Roman Holowinsky

A sieve method for shifted convolution sums

We study the average size of shifted convolution summation terms related to the problem of Quantum Unique Ergodicity on

THE AMPLIFICATION METHOD IN THE GL(3) HECKE ALGEBRA

{\rm SL}_2 (\mathbbm{Z})\backslash \mathbbm{H}

Mass equidistribution for Hecke eigenforms

. Establishing an upper-bound sieve method for handling such sums, we achieve an unconditional result which suggests that the average size of the summation terms should be sufficient in application to Quantum Unique Ergodicity. In other words, cancellations among the summation terms, although welcomed, may not be required. Furthermore, the sieve method may be applied to shifted sums of other multiplicative functions with similar results under suitable conditions.

Bounding sup-norms of cusp forms of large level

This article contains all of the technical ingredients required to implement an effective, explicit and unconditional amplifier in the context of GL(3) automorphic forms. In particular, several coset decomposition computations in the GL(3) Hecke algebra are explicitly done.