##
**Integral solutions of Apéry-like recurrence equations.**
*(English)*
Zbl 1244.11042

Harnad, John (ed.) et al., Groups and symmetries. From Neolithic Scots to John McKay. Selected papers of the conference, Montreal, Canada, April 27–29, 2007. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4481-6/pbk). CRM Proceedings and Lecture Notes 47, 349-366 (2009).

From the text: In [Math. Z. 241, No. 2, 425–444 (2002; Zbl 1023.34081)], F. Beukers studied the differential equation
\[
((t^3 + at^2 + bt)F'(t))'+ (t-\lambda)F(t) = 0 \tag{*}
\]
where \(a, b\) and \(\lambda\) are rational parameters, and asks for which values of these parameters this equation has a solution in \(\mathbb Z[[t]]\), the motivating example being the Apéry sequence with \(a=11\), \(b=-1\), \(\lambda=-3\). We describe a search over a suitably chosen domain of 100 million triples \((a,b,\lambda)\). In this domain there are 36 triples yielding integral solutions of (*). These can be further subdivided into members of four infinite classes, two of which are degenerate special cases of the other two, and seven sporadic solutions. Of these solutions, twelve, including all the sporadic ones, have parametrizations of Beukers’ type in terms
of modular forms and functions. These solutions are related to elliptic curves over \(\mathbb P^1\) with four singular fibres.

(Final remarks) The numerical experiments described in the first sections of this paper were carried out in 1997/98, inspired by the beautiful lecture given by Beukers on his work cited above at the conference in honor of Schinzel in Zakopane in 1997. A first version of the paper was written in 2000 and has been circulating as an informal preprint ever since, with few changes except for the addition of §6 on the periods associated to Apéry-like recursions (which was written in answer to a question posed by A. Connes during my course at the Collège de France in 2001) and some additions to the discussion of the connection with Beauville’s classification in §7. The main reason for not publishing it earlier was that I thought that more explanations of the connections to geometry should be added, but did not understand these well enough. In the meantime, several other related papers, both on the geometric and on the purely differential equations/modular forms side have appeared (in some cases quoting the informal preprint version of this paper), and part of what is presented here is perhaps obsolete. I have nevertheless kept the entire text since this paper in any case contains no real theorems but is to be seen more as an informal discussion of various experimental and theoretical aspects of the three-way connection between algebraic geometry, linear differential equations, and the theory of modular forms.

For the entire collection see [Zbl 1167.20300].

(Final remarks) The numerical experiments described in the first sections of this paper were carried out in 1997/98, inspired by the beautiful lecture given by Beukers on his work cited above at the conference in honor of Schinzel in Zakopane in 1997. A first version of the paper was written in 2000 and has been circulating as an informal preprint ever since, with few changes except for the addition of §6 on the periods associated to Apéry-like recursions (which was written in answer to a question posed by A. Connes during my course at the Collège de France in 2001) and some additions to the discussion of the connection with Beauville’s classification in §7. The main reason for not publishing it earlier was that I thought that more explanations of the connections to geometry should be added, but did not understand these well enough. In the meantime, several other related papers, both on the geometric and on the purely differential equations/modular forms side have appeared (in some cases quoting the informal preprint version of this paper), and part of what is presented here is perhaps obsolete. I have nevertheless kept the entire text since this paper in any case contains no real theorems but is to be seen more as an informal discussion of various experimental and theoretical aspects of the three-way connection between algebraic geometry, linear differential equations, and the theory of modular forms.

For the entire collection see [Zbl 1167.20300].

### MSC:

11F23 | Relations with algebraic geometry and topology |

11F20 | Dedekind eta function, Dedekind sums |

14J33 | Mirror symmetry (algebro-geometric aspects) |

33C80 | Connections of hypergeometric functions with groups and algebras, and related topics |