Resumen de Subconvexity for twisted L-functions on GL(3)

Valentin Blomer

• Let $q$ be a large prime and $\chi$ the quadratic character modulo $q$. Let $\phi$ be a self-dual cuspidal Hecke eigenform for ${\rm SL}(3, \Bbb{Z})$, and $f$ a Hecke-Maa{\ss} cusp form for $\Gamma_0(q) \subseteq {\rm SL}_2(\Bbb{Z})$. We consider the twisted $L$-functions $L(s, \phi \times f \times \chi)$ and $L(s, \phi \times \chi)$ on ${\rm GL}(3)\times {\rm GL}(2)$ and ${\rm GL}(3)$ with conductors $q^6$ and $q^3$, respectively. We prove the subconvexity bounds $$ L(1/2, \phi \times f \times \chi) \ll_{\phi, f, \varepsilon} q^{5/4+\varepsilon},\quad L(1/2+it, \phi \times \chi) \ll_{\phi, t, \varepsilon} q^{5/8+\varepsilon} $$ for any $\varepsilon > 0$.