Let A = F q [T ] be the polynomial ring over F q , and F be the field of fractions of A. Let φ be a Drinfeld A-module of rank r ≥ 2 over F . For all but finitely many primes p ✁ A, one can reduce φ modulo p to obtain a Drinfeld A-module φ ⊗ F p of rank r over F p = A/p. The endomorphism ring E p = End Fp (φ ⊗ F p ) is an order in an imaginary field extension K of F of degree r. Let O p be the integral closure of A in K, and let π p ∈ E p be the Frobenius endomorphism of φ ⊗ F p . Then we have the inclusion of orders A[π p ] ⊂ E p ⊂ O p in K. We prove that if End F alg (φ) = A, then for arbitrary non-zero ideals n, m of A there are infinitely many p such that n divides the index χ(E p /A[π p ]) and m divides the index χ(O p /E p ). We show that the index χ(E p /A[π p ]) is related to a reciprocity law for the extensions of F arising from the division points of φ. In the rank r = 2 case we describe an algorithm for computing the orders A[π p ] ⊂ E p ⊂ O p , and give some computational data.2010 Mathematics Subject Classification. 11G09, 11R58.

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

No abstract

No abstract

No abstract