Abe, T., Esnault, H.: A Lefschetz theorem for overconvergent isocrystals with Frobenius structure. Ann. Sci. Éc. Norm. Supér. 52(4), 1243–1264 (2019)

    MathSciNet  MATH  Google Scholar 

  8. Abe, T., Marmora, A.: Product formula for p-adic epsilon factors. J. Inst. Math. Jussieu 14(2), 275–377 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  9. Adolphson, A., Sperber, S.: Exponential sums and Newton polyhedra: cohomology and estimates. Ann. Math. 130, 367–406 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. André, Y.: Représentations galoisiennes et opérateurs de Bessel \(p\)-adiques. Ann. Inst. Fourier (Grenoble) 52(3), 779–808 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. André, Y.: Filtrations de type Hasse-Arf et monodromie \(p\)-adique. Invent. Math. 148, 285–317 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. André, Y., Baldassarri, F.: De Rham Cohomology of Differential Modules on Algebraic Varieties. Progress in Mathematics 189, Birkhäuser, Basel (2001)

    Book  MATH  Google Scholar 

  13. Baumann, P., Riche, S.: Notes on the geometric Satake equivalence. In: Relative Aspects in Representation Theory, Langlands Functoriality and Automorphic Forms (CIRM Jean-Morlet Chair, Spring 2016), Lecture Notes in Mathematics, vol. 2221, pp. 1–134. Springer, Cham (2018)

  14. Baldassarri, F.: Differential modules and singular points of \(p\)-adic differential equations. Adv. Math. 44(2), 155–179 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Baldassarri, F., Berthelot, P.: On Dwork cohomology for singular hypersurfaces. In: Adolphson, A., Baldassarri, F., Berthelot, P., Katz, N., Loeser, F. (eds.) Geometric Aspects of Dwork Theory, vol. I, pp. 177–244. De Gruyter, Berlin (2004)

    Chapter  MATH  Google Scholar 

  16. Beilinson, A., Bernstein, J., Deligne, P., Gabber, O.: Faisceaux pervers. Astérisque 100(1982)

  17. Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke eigensheaves.

  18. Berthelot, P.: Cohomologie rigide et théorie de Dwork: le cas des sommes exponentielles, in \(P\)-adic cohomology. Astérisque No. 119–120(3), 17–49 (1984)

  19. Berthelot, P.: Cohomologie rigide et cohomologie ridige à supports propres, première partie. Preprint (1996)

  20. Berthelot, P.: \(\mathscr {D}\)-modules arithmétiques. I. Opérateurs différentiels de niveau fini. Ann. Sci. École Norm. Sup. 29(2), 18–272 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  21. Berthelot, P.: Introduction à la théorie arithmétique des \(\mathscr {D}\)-modules. In: Berthelot, P., Fontaine, J.-M., Illusie, L., Kato, K., Rapoport, M. (eds.) Cohomologies \(p\)-adiques et applications arithmétiques II. Astérique, vol. 279, pp. 1–80 (2002)

  22. Braden, T.: Hyperbolic localization of intersection cohomology. Transf. Groups 8(3), 209–216 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  23. Braverman, A., Gaitsgory, D.: Geometric Eisenstein series. Invent. Math. 150, 287–384 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Carlitz, L.: A note on exponential sums. Pac. J. Math. 30, 35–37 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  25. Caro, D.: \(\mathscr {D}\)-modules arithmétiques surholonomes. Ann. Sci. Éc. Norm. Supér. 42(1), 141–192 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Caro, D., Tsuzuki, N.: Overholonomicity of overconvergent \(F\)-isocrystals over smooth varieties. Ann. Math. 176(2), 747–813 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Cass, R.: Perverse \({\mathbb{F}}_p\) sheaves on the affine Grassmannian. Preprint arXiv:1910.03377 (2019)

  28. Chiarellotto, B., Tsuzuki, N.: Logarithmic growth and Frobenius filtrations for solutions of \(p\)-adic differential equations. J. Inst. Math. Jussieu 8(3), 465–505 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Christol, G., Mebkhout, Z.: Sur le théorème de l’indice des équations différentielles p-adiques III. Ann. Math. 151, 385–457 (2000)

  30. Crew, R.: Specialization of crystalline cohomology. Duke Math. J. 53(3), 749–757 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  31. Crew, R.: \(F\)-isocrystals and their monodromy groups. Ann. Sci. École Norm. Sup. 25(4), 429–464 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  32. D’Addezio, M.: The monodromy groups of lisse sheaves and overconvergent F-isocrystals. Sel. Math. (N.S.) 26(45), 1–41 (2020)

  33. Deligne, P.: Applications de la formule des traces aux sommes trigonométriques, In: Cohomologie Étale (SGA \(4\frac{1}{2}\)), Lecture Notes in Mathematics, 569 (1977)

