##
**Subconvex equidistribution of cusp forms: reduction to Eisenstein observables.**
*(English)*
Zbl 1428.11093

The subconvexity problems for families of the automorphic \(L\)-functions are closely related to important arithmetical equidistribution problems. In the paper under the review, the author shows that, under certain assumptions, the subconvexity problem for the family \(L(\mathrm{ad}(\pi) \otimes \tau, \frac{1}{2})\), with \(\tau\) on \(\mathrm{PGL}_2\) fixed and \(\pi\) of \(\mathrm{GL}_2\) varying, can be reduced to the subconvexity problem for \(L(\pi \otimes \overline{\pi} \otimes \chi, \frac{1}{2})\), with \(\chi\) on \(\mathrm{GL}_1\) fixed and \(\pi\) on \(\mathrm{GL}_2\) varying.

More precisely, if \(\pi\) is a sequence of cuspidal automorphic representations of \(\mathrm{GL}_2\) with unramified central character, large prime and bounded infinity type, then the subconvexity for \(\mathrm{GL}_1\)-twists of the adjoint \(L\)-function of \(\pi\) implies the subconvexity for \(\mathrm{PGL}_2\)-twists of the adjoint \(L\)-function of \(\pi\) (with polynomial dependence upon the conductor of the twists).

More precisely, if \(\pi\) is a sequence of cuspidal automorphic representations of \(\mathrm{GL}_2\) with unramified central character, large prime and bounded infinity type, then the subconvexity for \(\mathrm{GL}_1\)-twists of the adjoint \(L\)-function of \(\pi\) implies the subconvexity for \(\mathrm{PGL}_2\)-twists of the adjoint \(L\)-function of \(\pi\) (with polynomial dependence upon the conductor of the twists).

Reviewer: Ivan Matić (Osijek)

### MSC:

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

11F27 | Theta series; Weil representation; theta correspondences |

58J51 | Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity |

### References:

[1] | J. Bernstein and A. Reznikov, Subconvexity bounds for triple \(L\)-functions and representation theory, Ann. of Math. (2), 172 (2010), no. 3, 1679-1718. · Zbl 1225.11068 |

[2] | V. Blomer and F. Brumley, On the Ramanujan conjecture over number fields, Ann. of Math. (2) 174 (2011), no. 1, 581-605. · Zbl 1322.11039 |

[3] | V. Blomer and G. Harcos, Twisted \(L\)-functions over number fields and Hilbert’s eleventh problem, Geom. Funct. Anal. 20 (2010), no. 1, 1-52. · Zbl 1221.11121 |

[4] | V. A. Bykovskiĭ, A trace formula for the scalar product of Hecke series and its applications (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 226 (1996), 14-36, Anal. Teor. Chisel i Teor. Funktsiĭ. 13, 235-236; English translation in J. Math. Sci. 89 (1998), no. 1, 915-932. · Zbl 0898.11017 |

[5] | M. Cowling, U. Haagerup, and R. Howe, Almost \(L^2\) matrix coefficients, J. Reine Angew. Math. 387 (1988), 97-110. · Zbl 0638.22004 |

[6] | W. Duke, J. B. Friedlander, and H. Iwaniec, The subconvexity problem for Artin \(L\)-functions, Invent. Math. 149 (2002), no. 3, 489-577. · Zbl 1056.11072 |

[7] | M. Einsiedler, E. Lindenstrauss, P. Michel, and A. Venkatesh, Distribution of periodic torus orbits and Duke’s theorem for cubic fields, Ann. of Math. (2) 173 (2011), no. 2, 815-885. · Zbl 1248.37009 |

[8] | É. Fouvry, E. Kowalski, and P. Michel, Algebraic twists of modular forms and Hecke orbits, Geom. Funct. Anal. 25 (2015), no. 2, 580-657. · Zbl 1344.11036 |

[9] | W. T. Gan, The shimura correspondence à la Waldspurger, preprint, http://www.math.nus.edu.sg/ matgwt/postech.pdf (accessed 27 May 2019). |

[10] | S. Gelbart and H. Jacquet, A relation between automorphic representations of \(\text{GL}(2)\) and \(\text{GL}(3)\), Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 4, 471-542. · Zbl 0406.10022 |

[11] | P. Gérardin and J.-P. Labesse, “The solution of a base change problem for \(\text{GL}(2)\) (following Langlands, Saito, Shintani)” in Automorphic forms, representations and \(L\)-functions (Corvallis, 1977), Part 2, Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc., Providence, 1979, 115-133. · Zbl 0412.10018 |

