Differential equations and arithmetic (study group)

We meet at IMPAN, Śniadeckich 8, room 6 from 10:00 to 12:00. To give a talk on one of the topics please write to Masha Vlasenko.

October 15An overiew of Fuchsian theory of linear ordinary differential operators [1, § 2-3]Bidisha Roy
October 29Monodromy of hypergeometric differential equations (Levelt's theorem) [2, § 1.2] Feliks Rączka
November 12Eisenstein's theorem Semen Słobodianuk
November 26                   and algebraic hypergeometric functions [3]notes: [part I][part II]
December 10Rigidity [2, § 1.4]Mieszko Zimny
Fuchsian theory in detail:
January 7                    regular singularities and local monodromy [5, § 1]Michał Łupiński
Criterion of algebraicity of hypergeometric functions [2, § 1.3], optionally [4, § 4]Gabriela Guzman
January 21p-adic Frobenius structure and its arithmetic consequences [6]Daniel Vargas Montoya
February 4Algebraicity of solutions modulo p [6]Daniel Vargas Montoya
Febryary 18Existence of Frobenius structure for rigid differential equations [6] [notes]Daniel Vargas Montoya
p-adic Frobenius structure and monodromy [slides]Masha Vlasenko
March ?Finishing the proof of existenceDaniel Vargas Montoya
Applications to integrality questions [8]Masha Vlasenko
March ?Dwork's construction of hypergeometric Frobenius structures [9]Bidisha Roy


Literature:
  1. F. Beukers, Gauss' hypergeometric function, notes from an MRI course given in 1993, Progr. Math. 260, 23-42
  2. F. Beukers, Hypergeometric functions of one variable, notes from MRI springschool 1999 held in Groningen
  3. F. Rodriguez Villegas, Integral ratios of factorials and algebraic hypergeometric functions, Summary of talk at 2005 Explicit Methods in Number Theory, Oberwolfach, arXiv:math/0701362
  4. F. Beukers, H. Heckman, Monodromy for the hypergeometric function _nF_n-1, Invent. Math., 95 (1989) 325-354
  5. A. Haefliger, Local theory of meromorphic connections in dimension 1 (Fuchs theory), Chapter III in Borel et al, Algebraic D-modules
  6. D. Vargas Montoya, Algébricité modulo p, séries hypergéométriques et structures de Frobenius forte, to appear in Bull. Soc. Math. France, arXiv:1911.09486
  7. K. Kedlaya, p-adic differential equations, Cambridge University Press, 2010
  8. F. Beukers, M. Vlasenko, On p-integrality of instanton numbers, Sections 1-2, arXiv:2109.10427
  9. K. Kedlaya, Frobenius structures on hypergeometric equations, arXiv:1912.13073