Mathematics Colloquia and Seminars

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This talk will be comprised of two parts.  Let A,B be integers such that 4A^3 - 27B^2 is nonzero and let Q(u,v) be a positive-definite quadratic form. The first part will focus on deriving an asymptotic for the number of rational points on the elliptic surface y^2 = x^3 + AxQ(u,v)^2 + BQ(u,v)^3 that are integral with respect to the singularity. This method utilizes Mordell's parameterization of integral points on quadratic twists on elliptic curves, which is based on a syzygy for invariants of binary quartic forms.  In the second part, we study sums of absolute values of Hecke eigenvalues of GL(2) representations that are tempered at all finite places. We show that these sums exhibit logarithmic savings over the trivial bound if and only if the representation is cuspidal. Further, we connect the problem of studying the sums of Hecke eigenvalues along polynomial values to the base change problem for GL(2).  While seemingly disconnected, the second part will be essential for deriving the asymptotic in the first part.