Dominic Lanphier

Values of symmetric cube

We obtain formulas for certain weighted sums of values of the symmetric square and triple product L-functions. As a consequence, we get exact values at the right critical point for the symmetric square and symmetric cube L-functions attached to certain cuspforms. We also give applications to Fourier coefficients of modular forms.

L

We give a decomposition theorem for Platonic graphs over finite fields and use this to determine the spectrum of these graphs. We also derive estimates for the isoperimetric numbers of the graphs.

-functions and Fourier coefficients of Siegel Eisenstein series of degree-3

Let GN be either a symplectic, unitary or Hermitian orthogonal group of rank 2N = 2m+2n with m ≤ n. We show that the restriction of a Siegel-Hilbert Eisenstein series on GN to the diagonally embedded group Gm×Gn has a nontrivial cuspidal component in the smaller variable. As a consequence, we explicitly construct classes of Siegel-Hilbert cuspforms with rational-valued Fourier coefficients.

The spectrum of Platonic graphs over finite fields

Abstract We study several measures of vulnerability of nearest neighbor graphs. We obtain estimates on the toughness, the tenacity, the scattering number, and the rupture degree of nearest neighbor graphs. The method is to recursively apply a certain separator theorem an appropriate number of times, depending on the particular measure studied. We also apply these methods to obtain estimates of these measures for several other classes of graphs.