Mass equidistribution for automorphic forms of cohomological type on 𝐺𝐿₂

Abstract

We extend Holowinsky and Soundararajan’s proof of quantum unique ergodicity for holomorphic Hecke modular forms on S L ( 2 , Z ) SL(2,\mathbb {Z}) , by establishing it for automorphic forms of cohomological type on G L 2 GL_2 over an arbitrary number field which satisfy the Ramanujan bounds. In particular, we have unconditional theorems over totally real and imaginary quadratic fields. In the totally real case we show that our result implies the equidistribution of the zero divisors of holomorphic Hecke modular forms, generalising a result of Rudnick over Q \mathbb {Q} .

Equidistribution of Zeros of Holomorphic Sections in the Non-compact Setting

“…If one considers Hecke modular forms, by a result of Rudnick [34] and its generalization by Marshall [26], the zeros of all sequences of Hecke modular forms are equidistributed. The method used in [26,34] follows the seminal paper of Nonnenmacher-Voros [32] and consists in showing the equidistribution of masses of Hecke forms. Rudnick [34] invoked for this purpose the Generalized Riemann Hypothesis and this hypothesis was later removed by Holowinsky and Soundararajan [22].…”

“…By replacing Laplace eigenfunctions with modular forms one is lead to study of the equidistribution of zeros of Hecke modular forms. This was done by Rudnick [34], Holowinsky and Soundararajan [22] and generalized by Marshall [26] and Nelson [31].…”

Erratum to “Mass equidistribution for automorphic forms of cohomological type on $GL_{2}$”

“…We may still draw interesting conclusions from Theorem 1 of [4] in the imaginary quadratic case. Cohomological forms on GL 2 /K which are base changes from Q will satisfy Ramanujan, and so Theorem 1 establishes their equidistribution as their weight becomes large.…”

“…Corollary 3 of [4] is not known unconditionally, as cohomological automorphic forms on GL 2 over an imaginary quadratic field are not known to satisfy the Ramanujan conjecture. We shall briefly describe the reason for this and discuss what information Theorem 1 of [4] does give in the case of imaginary quadratic fields.…”

“…We shall briefly describe the reason for this and discuss what information Theorem 1 of [4] does give in the case of imaginary quadratic fields.…”