On the $P(t)$-adic Littlewood Conjecture in Characteristics $\ell \equiv 3(\mathrm{mod}4)$

Abstract

Given a prime $p$, the $p$-adic Littlewood Conjecture stands as a well-known arithmetic variant of the celebrated Littlewood Conjecture in Diophantine Approximation. In the same way as the latter, it admits a natural function field analogue depending on the choice of an irreducible polynomial $P(t)$ with coefficients in a field $\mathbb{K}$. This analogue is referred to as the $P(t)$-adic Littlewood Conjecture ($P(t)$-LC for short). $P(t)$-LC is proved to fail for any choice of irreducible polynomial $P(t)$ over any ground field $\mathbb{K}$ with characteristic $\ell \equiv 3(\mathrm{mod}4)$. The counterexample refuting it is shown to present a local arithmetic obstruction emerging from the fact that -1 is not a quadratic residue modulo a prime $\ell \equiv 3(\mathrm{mod}4)$. The theory developed elucidates and generalises all previous approaches towards refuting the conjecture. They were all based on the computer-assisted method initiated by Adiceam, Nesharim and Lunnon (2021) which has been able to establish that $P(t)$-LC fails in some small characteristics (essentially up to 11). This computer-assisted method is, however, unable to provide a general statement as it relies on ad hoc computer verifications which, provided they terminate, refute $P(t)$--LC in a given characteristic. This limitation is overcome by exhibiting an arithmetic obstruction to the validity of $P(t)$--LC in infinitely many characteristics. The existence of arithmetic obstructions within the context of $P(t)$--LC leaves the remaining case of odd characteristics $\ell \equiv 1(\mathrm{mod}4)$ dependent on their determination. This is shown to hold in an effective and explicit way.