  34. Deligne, P.: La conjecture de Weil. II. Publ. Math. Inst. Hautes Études Sci. No. 52, pp. 137–252 (1980)

  35. Deligne, P.: Catégories tannakiennes. In: The Grothendieck Festschrift. Modern Birkhäuser Classics. Birkhäuser, Boston (2007)

  36. Deligne, P., Milne, J.S.: Tannakian Categories, Hodge Cycles, Motives, and Shimura Varieties. Lecture Notes in Mathematics, vol. 900. Springer, Berlin (1982)

    Book  MATH  Google Scholar 

  37. Drinfeld, V., Gaitsgory, D.: On a theorem of Braden. Transform. Groups 19(2), 313–358 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Drinfeld, V., Kedlaya, K.: Slopes of indecomposable F-isocrystals. Pure Appl. Math. Q. 13(1), 131–192 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  39. Dwork, B.: \(P\)-adic cycles. Publ. Math. Inst. Hautes Études Sci. No. 37, 27–115 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  40. Dwork, B.: Bessel functions as p-adic functions of the argument. Duke Math. J. 41, 711–738 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  41. Étesse, J.-Y., Le Stum, B.: Fonctions L associées aux F-isocristaux surconvergent I. Interprétation cohomologique. Math. Ann. 296, 557–576 (1993)

  42. Frenkel, E., Gross, B.: A rigid irregular connection on the projective line. Ann. Math. 170, 1469–1512 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  43. Fu, L., Wan, D.: L-functions for symmetric products of Kloosterman sums. J. Reine Angew. Math. 589, 79–103 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  44. Gaitsgory, D.: Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. Math. 144, 253–280 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  45. Görtz, U., Haines, T.J.: The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties. J. Reine Angew. Math. 609, 161–213 (2007)

    MathSciNet  MATH  Google Scholar 

  46. Gross, B., Reeder, M.: Arithmetic invariants of discrete Langlands parameters. Duke Math. J. 154, 431–508 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  47. Heinloth, J., Ngô, B.-C., Yun, Z.: Kloosterman sheaves for reductive groups. Ann. Math. 177(1), 241–310 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Huyghe, C.: \(\mathscr {D}^{\dagger }(\infty )\)-affinité des schémas projectifs. Ann. Inst. Fourier (Grenoble) 48(4), 913–956 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  49. Katz, N.: On the calculation of some differential Galois groups. Invent. Math. 87, 13–61 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  50. Katz, N., Sums, Gauss: Kloosterman Sums, and Monodromy Groups. Annals in Mathematics Studies 116, Princeton University Press, Princeton, NJ (1988)

    Book  Google Scholar 

  51. Katz, N.: Exponential Sums and Differential Equations. Annals in Mathematics Studies 124, Princeton University Press, Princeton, NJ (1990)

    Book  MATH  Google Scholar 

  52. Katz, N.: From Clausen to Carlitz: low-dimensional spin groups and identities among character sums. Mosc. Math. J. 9(1), 57–89 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  53. Kedlaya, K.: Semistable reduction for overconvergent F-isocrystals. I. Unipotence and logarithmic extensions. Compos. Math. 143(5), 1164–1212 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  54. Kedlaya, K.: p-Adic Differential Equations. Cambridge Studies in Advanced Mathematics, 125, Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  55. Kedlaya, K.: Semistable reduction for overconvergent F-isocrystals, IV: local semistable reduction at nonmonomial valuations. Compos. Math. 147, 467–523 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  56. Kedlaya, K.: Notes on isocrystals. Preprint (2016)

  57. Lafforgue, V.: Estimées pour les valuations \(p\)-adiques des valeurs propres des opérateurs de Hecke. Bull. Soc. Math. Fr. 139(4), 455–477 (2011)

    Article  MATH  Google Scholar 

  58. Lam, T., Templier, N.: The mirror conjecture for minuscule flag varieties, preprint, arXiv:1705.00758 (2017)

  59. Liu, R., Zhu, X.: Rigidity and a Riemann–Hilbert correspondence for p-adic local systems. Invent. Math. 207, 291–343 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  60. Matsuda, S.: Katz correspondence for quasi-unipotent overconvergent isocrystals. Compos. Math. 134(1), 1–34 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  61. Miyatani, K.: P-adic generalized hypergeometric functions from the viewpoint of arithmetic D-modules. Am. J. Math. 142(4), 1017–1050 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  62. Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. Math. 166, 95–143 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  63. Ngô, B.C., Polo, P.: Résolutions de Demazure affines et formule de Casselman–Shalika géométrique. J. Algebr. Geom. 10(3), 515–547 (2001)

