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Algebraic solutions of ODE using p-adic numbers - Katz N.M.

Название: Algebraic solutions of ODE using p-adic numbers
Автор: Katz N.M.
Категория: Математика
Тип: Книга
Дата: 30.12.2008 15:40:11
Скачано: 49
Описание: This paper grew out of an attempt to answer the following question, first raised by Grothendieck. Consider a linear homogeneous nxn system of first-order differential equations -L(Y)=A{z)Y az in which A{z). is an n x n matrix of rational functions of z. To fix ideas, suppose that the coefficients of the entries of A(z) all lie in an algebraic number field K. Then for almost all primes p of K, it makes sense to reduce this equation modulo p, obtaining a differential equation over F,(z). (I) Suppose that for almost all primes p, the reduced equation has a full set of solutions (i.e., has n solutions in (F,(z))" which are linearly independent over Fq(z)). Does the original equation admit a full set of solutions in algebraic functions of z? For example, the equation d /л l with aeZ may be reduced modulo p for all those primes p not dividing a, and the reduced equation admits the solution z* for any integer b such that absl modulo p. The original equation has for its solution the function z11". Of course, (I) may be reformulated in greater apparent generality. Let R be a subring of С which, as a ring, is finitely generated over Z. Let S be a smooth R-scheme with geometrically connected fibres, and consider a differential equation (M, V) on S/R, by which we understand a locally free sheaf M on S of finite rank together with an R-linear integrable connection V: M-* Qls/R®M. (We considered above the case K = 0[l/n], & the ring of integers in an algebraic number field, S an open set in P},, М = Й£, and V: M-*Q\jR®M given by Vm = dm-dz®A(z)m.) I Inventioncs math., Vol. 18
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