For a family of tempered cuspidal automorphic representations pi of GL_2(Q) with bounded archimedean parameters and conductor q(pi) tending to infinity, determine whether there exists a constant delta>0 such that, uniformly in the level aspect, L!(frac12,Sym^2 pi)≪_varepsilon,pi_∞q(Sym^2 pi)^frac14-delta+varepsilon for every varepsilon>0.

Subconvexity problem for symmetric-square L-functions in the level aspect

The problem asks for a bound beating the convexity estimate for symmetric-square $L$ -functions as the level grows. A 2017 preprint already gave a qualitative improvement for prime-level holomorphic forms, while the latest claim concerns a more specific local-ramification class.

Known results

In 2017, a preprint proved $L(1/2,\operatorname{Sym}^2 f)\ll p^{1/2-\delta}$ for prime level $p$ , with some computable $\delta>0$ , for fixed-weight holomorphic forms with trivial nebentypus.
A 2023 result was subconvex in the spectral parameter but remained nearly convex in the level aspect.

September 2026 claimed level-aspect bound

A September 3, 2026 report cites arXiv:2609.04155 as proving exponent $1/4-1/168+o(1)$ for the stated conductor and local-ramification class, described as the first reported bound of this strength for this $\mathrm{GL}_3$ setting. The preprint is unrefereed, so the claim remains unverified.

Current status (as of September 2026): A restricted level-aspect subconvexity claim is available, but it is unrefereed and unverified; no broader resolution is established.

Subconvexity problem for symmetric-square L-functions in the level aspect

Primary source

Additional references

Progress summary

Solutions 0