Relative Trace Formula And Simultaneous Nonvanishing for GL_3 x GL_2 and GL_3 x GL_1 L-functions

Fix a Dirichlet character $\chi$ and a cuspidal GL$(2)$ eigenform $\phi$ with relatively prime conductors. Then we show that there are infinitely many cusp forms $\pi$ on GL$(3)$ such that $L(1\mathrm{/}2,\pi \times \chi )$ and $L(1\mathrm{/}2,\pi \times \phi )$ are simultaneously non-zero. We achieve this by use of Jacquet’s Relative Trace Formula. We derive an expression of the average over the GL$(3)$ cuspidal spectrum as a sum of a non-zero main term and two subsidiary terms which are forced to be zero for large enough level by use of a suitable test function. This modest article is dedicated to the memory of Harish Chandra, on the occasion of his hundredth birthday.