    MATH  Google Scholar 

  64. Ogus, A.: F-isocrystals and de Rham cohomology. II. Convergent isocrystals. Duke Math. J. 51(4), 765–850 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  65. Ohkubo, S.: Logarithmic growth filtrations for \((\varphi,\nabla )\)-modules over the bounded Robba ring. Compos. Math. 157(6), 1265–1301 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  66. Pech, C., Rietsch, K., Williams, L.: On Landau-Ginzburg models for quadrics and flat sections of Dubrovin connections. Adv. Math. 300, 275–319 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  67. Richarz, T.: A new approach to the geometric Satake equivalence. Doc. Math. 19, 209–246 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  68. Richarz, T., Scholbach, J.: The motivic Satake equivalence. Math. Ann. 380, 1595–1653 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  69. Richarz, T., Zhu, X.: Appendix to [78]

  70. Shiho, A.: Crystalline fundamental groups. II. Log convergent cohomology and rigid cohomology. J. Math. Sci. Univ. Tokyo 9(1), 1–163 (2002)

    MathSciNet  MATH  Google Scholar 

  71. Sperber, S.: \(p\)-adic hypergeometric functions and their cohomology. Duke Math. J. 44(3), 535–589 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  72. Sperber, S.: Congruence properties of the hyper-Kloosterman sum. Compos. Math. 40(1), 3–33 (1980)

    MathSciNet  MATH  Google Scholar 

  73. Tsuzuki, N.: The local index and the Swan conductor. Compos. Math. 111, 245–288 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  74. Tsuzuki, N.: Minimal slope conjecture of \(F\)-isocrystals. Preprint arXiv:1910.03871 (2019)

  75. Fresnel, J., van der Put, M.: Rigid Analytic Geometry and Its Applications. Progress in Mathematics, 218, Birkhäuser Boston, Inc., Boston (2004)

    Book  MATH  Google Scholar 

  76. Wan, D.: Newton polygons of zeta functions and L-functions. Ann. Math. 137, 247–293 (1993)

    Article  MathSciNet  Google Scholar 

  77. Wan, D.: Variation of p-adic Newton polygons for L-functions of exponential sums. Asian J. Math. 8(3), 427–472 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  78. Zhu, X.: The geometric Satake correspondence for ramified groups. Ann. Sci. Éc. Norm. Supér. 48, 409–451 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  79. Zhu, X.: An introduction to affine Grassmannians and the geometric Satake equivalence. IAS/Park City Math. Ser. 24, 59–154 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  80. Zhu, X.: Frenkel–Gross’ irregular connection and Heinloth–Ngô–Yun’s are the same. Sel. Math. (N.S.) 23(1), 245–274 (2017)

  81. Zhu, X.: Geometric Satake, categorical traces, and arithmetic of Shimura varieties. Curr. Dev. Math. 2016, 145–206 (2016)

    Article  MATH  Google Scholar 

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We would like to thank Benedict Gross, Shun Ohkubo, Daqing Wan, Liang Xiao and Zhiwei Yun for valuable discussions. We are also grateful to an anonymous referee for his/her careful reading and valuable comments. X. Z. is partially supported by the National Science Foundation under agreement Nos. DMS-1902239 and a Simons Fellowship.

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Correspondence to Xinwen Zhu.

Appendix A. A 2-adic proof of Carlitz’s identity and its generalization

Appendix A. A 2-adic proof of Carlitz’s identity and its generalization

As mentioned in introduction, Carlitz [24] proved the following identity between Kloosterman sums:

$$\begin{aligned} {{\,\mathrm{Kl}\,}}(3;a)={{\,\mathrm{Kl}\,}}(2;a)^2-1,\qquad \forall ~ a\in {\mathbb {F}}_{2^s}^{\times }. \end{aligned}$$

In this appendix, we reprove and generalize this identity by establishing an isomorphism between two Bessel F-isocrystals \({{\,\mathrm{Be}\,}}_{2n+1}^{\dagger }\) and \({{\,\mathrm{Be}\,}}_{{{\,\mathrm{SO}\,}}_{2n+1},{{\,\mathrm{Std}\,}}}^{\dagger }\). The following is a restatement of Proposition 5.2.10(ii).

Proposition A1

There exists an isomorphism between following two overcovergent F-isocrystals on \({\mathbb {G}}_{m,{\mathbb {F}}_2}\):

Our strategy is first to show that their maximal slope quotient convergent F-isocrystals are isomorphic. Then we conclude the proposition by a dual version of a minimal slope conjecture (proposed by Kedlaya [56] and recently proved by Tsuzuki [74]) that we briefly recall in the following.

Let X be a smooth k-scheme and \(\mathscr {M}^{\dagger }\) an overconvergent F-isocrystal on X/K. We denote the associated convergent F-isocrystal on X/K by \(\mathscr {M}\). When the (Frobenius) Newton polygons of \(\mathscr {M}\) are constant on X, \(\mathscr {M}\) admits a (dual) slope filtration, that is a decreasing filtration

of convergent F-isocrystals on X/K such that

  • \(\mathscr {M}^{i}/\mathscr {M}^{i+1}\) is isoclinic of slope \(s^i\) and

  • \(s^0>s^1>\cdots >s^{r-1}\).

