If \(f\) is a cusp form (holomorphic or Maass) on \(\text{GL}(2)\) of level \(q\) and \(L(f,s)\) denotes its \(L\)-function, then an inequality of the form \[L(f,s)~\ll~ q^{\frac{1}{4}+\epsilon}\] on \(\text{Re}\,s = 1/2\) is the so-called “convexity bound” in this context. This is the trivial bound provided by the standard Phragmén-Lindelöf convexity principle.
For many important applications, however, one requires a better subconvex bound on \(L(f,s)\),i.e., one that breaks the convexity bound by some positive power of \(q\) (subconvexity in the level aspect). Duke-Friedlander-Iwaniec proved a subconvex bound for general automoprhic forms on \(\text{GL}(2)\). If \(f\) is primitive with a primitive nebentypus and archimedean parameter \(t_{f}\), then they proved that \[ L(f,s) \ll (|s|+|t_{f}|)^{\frac{19}{2}} q^{\frac{1}{4} - \frac{1}{23041}}. \]
This article improves on the Duke-Friedlander-Iwaniec result. The authors prove that if \(f\) is primitive with non-trivial nebentypus, then \[ L(f,s) \ll (|s|+|t_{f}|)^{A} q^{\frac{1}{4} - \frac{1}{1889}}, \] where \(A\) is an absolute constant (which the authors say can probably be arranged to be 2 with some work). Not only does this result improve on the exponent in the Duke-Friedlander-Iwaniec result, it also allows for more general nebentypus, a feature that turns out to play an important role in the applications that the authors present.

Reviewer: Mahdi Asgari (Stillwater)

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