[12] | A. Ghosh, A. Reznikov, and P. Sarnak, Nodal domains of Maass forms, I, Geom. Funct. Anal. 23 (2013), no. 5, 1515-1568. · Zbl 1328.11044 |

[13] | D. Goldberg and D. Szpruch, Plancherel measures for coverings of \(p\)-adic \(\text{SL}_2(F)\), Int. J. Number Theory 12 (2016), no. 7, 1907-193. · Zbl 1367.11052 |

[14] | G. Harcos and P. Michel, The subconvexity problem for Rankin-Selberg \(L\)-functions and equidistribution of Heegner points, II, Invent. Math. 163 (2006), no. 3, 581-655. · Zbl 1111.11027 |

[15] | J. Hoffstein and D. Ramakrishnan, Siegel zeros and cusp forms, Internat. Math. Res. Notices 1995, no. 6, 279-308. · Zbl 0847.11043 |

[16] | R. Holowinsky, Sieving for mass equidistribution, Ann. of Math. (2) 172 (2010), no. 2, 1499-1516. · Zbl 1214.11054 |

[17] | R. Holowinsky and K. Soundararajan, Mass equidistribution for Hecke eigenforms, Ann. of Math. (2) 172 (2010), no. 2, 1517-1528. · Zbl 1211.11050 |

[18] | R. Holowinsky, R. Munshi, and Z. Qi, Hybrid subconvexity bounds for \(L(\frac{1}{2},\text{Sym}^2f\otimes g)\), Math. Z. 283 (2016), no. 1-2, 555-579. · Zbl 1408.11045 |

[19] | Y. Hu, Triple product formula and mass equidistribution on modular curves of level \(N\), Int. Math. Res. Not. IMRN 2018, no. 9, 2899-2943. · Zbl 1444.11099 |

[20] | A. Ichino, Trilinear forms and the central values of triple product \(L\)-functions, Duke Math. J. 145 (2008), no. 2, 281-307. · Zbl 1222.11065 |

[21] | H. Iwaniec and P. Michel, The second moment of the symmetric square \(L\)-functions, Ann. Acad. Sci. Fenn. Math. 26 (2001), no. 2, 465-482. · Zbl 1075.11040 |

[22] | H. Iwaniec and P. Sarnak, “Perspectives on the analytic theory of \(L\)-functions” in GAFA 200(Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part II, Birkhäuser, Basel, 2000, 705-741. · Zbl 0996.11036 |

[23] | J. Jung, Quantitative quantum ergodicity and the nodal domains of Hecke-Maass cusp forms, Comm. Math. Phys. 348 (2016), no. 2, 603-653. · Zbl 1388.58021 |

[24] | B. Kahn, “Le groupe des classes modulo \(2\), d’après Conner et Perlis” in Seminar on number theory, 1984-1985 (Talence, 1984/1985), Univ. Bordeaux I, Talence, 1985. · Zbl 0589.10022 |

[25] | H. H. Kim, Functoriality for the exterior square of \(\text{GL}_4\) and the symmetric fourth of \(\text{GL}_2 \), with appendix 1 “A descent criterion for isobaric representations“ by D. Ramakrishnan and appendix 2 “Refined estimates towards the Ramanujan and Selberg conjectures” by H. Kim and P. Sarnak, J. Amer. Math. Soc. 16 (2003), no. 1, 139-183. · Zbl 1018.11024 |

[26] | E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2) 163 (2006), no. 1, 165-219. · Zbl 1104.22015 |

[27] | W. Luo and P. Sarnak, Mass equidistribution for Hecke eigenforms, Comm. Pure Appl. Math. 56 (2003), no. 7, 874-891. · Zbl 1044.11022 |

[28] | P. Michel, The subconvexity problem for Rankin-Selberg \(L\)-functions and equidistribution of Heegner points, Ann. of Math. (2) 160 (2004), no. 1, 185-236. · Zbl 1068.11033 |

[29] | P. Michel, “Analytic number theory and families of automorphic \(L\)-functions” in Automorphic Forms and Applications, IAS/Park City Math. Ser. 12, Amer. Math. Soc., Providence, 2007, 181-295. · Zbl 1168.11016 |

[30] | P. Michel and A. Venkatesh, “Equidistribution, \(L\)-functions and ergodic theory: on some problems of Yu. Linnik” in International Congress of Mathematicians, Vol. II, Eur. Math. Soc., Zürich, 2006, 421-457. · Zbl 1157.11019 |

[31] | P. Michel and A. Venkatesh, The subconvexity problem for \(\text{GL}_2 \), Publ. Math. Inst. Hautes Études Sci. 111 (2010), 171-271. · Zbl 1376.11040 |