Theorem A2

(Tsuzuki, [74] theorem 1.3) Let X be a smooth connected curve over k. Let \(\mathscr {M}^{\dagger },\mathscr {N}^{\dagger }\) be two irreducible overconvergent F-isocrystals such that the corresponding convergent F-isocrystals \(\mathscr {M},\mathscr {N}\) admit the slope filtrations \(\{\mathscr {M}^i\}\), \(\{\mathscr {N}^i\}\) respectively. Suppose there exists an isomorphism \(h:\mathscr {N}/\mathscr {N}^1\xrightarrow {\sim } \mathscr {M}/\mathscr {M}^1\) of convergent F-isocrystals between the maximal slope quotients. Then there exists a unique isomorphism \(g^{\dagger }:\mathscr {N}^{\dagger }\xrightarrow {\sim } \mathscr {M}^{\dagger }\) of overconvergent F-isocrystals, which is compatible with h as morphisms of convergent F-isocrystals.

A3 Following Dwork’s strategy [39, § 1-3], we study the maximal slope quotients of \({{\,\mathrm{Be}\,}}_{2n+1}^{\dagger }\) and of \({{\,\mathrm{Be}\,}}_{{{\,\mathrm{SO}\,}}_{2n+1},{{\,\mathrm{Std}\,}}}^{\dagger }\) in terms of their unique solutions at 0.

In the following, we assume \(k={\mathbb {F}}_p\). We first recall Dwork’s congruences and show a refinement of his result in the 2-adic case. Consider for every \(i\ge 0\), a map \(B^{(i)}(-):{\mathbb {Z}}_{\ge 0}\rightarrow K^{\times }\) and the following congruence relation for \(0\le a<p\) and \(n,m,s \in {\mathbb {Z}}_{\ge 0}\):

  1. (a)

    \(B^{(i)}(0)\) is a p-adic unit for all \(i\ge 0\),

  2. (b)

    \( \displaystyle \frac{B^{(i)}(a+np)}{B^{(i+1)}(n)} \in R\) for all \(i\ge 0\),

  3. (c)

    \( \displaystyle \frac{B^{(i)}(a+np+mp^{s+1})}{B^{(i+1)}(n+mp^{s})} \equiv \frac{B^{(i)}(a+np)}{B^{(i+1)}(n)} \mod p^{s+1}\) for all \(i\ge 0\).

  4. (c’)

    When \(p=2\), \( \displaystyle \frac{B^{(i)}(a+n2+m2^{s+1})}{B^{(i+1)}(n+m2^{s})} \equiv u(i,s,m) \frac{B^{(i)}(a+n2)}{B^{(i+1)}(n)} \mod 2^{s+1}\) for all \(i\ge 0\), where \(u(i,s,m)=1\) if \(s\ne 1\) and \(u(i,1,m)=1\) or \(-1\) depending on i and m.

If conditions (a–c) (or (a,b,c’)) are satisfied, then \(B^{(i)}(n)\in R\) for all \(i,n\ge 0\). We set

$$\begin{aligned}&F^{(i)}(x)=\sum _{j=0}^{\infty } B^{(i)}(j) x^j ~ \in K \llbracket x \rrbracket , \\&F^{(i)}_{m,s}(x)=\sum _{j=mp^s}^{(m+1)p^{s}-1} B^{(i)}(j) x^j~ \in K[x],\quad s\ge 0. \end{aligned}$$

We write \(F^{(i)}_{0,s}\) by \(F^{(i)}_{s}\) for simplicity.

Theorem A4

(i) [39, theorem 2] If conditions (a–c) are satisfied, then

(i\(^{\prime }\)) If conditions (a,b,c\(^{\prime }\)) are satisfied (in particular \(p=2\)), then

(ii) [39, theorem 3] Under the assumption of (i) or (i’) and suppose moreover that

  1. (d)

    \(B^{(i)}(0)=1\) for \(i\ge 0\).

  2. (e)

    \(B^{(i+r)}=B^{(i)}\) for all \(i\ge 0\) and some fixed \(r\ge 1\).

Let U be the open subscheme of \({\mathbb {A}}^1_{k}\) defined by \(F_{1}^{(i)}(x)\ne 0\), for \(i=0,1,\ldots ,r-1\). Then the limit

defines a global function on the formal open subscheme \(\mathfrak {U}\) of \(\widehat{{\mathbb {A}}}^{1}_R\) associated to U, which takes p-adic unit value at each rigid point of \(\mathfrak {U}^{{{\,\mathrm{rig}\,}}}\).

We prove assertion (i’) in the end (A11). We briefly explain Dwork’s result (ii) in the language of formal schemes. The assumption implies that \(F_s^{(i)}\ne 0\) on U (cf. [39] 3.4). For \(s\ge 1\), the congruences (A41) or ((A42)) imply that

$$\begin{aligned} F_{s+1}^{(0)}(x)/ F_s^{(1)}(x^p)= F_s^{(0)}(x)/F_{s-1}^{(1)}(x^p) \quad \in \Gamma (\mathfrak {U},\mathscr {O}_{\mathfrak {U}}/p^{s-1}\mathscr {O}_{\mathfrak {U}}). \end{aligned}$$

This allows us to use (A43) to define a global function f of \(\mathscr {O}_{\mathfrak {U}}\).

