When Apéry announced his proof of the irrationality of ζ(3) in 1978, he was reportedly asked where his formulas came from and replied that they “grow in my garden.” Over the following decades, mathematicians developed a range of constructions in search of further irrationality proofs in Apéry’s garden. In this talk, we revisit this garden and present a construction proving that 1, ζ(3) and ζ(4) are linearly independent over the rationals.

Let p > 3 be a prime number. If A and B are two principally polarized superspecial abelian surfaces over F_p² with p²-Frobenius [−p], we prove that there exists a separable polarized isogeny between them with multiplier at most p/√2. The multiplier bound is uniform in the pair and asymptotically optimal up to a small multiplicative constant. Given quaternionic coordinates, we provide a deterministic algorithm that satisfies this upper bound in polynomial time. For unrestricted multipliers, KLPT² provides a heuristic algorithm with a multiplier upper bound of order p^6+o(1) for arbitrary pairs and p^3+o(1) when one polarization matrix is the identity matrix. Our algorithm’s improvement is an explicit upper bound below p with an unconditional deterministic guarantee.

Let K be a field and let \phi be a quadratic form over K. Then \phi gives rise to a linear algebraic group, called the group of proper projective similitudes PSim^+(\phi). Such groups, for even-dimensional forms, constitute a family of classical linear algebraic groups of type D. While investigating which linear algebraic groups are rational varieties, A. Merkurjev showed that PSim^+(\phi) is R-trivial precisely when a solely quadratic form theoretic group attached to the form \phi is trivial over all field extensions of K. This means in particular that with tools from quadratic form theory one can try to understand for which forms \phi the group PSim^+(\phi) is not R-trivial, and therefore not a rational variety. In this talk, we will discuss the way in which a mild assumption on the even Clifford algebra of \phi, a classical invariant for quadratic forms, assures that the group PSim^+(\phi) is not R-trivial. Towards the end we will also glimpse at the more general situation of PSim^+(A, \sigma) for a K-algebra with involution (A, \sigma) and discuss how much of the strategies carry over.