Joint IAS/Princeton University Number Theory Seminar

In this talk I will describe some new arithmetic invariants for pairs of torus orbits on inner forms of $\mathbf{PGL}_n$ and $\mathbf{SL}_n$. These invariants allow us to significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank $S$-arithmetic quotients. An important aspect of our method is that it applies to packets of periodic orbits of maximal tori which are only partially split. Packets of periodic torus orbits are natural collections of torus orbits coming from a single rational adelic torus and are closely related to class groups of number fields. The distribution of these orbits is akin to the distribution of integral points on homogeneous algebraic varieties with a torus stabilizer. The distribution of packets of periodic torus orbit has been studied using dynamical methods in the pioneering work of Linnik in the rank 1 case (equidistribution on the 2-sphere) and by Einsiedler, Lindenstrauss, Michel and Venkatesh (ELMV) in higher rank. We note that in rank 1, stronger equidistribution results for packets of periodic orbits have been established by Duke and Iwaniec using the theory of automorphic functions. The dynamical approach typically consists of two main ingredients: an arithmetic one, which implies that the toral packets have high asymptotic metric entropy, and a measure rigidity argument, which deduces from the entropy result a statement regarding the limit distribution of the orbits. While thanks to Einsiedler, Katok and Lindenstrauss we have very powerful measure rigidity tools for higher rank toral actions, we know much less regarding the arithmetic of these packets in higher rank. A notable exception is the work of ELMV which synergies the dynamical approach with harmonic analysis to prove an equidistribution theorem similar to Linnik's in the split rank 2 case. In the other cases the current known results, due to the same authors, are significantly weaker. Our methods generalize Linnik's original arithmetic approach in a different direction. We derive new invariants, akin to the discriminant inner product used by Linnik. These invariants come from studying double quotients of a reductive group by a torus using geometric invariant theory. We then derive a sharper lower bound for the entropy from these invariants using the action of the Galois group of the torus' splitting field and the algebraic relations between the invariants. This lower bound gives new qualitative restrictions on the possible limit measures and applies also to partially split maximal tori.