A5 Let \(F(x)=\sum _{j\ge 0} B(j) x^{j}\) be a formal power series in \(R\llbracket x \rrbracket \). We say F satisfies Dwork’s congruences if by setting \(B^{(i)}(j)=B(j)\) for every \(i\ge 0\), conditions of Theorem A4(ii) are satisfied.

We take such a function F and then we obtain a function \(f\in \Gamma (\mathfrak {U},\mathscr {O}_{\mathfrak {U}})\) coinciding with \(F(x)/F(x^p)\) in \(K\{x\}\) ( (i.e. the open unit disc). Moreover, by [39, lemma 3.4(ii)], there exists a function \(\eta \in \Gamma (\mathfrak {U},\mathscr {O}_{\mathfrak {U}})\) coinciding with \(F'(x)/F(x)\) in \(K\{x\}\) defined by

$$\begin{aligned} \eta (x)\equiv F'_{s+1}(x)/F_{s+1}(x) \mod p^{s}. \end{aligned}$$

The functions f(x) and \(\eta (x)\) satisfy a differential equation:

$$\begin{aligned} \frac{f'(x)}{f(x)}+px^{p-1}\eta (x^{p})=\eta (x). \end{aligned}$$

Note that \(f(0)=F(0)/F(0)=1\). Then we deduce that the following corollary.

Corollary A6

The connection \(d- \eta \) on the trivial bundle \(\mathscr {O}_{\mathfrak {U}^{{{\,\mathrm{rig}\,}}}}\) and the function f form a unit-root convergent F-isocrystal \(\mathscr {E}_F\) on U/K, whose Frobenius eigenvalue at 0 is 1.

A7. Let \(\mathscr {M}^{\dagger }\) be an overconvergent F-isocrystal on \({\mathbb {G}}_{m,k}\) over K of rank r whose underlying bundle is trivial and the connection is defined by a differential equation:

$$\begin{aligned} P(\delta )=\delta ^r+p_{r}\delta ^{r-1}+\cdots + p_{1}=0, \end{aligned}$$

where \(\delta =x\frac{d}{dx}\), \(p_i\in \Gamma (\widehat{{\mathbb {A}}}_R^1,\mathscr {O}_{\widehat{{\mathbb {A}}}_R^1})[\frac{1}{p}]\). We assume moreover that \(\mathscr {M}^{\dagger }\) is unipotent at 0 with a maximal unipotent local monodromy. Then \(\mathscr {M}^{\dagger }\) extends to a log convergent F-isocrystal \(\mathscr {M}^{\log }\) on \(({\mathbb {A}}^1,0)\) and its Frobenius slopes at 0 are

$$\begin{aligned} s^0>s^1=s^0-1>\cdots > s^{r-1}=s^0-(r-1). \end{aligned}$$

Note that \(\mathscr {M}^{\dagger }\) is indecomposable in \(F\text {-}{{\,\mathrm{Isoc}\,}}^{\dagger }({\mathbb {G}}_{m,k}/K)\) and so is \(\mathscr {M}\) in \(F\text {-}{{\,\mathrm{Isoc}\,}}({\mathbb {G}}_{m,k}/K)\). Then by Drinfeld–Kedlaya’s theorem on the generic Frobenius slopes [38], we deduce property (i):

(i) The generic Frobenius slopes (mult-)set is \(\{s^0,\ldots ,s^{r-1}\}\) with \(s^i=s^0-(i-1)\).

(ii) In view of (, the differential equation \(D=0\) admits a unique solution at 0:

$$\begin{aligned} F(x)=\sum _{n\ge 0} A(n) x^n \quad \in K\{x\},\quad \text { with A(0)=1}. \end{aligned}$$

Proposition A8

Suppose the function F(x) satisfies Dwork’s congruences (A5) and let \(\mathscr {E}_F\) be the associated unit-root convergent F-isocrystal on \(U \subset {\mathbb {A}}^1_k\). Then

(i) There exists an epimorphism of log convergent isocrystals \(\mathscr {M}^{\log }\rightarrow \mathscr {E}_F\) on (U, 0).

(ii) As convergent isocrystals, \(\mathscr {E}_F\) coincides with the maximal slope quotient \(\mathscr {M}^{\log }/\mathscr {M}^{\log ,1}\) of \(\mathscr {M}^{\log }\) (A12).


