On the size of 𝑝-adic Whittaker functions

Assing, Edgar

doi:10.1090/tran/7685

On the size of $p$-adic Whittaker functions
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Trans. Amer. Math. Soc. 372 (2019), 5287-5340 Request permission

Abstract:

In this paper we tackle a question raised by N. Templier and A. Saha concerning the size of Whittaker new vectors appearing in infinite-dimensional representations of $\textrm {GL}_2$ over nonarchimedean fields. We derive precise bounds for such functions in all possible situations. Our main tool is the $p$-adic method of stationary phase.

References

Valentin Blomer and Djordje Milićević, $p$-adic analytic twists and strong subconvexity, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 3, 561–605 (English, with English and French summaries). MR 3377053, DOI 10.24033/asens.2252
F. Brumley and N. Templier, Large values of cusp forms on GL(n), arXiv:1411.4317 (2014).
Todd Cochrane and Zhiyong Zheng, Exponential sums with rational function entries, Acta Arith. 95 (2000), no. 1, 67–95. MR 1787206, DOI 10.4064/aa-95-1-67-95
Todd Cochrane and Zhiyong Zheng, A survey on pure and mixed exponential sums modulo prime powers, Number theory for the millennium, I (Urbana, IL, 2000) A K Peters, Natick, MA, 2002, pp. 273–300. MR 1956230
James W. Cogdell, Bessel functions for $GL_2$, Indian J. Pure Appl. Math. 45 (2014), no. 5, 557–582. MR 3286079, DOI 10.1007/s13226-014-0081-8
A. Corbett and A. Saha, On the order of vanishing of newforms at cusps, arXiv:1609.08939; Math. Res. Lett. (to appear).
Romuald Dąbrowski and Benji Fisher, A stationary phase formula for exponential sums over $\mathbf Z/p^m\mathbf Z$ and applications to $\textrm {GL}(3)$-Kloosterman sums, Acta Arith. 80 (1997), no. 1, 1–48. MR 1450415, DOI 10.4064/aa-80-1-1-48
Stephen Gelbart and Hervé Jacquet, Forms of $\textrm {GL}(2)$ from the analytic point of view, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 213–251. MR 546600
Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214, DOI 10.1090/coll/053
H. Jacquet and R. P. Langlands, Automorphic forms on $\textrm {GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654, DOI 10.1007/BFb0058988
A. Knightly and C. Li, Kuznetsov’s trace formula and the Hecke eigenvalues of Maass forms, Mem. Amer. Math. Soc. 224 (2013), no. 1055, vi+132. MR 3099744, DOI 10.1090/S0065-9266-2012-00673-3
E. Kowalski, Some aspects and applications of the Riemann hypothesis over finite fields, Milan J. Math. 78 (2010), no. 1, 179–220. MR 2684778, DOI 10.1007/s00032-010-0111-x
Abhishek Saha, Large values of newforms on $\textrm {GL}(2)$ with highly ramified central character, Int. Math. Res. Not. IMRN 13 (2016), 4103–4131. MR 3544630, DOI 10.1093/imrn/rnv259
Abhishek Saha, Hybrid sup-norm bounds for Maass newforms of powerful level, Algebra Number Theory 11 (2017), no. 5, 1009–1045. MR 3671430, DOI 10.2140/ant.2017.11.1009
Ralf Schmidt, Some remarks on local newforms for $\rm GL(2)$, J. Ramanujan Math. Soc. 17 (2002), no. 2, 115–147. MR 1913897
N. Templier, Voronoï summation for $\operatorname {GL}(2)$, Representation Theory, Automorphic Forms, and Complex Geometry, Proceedings Volume in Honor of Wilfried Schmid, International Press, (to appear), http://pi.math.cornell.edu/~templier/papers/voronoi-gl2.pdf.
Nicolas Templier, Large values of modular forms, Camb. J. Math. 2 (2014), no. 1, 91–116. MR 3272013, DOI 10.4310/CJM.2014.v2.n1.a3
André Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A. 34 (1948), 204–207. MR 27006, DOI 10.1073/pnas.34.5.204