[32] | R. Munshi, The circle method and bounds for \(L\)-functions—IV: Subconvexity for twists of \(\text{GL}(3) L\)-functions, Ann. of Math. (2) 182 (2015), no. 2, 617-672. · Zbl 1333.11046 |

[33] | R. Munshi, Subconvexity for symmetric square \(L\)-functions, preprint, arXiv:1709.05615 [math.NT]. · Zbl 1443.11074 |

[34] | P. D. Nelson, Equidistribution of cusp forms in the level aspect, Duke Math. J. 160 (2011), no. 3, 467-501. · Zbl 1273.11069 |

[35] | P. D. Nelson, Evaluating modular forms on Shimura curves, Math. Comp. 84 (2015), no. 295, 2471-2503. · Zbl 1323.11021 |

[36] | P. D. Nelson, Quantum variance on quaternion algebras, I, preprint, arXiv:1601.02526 [math.NT]. |

[37] | P. D. Nelson, The spectral decomposition of \(|\theta|^2\), preprint, arXiv:1601.02529 [math.NT]. |

[38] | P. D. Nelson, Quantum variance on quaternion algebras, II, preprint, arXiv:1702.02669 [math.NT]. |

[39] | P. D. Nelson, A. Pitale, and A. Saha, Bounds for Rankin-Selberg integrals and quantum unique ergodicity for powerful levels, J. Amer. Math. Soc. 27 (2014), no. 1, 147-191. · Zbl 1322.11051 |

[40] | H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J. 113 (2002), no. 1, 133-192. · Zbl 1011.22007 |

[41] | Y. Qiu, The Whittaker period formula on metaplectic \(SL_2\), Trans. Amer. Math. Soc. 371 (2019), no. 2, 2013. |

[42] | Y. Qiu, Generalized formal degree, Int. Math. Res. Not. IMRN 2012, no. 2, 239-298. · Zbl 1238.22010 |

[43] | Y. Qiu, Periods of Saito-Kurokawa representations, Int. Math. Res. Not. IMRN 2014, no. 24, 6698-6755. · Zbl 1370.11060 |

[44] | B. Roberts and R. Schmidt, “On the number of local newforms in a metaplectic representation” in Arithmetic Geometry and Automorphic Forms, Adv. Lect. Math. (ALM) 19, Int. Press, Somerville, 2011, 505-530. · Zbl 1276.11068 |

[45] | P. Sarnak, “Arithmetic quantum chaos” in The Schur Lectures (Tel Aviv, 1992), Israel Math. Conf. Proc. 8, Bar-Ilan Univ., Ramat Gan, 1995, 183-236. · Zbl 0831.58045 |

[46] | P. Sarnak, Estimates for Rankin-Selberg \(L\)-functions and quantum unique ergodicity, J. Funct. Anal. 184 (2001), no. 2, 419-453. · Zbl 1006.11022 |

[47] | P. Sarnak, Recent progress on the quantum unique ergodicity conjecture, Bull. Amer. Math. Soc. (N.S.) 48 (2011), no. 2, 211-228. · Zbl 1234.58007 |

[48] | R. Schmidt, Some remarks on local newforms for \(\text{GL}(2)\), J. Ramanujan Math. Soc. 17 (2002), no. 2, 115-147. · Zbl 0997.11040 |

[49] | K. Soundararajan, Quantum unique ergodicity for \(\text{SL}_2(\mathbb{Z})\backslash\mathbb{H} \), Ann. of Math. (2) 172 (2010), no. 2, 1529-1538. · Zbl 1209.58019 |

[50] | D. Szpruch, Computation of the local coefficients for principal series representations of the metaplectic double cover of \(\text{SL}_2(\mathbb{F})\), J. Number Theory 129 (2009), no. 9, 2180-2213. · Zbl 1245.11065 |

[51] | D. Szpruch, On the existence of a \(p\)-adic metaplectic Tate-type \(\tilde{\gamma} \)-factor, Ramanujan J. 26 (2011), no. 1, 45-53. · Zbl 1245.11066 |

[52] | D. Szpruch, A short proof for the relation between Weil indices and \(\epsilon \)-factors, Comm. Algebra 46 (2018), no. 7, 2846-2851. · Zbl 1439.11281 |

[53] | J. B. Tunnell, On the local Langlands conjecture for \(GL(2)\), Invent. Math. 46 (1978), no. 2, 179-200. · Zbl 0385.12006 |

[54] | A. Venkatesh, Sparse equidistribution problems, period bounds and subconvexity, Ann. of Math. (2) 172 (2010), no. 2, 989-1094. · Zbl 1214.11051 |

[55] | J. L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), no. 3, 219-307. · Zbl 0724.11026 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.