(i) We set \(A=\Gamma (\mathfrak {U},\mathscr {O}_{\mathfrak {U}})[\frac{1}{p}]\). We claim that there exists a decomposition of differential operators:

Indeed, by the Euclidean algorithm [54, 5.5.2], there exists \(r\in A\) such that \(P=Q(\delta -x\eta )+r\). By evaluating the above identity at F (in the ring \(K\{x\}\) containing A), we obtain

$$\begin{aligned} P(\delta )(F)=0=Q(\delta )(\delta -x\eta )(F)+rF=rF. \end{aligned}$$

Then we deduce \(r=0\) and (A81) follows.

Let \(e_1,\ldots ,e_r\) be a basis of \(\mathscr {M}\) such that \(\nabla _{\delta }(e_i)=e_{i+1}, 1\le i\le r-1\) and \(\nabla _{\delta }(e_r)=-(p_r e_r+\cdots +p_1 e_1)\). We consider a free \(\mathscr {O}_{\mathfrak {U}^{{{\,\mathrm{rig}\,}}}}\)-module with a log connection \(\mathscr {N}\) with a basis \(f_1,\ldots ,f_{r-1}\) and the connection defined by \(\nabla _{\delta }(f_i)=f_{i+1}\), \(\nabla _{\delta }(f_{r-1})=-(q_{r-1} f_{r-1}+\cdots +q_{1}f_{1})\). By (A81), the morphism \(f_1\mapsto e_2- x\eta e_1\) induces a horizontal monomorphism \(\mathscr {N}\rightarrow \mathscr {M}^{\log }\) whose cokernel is isomorphic to \(\mathscr {E}_F\).

(ii) Note that \({{\,\mathrm{Pic}\,}}(\mathfrak {U}^{{{\,\mathrm{rig}\,}}})\simeq {{\,\mathrm{Pic}\,}}(U)\) [75, 3.7.4] is trivial. Then the rank one convergent isocrystal \(\mathscr {M}^{\log }/\mathscr {M}^{\log ,1}\) can be represented as a connection \(d-\lambda \) on the trivial bundle \(\mathscr {O}_{\mathfrak {U}^{{{\,\mathrm{rig}\,}}}}\).

Since \(\mathscr {M}^{\log }\) has a maximal unipotent at 0, the rank one quotient of the restriction \(\mathscr {M}^{\log }|_0\) of \(\mathscr {M}^{\log }\) at the open unit disc around 0 is unique (2.3.1). In particular, \(d-\lambda \) kills the unique solution F of \(P(\delta )=0\). By analytic continuation, we have \(\lambda =\eta \) and the assertion follows. \(\square \)

Remark A9

The unique solution F(x) belongs to the ring \(K\llbracket x \rrbracket _0 = R \llbracket x \rrbracket \otimes _R K\) of bounded functions on open unit disc, which is a subring of \(K\{x\}\). Assertion (ii) can be viewed as an example of Dwork–Chiarellotto–Tsuzuki conjecture on the comparison between the log-growth filtration (of solutions) and Frobenius slope filtration [28]. This conjecture was recently proved by Ohkubo [65].

Proof of Proposition A1

We set \(k={\mathbb {F}}_2\) and apply the above discussions to overconvergent F-isocrystals \(\mathscr {M}^{\dagger }={{\,\mathrm{Be}\,}}_{2n+1}^{\dagger }\) and \(\mathscr {N}^{\dagger }={{\,\mathrm{Be}\,}}_{{{\,\mathrm{SO}\,}}_{2n+1},{{\,\mathrm{Std}\,}}}^{\dagger }\) on \({\mathbb {G}}_{m,{\mathbb {F}}_2}/K\) (A11). Their unique solutions at 0 are:

$$\begin{aligned} F(x)= \sum _{r\ge 0} \frac{(-2)^{(2n+1)r}}{(r!)^{2n+1}} x^r,\qquad G(x)=\sum _{r\ge 0} \frac{2^{(2n+1)r}(2r-1)!!}{(r!)^{2n+1}} x^r. \end{aligned}$$

In the following lemma, we show that F and G satisfy Dwork’s congruences and that the associated maximal slope quotients \(\mathscr {E}_F\) and \(\mathscr {E}_G\) (A8) are isomorphic. Then Proposition A1 follows from theorem A2 and the following lemma. \(\square \)

Lemma A10

(i) The functions F(x) and G(x) satisfy Dwork’s congruences (A5) and define unit-root convergent F-isocrystals \(\mathscr {E}_F\) and \(\mathscr {E}_G\) on \({\mathbb {A}}_k^1\) respectively.

(ii) The function F(x)/G(x) extends to a global function of \(\widehat{{\mathbb {A}}}^1_R\) and induces an isomorphism \(\mathscr {E}_G\xrightarrow {\sim } \mathscr {E}_F\).


