Algebraic twists of GL(2) automorphic forms

In this thesis we consider the problem of estimating the correlation of Hecke eigenvalues of GL2 automorphic forms with a class of functions of algebraic origin defined over finite fields called trace functions. The class of trace functions is vast and includes many standard exponential sums including Gauss sums, Kloosterman sums, Hyperkloosterman sums etc. In particular we prove a Burgess type power saving (of exponent 1/8) over the trivial bound. This generalizes the results of [FKM15] to the case of number fields with a slightly more restrictive assumption on the automorphism group attached to the trace function. We work using the language of adeles which makes the analysis involved softer and makes the generalisation to number fields more natural. The proof proceeds by studying the amplified second moment spectral average of the correlation sum using the relative trace formula. This, like in the case of [FKM15] leads us to use square root cancellation of the autocorrelation sums of the trace function.