A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple. In this paper, we prove that there exist infinitely many rational Diophantine sextuples. set 1 16 , 33 16 , 17 4 , 105 16 found by Diophantus (see [4]). Euler found infinitely many rational Diophantine quintuples (see [20]), e.g. he was able to extend the integer Diophantine quadruple {1, 3, 8, 120} found by Fermat, to the rational quintuple 1, 3, 8, 120, 777480 8288641 . Let us note that Baker and Davenport [2] proved that Fermat's set cannot be extended to an integer Diophantine quintuple, while Dujella and Pethő [15] showed that there is no integer Diophantine quintuple which contains the pair {1, 3}. For results on the existence of infinitely many rational D(q)-quintuples, i.e. sets in which xy + q is always a square, for q = 1 see [12].

A set of m distinct nonzero rationals {a1,a2,…,am} such that aiaj+1 is a perfect square for all 1 ≤ i < j ≤ m, is called a rational Diophantine m-tuple. It is proved recently that there are infinitely many rational Diophantine sextuples. In this paper, we construct infinite families of rational Diophantine sextuples with special structure, namely the sextuples containing quadruples and quintuples of certain type.

In this paper we show that Atkin and Swinnerton-Dyer type of congruences hold for weakly modular forms (modular forms that are permitted to have poles at cusps). Unlike the case of original congruences for cusp forms, these congruences are nontrivial even for congruence subgroups. On the way we provide an explicit interpretation of the de Rham cohomology groups associated to modular forms in terms of "differentials of the second kind". As an example, we consider the space of cusp forms of weight 3 on a certain genus zero quotient of Fermat curve X N + Y N = Z N . We show that the Galois representation associated to this space is given by a Grossencharacter of the cyclotomic field Q(ζ N ). Moreover, for N = 5 the space does not admit a "p-adic Hecke eigenbasis" for (non-ordinary) primes p ≡ 2, 3 (mod 5), which provides a counterexample to Atkin and Swinnerton-Dyer's original speculation [2,8,9].

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are infinitely many rational Diophantine quintuples. In 1999, Gibbs found the first example of a rational Diophantine sextuple, and in 2016 Dujella, Kazalicki, Mikić and Szikszai proved that there are infinitely many of them. In this paper, we prove that there exist infinitely many rational Diophantine sextuples such that the denominators of all the elements in the sextuples are perfect squares.

A Diophantine m-tuple is a set of m positive integers with the property that the product of any two of its distinct elements is one less then a square. If a set of nonzero rationals has the same property, then it is called a rational Diophantine m-tuple. Diophantus of Alexandria found the first example of a rational Diophantine quadruple {1/16, 33/16, 17/4, 105/16}, while the first Diophantine quadruple in integers was found by Fermat, and it was the set {1, 3, 8, 120}. It is well-known that there exist infinitely many integer Diophantine quadruples (e.g. {k, k + 2, 4k + 4, 16k 3 + 48k 2 + 44k + 12} for k ≥ 1), while it was proved in [3] that an integer Diophantine sextuple does not exist and that there are only finitely many such quintuples. A folklore conjecture is that there does not exist an integer Diophantine quintuple. There is an even stronger conjecture which predicts that all integer Diophantine quadruples {a, b, c, d} satisfy the equation (a+b−c−d) 2 = 4(ab+1)(cd+1) (such quadruples are called regular). However, in the rational case, there exist larger sets with the same property. Euler found infinitely many rational Diophantine quintuples, e.g. he was able to extend the Fermat quadruple to the rational quintuple {1, 3, 8, 120, 777480/8288641}. Gibbs [5] found the first rational Diophantine sextuple {11/192, 35/192, 155/27, 512/27, 1235/48, 180873/16}, while Dujella, Kazalicki, Mikić and Szikszai [4] recently proved that there exist infinitely many rational Diophantine sextuples. No example of a rational Diophantine septuple is known. Moreover, we do not know any rational Diophantine quadruple which can be extended to two different rational Diophantine sextuples. On the other hand, by the construction from [4], we know that there exist infinitely many rational Diophantine triples, each of which can be extended to rational Diophantine sextuples in infinitely many ways. In particular, there are infinitely many rational Diophantine sextuples containing the triples {15/14, −16/21, 7/6} and {3780/73, 26645/252, 7/13140}. The construction from [4] uses elliptic curves induced by Diophantine triples, i.e. curves of the form y 2 = (x + ab)(x + ac)(x + bc) where {a, b, c} is a rational Diophantine triple, with torsion group Z/2Z × Z/6Z over Q. Piezas [7] studied Gibbs's examples of rational Diophantine sextuples which do not fit into the construction from [4] and realized that most of them follow a common pattern: they contain two regular subquadruples with two common elements (see Proposition 1). By studying sextuples