Differential Equations and Arithmetic
Monday , Fridays 10:30-11:30 (Rio de Janeiro's time)
During 2024 the GADEPs seminar will focus on arithmetic aspects of differential equations. We are going to read
the following articles/books:
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Yves André. G-functions and geometry, volume E13 of Aspects of Mathematics.
Friedr. Vieweg & Sohn, Braunschweig, 1989.
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A. Buium, Arithmetic Differential Equations, Mathematical Surveys and Monographs
Volume: 118; 2005.
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J.-B. Bost. Algebraic leaves of algebraic foliations over number fields. Publ. Math.
Inst. Hautes ´Etudes Sci., (93):161–221, 2001.
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N. M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of turrittin, Publications mathématiques de l’IHES 39 (1970), 175–232.
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Katz, Nicholas M. (1972). "Algebraic solutions of differential equations (p-curvature and the Hodge filtration)". Invent. Math. 18 (1–2): 1–118.
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W. Mendson, J. V. Pereira. Codimension one foliations in positive characteristic, Preprint 2023 (and the references therein).
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Y. Miyaoka and T. Peternell, Geometry of higher dimensional algebraic varieties, vol. 26, Springer, 1997.
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H. Movasati, A Course in Hodge Theory: With Emphasis on Multiple Integrals, Somerville, MA: International Press Boston, 2021.
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H. Movasati, Modular and Automorphic Forms & Beyond, Monographs in Number Theory, World Scientific, 2021.
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H. Movasati, R. Villaflor, A Course in Hodge Theory: Periods of Algebraic cycles, 33 Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, Brazil, 2021.
From the research point of view, the goal is to apply (or try to apply) arithmetic study of differential equations
to the study of Hodge loci. Hodge conjecture is one of the millennium conjectures for which the evidences are not so much:
It is proved for surfaces (Lefschetz (1,1) theorem) and surface type varieties (like cubic fourfolds),
and we do not know even it is true for all Fermat varieties (despite some partial result by T. Shioda and
his coauthors). The main goal is to introduce computational methods for finding new Hodge cycles
for hypersurfaces such that its verification is challenging in such cases! I will explain few results which
motivate us to claim that either the verification of Hodge conjecture for such cycles must be an easy
exercise in commutative algebra or they might be good candidates to be counterexamples.
For this we aim to study Hodge loci for hypersurfaces. We introduce larger parameter space and holomorphic
foliations on them such that Hodge loci become the leaves of such foliation. In our way we have to
develop a theory of holomorphic foliations on schemes such that the leaves also enjoy scheme structures
(we call them leaf schemes). We will also introduce Hasse principal or local-global conjectures for
these foliations which generalize the Katz-Grothendieck conjecture for linear differential equations.