TY - JOUR

T1 - Arithmetic of double torus quotients and the distribution of periodic torus orbits

AU - Khayutin, Ilya

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of PGLn over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus. Using the new invariants we significantly strengthen results toward the equidistribution of packets of periodic torus orbits on higher rank S-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adèlic 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 proof combines geometric invariant theory, Galois actions, local arithmetic estimates, and homogeneous dynamics.

AB - We describe new arithmetic invariants for pairs of torus orbits on groups isogenous to an inner form of PGLn over a number field. These invariants are constructed by studying the double quotient of a linear algebraic group by a maximal torus. Using the new invariants we significantly strengthen results toward the equidistribution of packets of periodic torus orbits on higher rank S-arithmetic quotients. Packets of periodic torus orbits are natural collections of torus orbits coming from a single adèlic 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 proof combines geometric invariant theory, Galois actions, local arithmetic estimates, and homogeneous dynamics.

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U2 - 10.1215/00127094-2019-0016

DO - 10.1215/00127094-2019-0016

M3 - Article

AN - SCOPUS:85074137207

VL - 168

SP - 2365

EP - 2432

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

SN - 0012-7094

IS - 12

ER -