One can attach $\ell$-adic Galois representations to holomorphic cuspidal newforms of weight $2$ on the upper half-plane.
If a newform is $L^2$-normalized, can one extract its maximum value from the Galois representation?
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3$\begingroup$ I think you can only attach a Galois-representation to cohomological cuspidal automorphic reprepresentations (and their newforms). The maximum value of a newform is a subtle problem, even for holomorphic newforms on the upper half-plane, and I don't know of any connection with Galois-representations. $\endgroup$– GH from MOCommented Apr 12, 2021 at 13:45
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$\begingroup$ I had holomorphic modular forms in mind. $\endgroup$– supCommented Apr 12, 2021 at 15:45
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1$\begingroup$ I am not sure I understand the question: isn't a holomorphic modular form (cuspidal or not) unbounded on the upper half-plane ? It is y^{k/2}|f(x+iy)| which is bounded. $\endgroup$– Henri CohenCommented Apr 12, 2021 at 16:33
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1$\begingroup$ @HenriCohen: In the sup-norm problem for holomorphic cusp forms, one studies $y^{k/2}|f(x+iy)|$, as you suggest. I am sure the OP had this function in mind (and so did I). $\endgroup$– GH from MOCommented Apr 12, 2021 at 16:40
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$\begingroup$ I added the adjective "holomorphic" to the post. Note that there are Maass forms of all weights on the upper half-plane, including those that do not come from holomorphic cusp forms (cf. Henri Cohen's remark). $\endgroup$– GH from MOCommented Apr 12, 2021 at 16:43
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