Large values of newforms on GL(2) with highly ramified central character

Abstract

We give a lower bound for the sup-norm of an $L^2$-normalized newform in an irreducible, unitary, cuspidal representation $\pi$ of $G{L}_2$ over a number field. When the central character of $\pi$ is sufficiently ramified, this bound improves upon the trivial bound by a positive power of $N$ where $N$ is the norm of the conductor of $\pi$. This generalizes a result of Templier, who dealt with the special case when the conductor of the central character equals the conductor of the representation. We also make a conjecture about the true size of the sup-norm in the $N$-aspect that takes into account this central character phenomenon. Our results depend upon some explicit formulas and bounds for the Whittaker newvector over a non-archimedean local field, which may be of independent interest.

Final version; to appear in IMRN

References in corpus (2)

• Hybrid sup-norm bounds for Maass newforms of powerful level
• Local bounds for $L^p$ norms of Maass forms in the level aspect

Cited by in corpus (8)

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• On sup-norm bounds part II: $GL(2)$ Eisenstein series
• On the global sup-norm of GL(3) cusp forms
• Sup norms of newforms on $G{L}_2$ with highly ramified central character
• An explicit conductor formula for ${\mathrm{GL}}_n\times {\mathrm{GL}}_1$
• Adelic Voronoi Summation and Subconvexity for GL(2) L-Functions in the depth Aspect