Geodesic
On the local $L^2$-Bound of the Eisenstein series
Forum of Mathematics, Sigma, Tome 12 (2024)

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We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. We derive a poly-logarithmic bound on an average, for a large class of reductive groups. The method is based on Arthur’s development of the spectral side of the trace formula, and ideas of Finis, Lapid and Müller.As applications of our method, we prove the optimal lifting property for $\mathrm {SL}_n(\mathbb {Z}/q\mathbb {Z})$ for square-free q, as well as the Sarnak–Xue [52] counting property for the principal congruence subgroup of $\mathrm {SL}_n(\mathbb {Z})$ of square-free level. This makes the recent results of Assing–Blomer [8] unconditional. 
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     author = {Subhajit Jana and Amitay Kamber},
     title = {On the local $L^2${-Bound} of the {Eisenstein} series},
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Subhajit Jana; Amitay Kamber. On the local $L^2$-Bound of the Eisenstein series. Forum of Mathematics, Sigma, Tome 12 (2024). doi: 10.1017/fms.2024.59

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