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Fractional Parts of Linear Polynomials and an Application to Hypergeometric Functions

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Fractional Parts of Linear Polynomials and an Application to Hypergeometric Functions J. Austral. Math. Soc. 70 (2001), 401^123 FRACTIONAL PARTS OF LINEAR POLYNOMIALS AND AN APPLICATION TO HYPERGEOMETRIC FUNCTIONS ROBERTO DVORNICICH and UMBERTO ZANNIER (Received 12 March 1998; revised 8 November 2000) Communicated by W. W. L. Chen Abstract Using a result on arithmetic progressions, we describe a method for finding the rational /i-tuples p = (pi,..., ph) such that all the multiples mp (for m coprime to a denominator of p) lie in a linear variety modulo I. We give an application to hypergeometric functions. 2000 Mathematics subject classification: primary 11J54, C35. 1. Introduction We consider a class of relations of the type (1) co where ((*)) = {x} — 1/2, x e K. The L,s are linear polynomials in Q[y], y = Cyi. • • •, yh), and the qs are rational numbers. We are interested in the rational h- tuples p = (p\,..., ph) with the following property. Let m be a common denominator for pi,... , ph. For all n coprime with m (this is a technical condition), we require that y = tip be a solution of (1). We expect that these multiples np will in general be well-distributed modulo lh. On the other hand, if they all satisfy the relation (1), then they have special properties which we try to analyse. We show that they in fact form a 'quasi-linear' set, being roughly described by finitely many linear equalities mod Z* (see Section 2-II and Theorem 1 in Section 2-III). Moreover, these sets can be effectively computed. © 2001 Australian Mathematical Society 0263-6115/2001 $A2.00 + 0.00 401 Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 28 Sep 2021 at 02:55:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870000241X 402 Roberto Dvornicich and Umberto Zannier [2] In the fundamental cases (which will be called indecomposable) such quasi-linear sets will turn out to be translations by J.h of finitely many lines plus or minus finitely many points. We shall give practical methods for computing the linear conditions which describe completely the above quasi-linear set. The problem we study was motivated by the work of Schwarz [15], who in 1873 classified the hypergeometric differential equations (2) D(a, 0, Y)y = x(l - x)~-2 + [y - (1 + a + £)*]£ - a? = 0 having a full system of algebraic solutions. We briefly recall Schwarz's work and restrict our considerations to the operators which are irreducible over C(x). Set X = 1 — y, ft = y — a — fi, v = a — /?. Looking at the singularities and at the reducibility of (2), it can be proved that for such operators a, fi, y € Q, while none of the numbers X, [i, v, a, 0, y — a, y — /? belongs to Z. Further, if D(a, fi, y)y = 0 has only algebraic solutions, one sees that the same is true for D(a + l, ft+m, y+n)y = 0 for any /, m,n € 2. Hence we may assume that 0 < A < 1, 0 < /J. < 1, 0 < M < 1. Schwarz also found a group of 24 transformations on (A., /x, v) isomorphic to y^, and preserving the algebraicity of solutions; this allows the further restriction A > M > v- Under these conditions Schwarz proved that, apart from an explicitly given finite number of sporadic cases (see also [13]), algebraic solutions occur if and only if or, equivalently, In this case the monodromy group is dihedral of order 2n, where n is the denominator of v. Moreover, Schwarz produced the full list of solutions. In 1904 and 1911 Landau ([11, 12]), applying Eisenstein's criterion to power series solutions of (2), obtained the following arithmetic condition for algebraic solutions in terms of the rational parameters a = —, P = —, Y = — ((a,b,c,m) = l). m m m For every integer n such that (n, m) = 1 we have (3) either na < nc < nb (mod m) or nb < nc < na (mod m), where the inequalities are referred to the minimal non-negative residue. In 1943 Errera [5] used Landau's condition to obtain Schwarz's list by an elementary arithmetic argument (an account of which is also given in [13]). Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 28 Sep 2021 at 02:55:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870000241X [3] Fractional parts of linear polynomials 403 The arithmetic condition (3) appears also when one assumes that (2) satisfies the hypotheses for the Grothendieck conjecture: this roughly states that, if a linear differential equation with polynomial coefficients has a full system of solutions modulo all large primes, then it has a full system of algebraic solutions. In fact, this connection between (3) and the reduction of (2) mod p has been observed by Katz, who verified in [9] the Grothendieck conjecture for Picard-Fuchs differential equations, so in particular for (2) (see [9, Section 6.4]). This part of Katz's paper is however independent of the rest and essentially self-contained. One may then combine this with the Landau- Errera Theorems and Schwarz's list to verify in a different and more elementary way the Grothendieck conjecture for equations (2). To connect with the problem described at the beginning, we shall interpret the two alternative inequalities appearing in (3) in the form of just one equality of type (1) with y = np, p = (a, ft, y), namely (4) 2{(ny)) = ((no)) + ((«£)) + ((n(y - a))) + «n(y - 0))) for (n, m) = 1, where ((JC)) = {*} - 1/2, assuming that y, a, fi, y -a, y - fi £ 1. To our knowledge, this formulation of (3) seems to be new. We give and explicit computation of the linear equations which describe completely the above quasi-linear set in the case (4) (see Theorem 2 and Section 2). In this procedure we shall use a result on arithmetic progressions, proved in [4], which we recall as Lemma 10. It is quite possible that these methods can be applied to more general hypergeometric equations. In Section 2 we study general relations of type (1), while Section 3 will be devoted to the attainment of Schwarz's list (apart from sporadic cases). 2. Fractional parts of linear polynomials I. Uniform distribution Let/ : W1 —>• K be a function periodic mod Z\ and define (5) 5 = 5/ = {xeR*|/(x) = 0}. Observe that 5/ is invariant under translations by Zk. Later on / will be a linear combination of functions of the form ((ai*i + • • • + ahxh + /?)), where ajeZJeK, and ((*)) = {*} - \- We shall be interested in characterizing the set Xf C Q* defined by (6) Xf = {p e Q" | np e 5/ Vn e (1/Rl)*, where R € N and Rp e I"}. REMARK 1. We observe that the definition of Xf does not depend on R, provided that Rp e 2h. In fact, a simple argument based on the Chinese Remainder Theorem proves that whenever R | R' every class in (l/RZ)* has a representative in (l/R'T)*. Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 28 Sep 2021 at 02:55:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870000241X 404 Roberto Dvornicich and Umberto Zannier [4] For technical reasons, which will appear later, it will be convenient to work with a set more general than X;. To be precise, for a congruence class a mod q where (a, q) — 1, define h (7) Xf{a, q) = {peQ \npeSfWn=a (mod q), (n, R) = 1), where R is any natural number such that Rp € Z*. We contend that the definition of Xf{a,q) is independent of the choice of R with the given property. To see this, it is enough to show that in (7) we can replace R with any of its multiples, or, equivalently, that if R\R' and n is such that (n, R) = 1, n = a (mod q), then there exists n! = a (mod q), («', R') = 1, ri = n (mod R). Now, the reduction (I/[q, /?'])* -• {1/[q, /?])* is surjective. But the class of n mod [q, R] is coprime with [q, R] since (a, q) = 1; hence it lies in the image of the reduction, which proves our contention. This important remark will be used repeatedly in the sequel: for instance we shall assume, replacing possibly R by qR, that q divides R. More generally, we may assume that R is divisible by any given integer. The technique below was introduced by Davenport and Schinzel [3], who dealt with a problem concerning roots of unity. Another approach can be found in [7] and [8]. REMARK 2. The condition (n, R) = 1 seems rather artificial. In fact, it may be replaced by restricting n to any subset ^ of (I/RZ) such that the exponential sums 5Zn6^r e(nc//?) can be estimated somewhat non trivially. A motivation for choosing this condition is provided, for instance, by the example coming from the hypergeometric equations and discussed in the next section. For q | R, (a, q) = 1, we set R — qR* and, for integral c, we put S(c,a,q,R)= ^ e(cm/R). m=a (mod q) LEMMA 1. Setting d — (c, R*) we have d>(R) \S(c,a,q,R)\ < '*' n~ <fi(R*/d) where <p is Euler'sfunction. PROOF. S(c,a,q,R)= Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.76, on 28 Sep 2021 at 02:55:17, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.
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