Let $L(\pi,s)=\sum_{n\geq 1}\lambda(n)/n^{s}$ be an automorphic $L$-function.

For $q$ a prime number and $\chi(q)$ a non-trivial multiplicative character, the $\chi$ twisted $L$-function is (essentially) given by

$$L(\pi.\chi,s)=\sum_{n\geq 1}\lambda(n)\chi(n)/n^{s}.$$

The subconvexity problem (in the $\chi$-aspect) aims at bounding non-trivially $L(\pi.\chi,s)$ when $\Re s=1/2$ and has now been resolved in a number of cases.

In this talk, we discuss a series of works joint with E. Fouvry, E. Kowalski, Y. Lin and W. Sawin regarding a generalisation of this problem when $\chi$ is replaced by a more general function

$$K:{\mathbb Z}/q :{\mathbb Z}\rightarrow {\mathbb C}$$

and $L(\pi.\chi,s)$ is replaced by the the $K$ algebraically twisted $L$-series

$$L(\pi.K,s)=\sum_{n\geq 1}\lambda(n)K(n)/n^{s}.$$