(i) Conditions (a,b,d,e) are easy to verified. The coefficients of F(x) (resp. G(x)) satisfy condition (c’) (resp. (c)), that is

$$\begin{aligned}&\frac{(-2)^{(2n+1)(a+\ell 2+m2^{s+1})}/((a+\ell 2+m2^{s+1})!)^{2n+1}}{(-2)^{(2n+1)(\ell +m2^{s})}/((\ell +m2^{s})!)^{2n+1}} \\&\quad \equiv u(s,m) \frac{ (-2)^{(2n+1)(a+\ell 2)}/ ((a+\ell 2)!)^{2n+1}}{(-2)^{(2n+1)\ell }/ (\ell !)^{2n+1}} \mod 2^{s+1}, \end{aligned}$$

where \(u(1,m)=(-1)^m\) and \(u(s,m)=1\) if \(s\ne 1\), and

$$\begin{aligned}&\frac{(2(a+\ell 2+m2^{s+1})-1)!! 2^{(2n+1)(a+\ell 2+m2^{s+1})}/((a+\ell 2+m2^{s+1})!)^{2n+1}}{(2(\ell +m2^{s})-1)!!2^{(2n+1)(\ell +m2^{s})}/((\ell +m2^{s})!)^{2n+1}} \\&\quad \equiv \frac{(2(a+\ell 2)-1)!!2^{(2n+1)(a+\ell 2)}/ ((a+\ell 2)!)^{2n+1}}{(2\ell -1)!!2^{(2n+1)\ell }/ (\ell !)^{2n+1}} \mod 2^{s+1}. \end{aligned}$$

Since \(F_1(x)\equiv G_1(x)\equiv 1\mod 2\), the F-isocrystals \(\mathscr {E}_{F},\mathscr {E}_G\) are defined over \({\mathbb {A}}_k^1\).

(ii) We set \(B^{(0)}(r)=\frac{(-2)^{(2n+1)r}}{(r!)^{2n+1}}\) and \(B^{(1)}(r)=\frac{2^{(2n+1)r}(2r-1)!!}{(r!)^{2n+1}}\) and \(B^{(i+2)}=B^{(i)}\). Then these sequences satisfy conditions (a,b,c’,d,e). For condition (c’), the constants u(i, 1, m) are given by

$$\begin{aligned} u(0,1,m)=1, \quad u(1,1,m)=(-1)^m,\quad u(i+2,1,m)=u(i,1,m). \end{aligned}$$

Since \(F_1(x)\equiv G_1(x)\equiv 1\mod 2\), \(F(x)/G(x^2)\) extends to a global function of \(\mathscr {O}_{\widehat{{\mathbb {A}}}_R^1}\) by Theorem A4 and so is F(x)/G(x). Then the assertion follows. \(\square \)

A11 Proof of Theorem A4(i’). We prove assertion (i’) by modifying the argument of [39, theorem 2]. Note that condition (c’) implies the following congruence relation:

When \(n<0\), we set \(B^{(i)}(n)=0\). We set \(A=B^{(0)}\), \(B=B^{(1)}\) and for \(a\in \{0,1\}\), \(j,N\in {\mathbb {Z}}\), we set

$$\begin{aligned}&U_{a}(j, N) =A(a+2(N-j)) B(j)-B(N-j) A(a+2j), \\&H_{a}(m, s, N) = \sum _{j=m2^s}^{(m+1)2^s-1} U_{a}(j, N). \end{aligned}$$

Then the assertion is equivalent to

By condition (b), we have \(A(a+2m)/B(m)\in R\) and hence

$$\begin{aligned} U_a(m,N) \equiv 0 \mod B(m). \end{aligned}$$

Then equation (A112) for \(s=0\) follows from the fact that \(H_a(m,0,N)=U_a(m,N)\).

We now prove by induction on s. We write the induction hypothesis

$$\begin{aligned} \alpha _s: H_a(m,u,N)\equiv 0 \mod 2^u B^{(u+1)}(m),\quad \text {for } u\in [0,s), m,N\ge 0. \end{aligned}$$

We may assume \(\alpha _{s}\) for fixed \(s\ge 1\). The main step is to show for \(0\le t\le s\) that

$$\begin{aligned}&\beta _{t,s}: v(s,t,m) H_a(m,s,N+m2^s)\\&\quad \equiv \sum _{j=0}^{2^{s-t}-1} B^{(t+1)}(j+m2^{s-t})H_a(j,t,N)/B^{(t+1)}(j)\mod 2^s B^{(s+1)}(m), \end{aligned}$$

where \(v(s,t,m)=1\) or \(-1\) depending on stm.

We list some elementary facts (cf. [39, 2.5–2.7])

We first prove \(\beta _{0,s}\). We have

By (A111), we have

$$\begin{aligned} A\left( a+2 j+m 2^{s+1}\right) = A(a+2j)B(j+m2^s)/B(j)+X_j 2^s B(j+m 2^s), \end{aligned}$$

where \(X_j\in R\). Then the right hand side of (A116) is

$$\begin{aligned} B\left( j+m 2^{s}\right) \biggl (U_{a}(j, N) / B(j)-2^{s} X_{j} B(N-j)\biggr ). \end{aligned}$$

Since \(U_{a}(j,N)=H_a(j,0,N)\), we obtain

$$\begin{aligned} H_{a}\left( m, s, N+m 2^{s}\right)= & {} \sum _{j=0}^{2^{s}-1} B\left( j+m 2^{s}\right) H_{a}(j, 0, N) / B(j)\\&-2^{s} \sum _{j=0}^{2^{s}-1} X_{j} B\left( j+m 2^{s}\right) B(N-j). \end{aligned}$$

Since \(X_jB(N-j)\in R\), it follows from (A115) (\(B=B^{(1)}\)) that the second sum is congruent to zero modulo \(2^s B^{(s+1)}(m)\). This proves \(\beta _{0,s}\) with \(v(s,0,m)=1\).

