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.