# On Deligne's conjecture for symmetric sixth $$L$$-functions of modular forms

08.06.2021 14:30 - 16:00

In this talk, we present our recent result on Deligne's conjecture for symmetric sixth $$L$$-functions of modular forms. We define automorphic periods associated to globally generic cuspidal automorphic representations of GSp$$_4$$ and show that the algebraicity of critical $$L$$-values for GSp$$_4 \times$$GL$$_2$$ can be expressed in terms of these periods. In the case of Kim–Ramakrishnan–Shahidi lifts of GL$$_2$$, we establish period relations for the automorphic periods and powers of Petersson norm of modular forms.The conjecture for symmetric sixth $$L$$-functions then follows from our previous work on the algebraicity of adjoint $$L$$-functions for GSp$$_4$$.