Number Theory Seminar

Via Artin formalism, if a complex Galois representation $V$ decomposes as a direct sum of $V_1$ and $V_2$, then the associated complex $L$-function also decomposes as a product of $L(V_1, s)$ and $L(V_2, s)$. Unfortunately, this can fail when one considers $p$-adic $L$-functions. I would recall some known cases where this holds and discuss what one can say about the case of the triple product of modular forms.