1
$\begingroup$

Sorry to disturb, the experts here. Recently, I read a paper of Nelson ("Subconvex equidistribution of cusp forms: reduction to Eisenstein observables--"https://arxiv.org/pdf/1702.02908.pdf). The paper says that the subconvexity for the twisted symmetric-square L-function $L(s,\text{sym}^2 f\otimes \chi)$ in the level aspect would imply the subconvexity for $GL_3\times GL_2$ L-functions $L(s, \text{sym}^2 f\otimes g)$ in the level aspect of the newform $f$, where $g$ is a fixed $GL_2$-newform.

My question is: if one has the analogy that the hybrid subconvexity for $L(s,\text{sym}^2 f\otimes \chi)$ in the level and conductor aspects implies that the hybrid subconvexity for $L(s, \text{sym}^2 f\otimes g)$ in both levels aspects of $f,g$?

If any expert here knows something about this, please show me a guide or certain references.

Many thanks in advance.

Improve this question
$\endgroup$
10
  • $\begingroup$ No, we don't know this, and there's no reason to expect this to be true. $\endgroup$
    – Peter Humphries
    Commented Aug 4, 2023 at 15:52
  • 2
    $\begingroup$ Okay, maybe this isn't completely accurate. Say $f$ has squarefree level $q$ and trivial central character, and $g$ is of level $r^2$ (where $(q,r) = 1$) and trivial central character such that there exists a primitive Dirichlet character $\psi$ modulo $r$ such that $g \otimes \psi$ has level $r$. Then it should be possible to prove using currect techniques that hybrid subconvexity for $L(s,\operatorname{sym}^2 f \otimes \chi)$ implies hybrid subconvexity for $L(s,\operatorname{sym}^2 f \otimes g)$. This would be a ton of work, however. $\endgroup$
    – Peter Humphries
    Commented Aug 4, 2023 at 16:14
  • 2
    $\begingroup$ On the other hand, if instead we assume that $g$ has squarefree level $r$ (rather than level $r^2$) and trivial central character, then current techniques no longer work. So there's no systematic treatment. $\endgroup$
    – Peter Humphries
    Commented Aug 4, 2023 at 16:16
  • $\begingroup$ @PeterHumphries Dear Humphries, could you please show some references on this, if $g$ is of level $r^2$ as you assumed? On the other hand, I also curious if Nelson's work can be adaptable to show that the level aspect subconvexity of $f$ for $L(\text{sym}^2 f\otimes \chi)$ implies the aspect subconvexity of $f$ for $L(\text{sym}^2 f\otimes g)$, if $\chi$ and $g$ are all allowed to be taken as certain special cases. Great thanks! $\endgroup$
    – hofnumber
    Commented Aug 7, 2023 at 1:26
  • $\begingroup$ There is no reference; this isn't in the literature. As I said, it would be a ton of work to prove, but is doable by current methods (the strategy is to combine Nelson's work with earlier work of Blomer on subconvexity for twists of selfdual $\mathrm{GL}_3$ $L$-functions, as well as recent work of Petrow-Young). As to your second question, Nelson's paper is quite clear about why his method works and how it does not generalise to other settings. $\endgroup$
    – Peter Humphries
    Commented Aug 7, 2023 at 2:51

0

Reset to default

You must log in to answer this question.