With s fixed, \(s\ge 1\), t fixed, \(0\le t \le s-1\), we show that \(\beta _{t,s}\) together with \(\alpha _s\) imply \(\beta _{t+1,s}\). To do this we put \(j=\mu +2i\) in the right side of \(\beta _{t,s}\) and write it in the form

$$\begin{aligned} \sum _{\mu =0}^{1} \sum _{i=0}^{2^{s-t-1}} B^{(t+1)}\left( \mu + 2i+m 2^{s-i}\right) H_{a}(\mu +2 i, t, N) / B^{(t+1)}(\mu +2i). \end{aligned}$$

By condition (c’), we have,

$$\begin{aligned}&B^{(t+1)}\left( \mu +2i+m2^{s-t}\right) \\&\quad = u(t+1,s-t-1,m)\left( B^{(t+1)}(\mu +2 i) B^{(t+2)}\left( i+m2^{s-t-1}\right) / B^{(t+2)}(i)\right) \\&\qquad +X_{i, \mu } 2^{s-t} B^{(t+2)}\left( i+m 2^{s-t-1}\right) , \end{aligned}$$

where \(X_{i,\mu }\in R\). Thus the general term in the above double sum is

$$\begin{aligned} u(t+1,s-t-1,m)\biggl (B^{(t+2)}(i+m2^{s-t-1})H_a(\mu +2i,t,N)/B^{(t+2)}(i)\biggr )+Y_{i,\mu }, \end{aligned}$$

where the error term:

$$\begin{aligned} Y_{i, \mu }=X_{i, \mu } 2^{s-t} B^{(t+2)}\left( i+m 2^{s-t-1}\right) H_{a}(\mu +2 i, t, N) / B^{(t+1)}(\mu +2i). \end{aligned}$$

For this error term, since \(t<s\), we can apply \(\alpha _s\) to conclude that

$$\begin{aligned} Y_{i,\mu }\equiv 0 \mod B^{(t+2)}(i+m2^{s-t-1})2^s. \end{aligned}$$

Then we can use (A115) to conclude that

$$\begin{aligned} Y_{i,\mu }\equiv 0 \mod 2^s B^{(s+1)}(m). \end{aligned}$$

After modulo \(2^s B^{(s+1)}(m)\), the right side of \(\beta _{t,s}\) is equal to

$$\begin{aligned}&u(t+1,s-t-1,m)\sum _{\mu =0}^{1} \sum _{i=0}^{2^{s-t-1}-1} B^{(t+2)}\left( i+m2^{s-t-1}\right) \\&\quad \times H_{a}(\mu +2 i, t, N) / B^{(t+2)}(i).\end{aligned}$$

By reversing the order of summation and using (A114), the above sum is the same as

$$\begin{aligned}&u(t+1,s-t-1,m)\sum _{i=0}^{2^{s-t-1}-1} B^{(t+2)}\left( i+m2^{s-t-1}\right) \\&\quad \times H_{a}(i, t+1, N) / B^{(t+2)}(i),\end{aligned}$$

which proves \(\beta _{t+1,s}\). In particular, we obtain \(\beta _{s,s}\), which states

We now consider the statement (with s fixed before)

$$\begin{aligned} \gamma _{N}:H_a(0,s,N)\equiv 0 \mod 2^s. \end{aligned}$$

We know that \(\gamma _N\) is true for \(N<0\). Let \(N'\) (if it exists) be the minimal value of N for which \(\gamma _{N'}\) fails. For \(m\ge 1\), since \(B^{(s+1)}(0)\) is a unit, we have by (A117)

$$\begin{aligned} H_a(m,s,N')\equiv & {} v(s,s,m) B^{(s+1)}(m)H_a(0,s,N'-m2^s)/B^{(s+1)}(0)\\\equiv & {} 0 \mod 2^s. \end{aligned}$$

Applying this to (A113), we obtain that

$$\begin{aligned} H_a(0,s,N')\equiv 0 \mod 2^s. \end{aligned}$$

Thus \(\gamma _N\) is valid for all N and Eq. (A117) implies \(\alpha _{s+1}\). This proves assertion (i\(^{\prime }\)). \(\square \)

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Xu, D., Zhu, X. Bessel F-isocrystals for reductive groups. Invent. math. 227, 997–1092 (2022).

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