UBC Theses and Dissertations
Escape of mass on Hilbert modular varieties Zaman, Asif Ali
Let F be a number field, G = PGL(2,F_∞), and K be a maximal compact subgroup of G. We eliminate the possibility of escape of mass for measures associated to Hecke-Maaß cusp forms on Hilbert modular varieties, and more generally on congruence locally symmetric spaces covered by G/K, hence enabling its application to the non-compact case of the Arithmetic Quantum Unique Ergodicity Conjecture. This thesis generalizes work by Soundararajan in 2010 eliminating escape of mass for congruence surfaces, including the classical modular surface SL(2,Z)\H², and follows his approach closely. First, we define M, a congruence locally symmetric space covered by G/K, and articulate the details of its structure. Then we define Hecke-Maass cusp forms and provide their Whittaker expansion along with identities regarding the Whittaker coefficients. Utilizing these identities, we introduce mock P-Hecke multiplicative functions and bound a key related growth measure following Soundararajan’s paper. Finally, amassing our results, we eliminate the possibility of escape of mass for Hecke-Maass cusp forms on M.
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