Abstract
We prove nontrivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Pólya-Vinogradov range). We then derive applications to the second moment of cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on $\mathrm{GL}_3$. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially $\ell$-adic cohomology and the Riemann Hypothesis.
-
[Bl]
V. Blomer, "Applications of the Kuznetsov formula on $\mathrm{GL}(3)$," Invent. Math., vol. 194, iss. 3, pp. 673-729, 2013.@ARTICLE{Bl, mrkey = {3127065},
number = {3},
issn = {0020-9910},
author = {Blomer, Valentin},
mrclass = {11F72 (11F66)},
doi = {10.1007/s00222-013-0454-3},
journal = {Invent. Math.},
zblnumber = {1292.11064},
volume = {194},
mrnumber = {3127065},
fjournal = {Inventiones Mathematicae},
mrreviewer = {Lei Yang},
title = {Applications of the {K}uznetsov formula on {$\mathrm{GL}(3)$}},
year = {2013},
pages = {673--729},
} -
[BFKMM] V. Blomer, É. Fouvry, E. Kowalski, P. Michel, and D. Milićević, "On moments of twisted $L$-functions," Amer. J. Math., vol. 139, iss. 3, pp. 707-768, 2017.
@article{BFKMM,
author = {Blomer, Valentin and Fouvry, {É}. and Kowalski, E. and Michel, Ph. and Mili{ć}evi{ć},
D.},
pages = {707--768},
year = {2017},
doi = {10.1353/ajm.2017/0019},
journal={Amer. J. Math.},
volume={139},
number={3},
title = {On moments of twisted {$L$}-functions},
} -
[2for6] V. Blomer, É. Fouvry, E. Kowalski, P. Michel, D. Milićević, and W. Sawin, On the non-vanishing of twisted $L$-functions.
@MISC{2for6,
author = {Blomer, Valentin and Fouvry, {É}. and Kowalski, E. and Michel, Ph. and Mili{ć}evi{ć},
D. and Sawin, W.},
note = {in preparation},
title = {On the non-vanishing of twisted {$L$}-functions},
} -
[BlMi] V. Blomer and D. Milićević, "The second moment of twisted modular $L$-functions," Geom. Funct. Anal., vol. 25, iss. 2, pp. 453-516, 2015.
@ARTICLE{BlMi, mrkey = {3334233},
number = {2},
issn = {1016-443X},
author = {Blomer, Valentin and Milićević, Djordje},
mrclass = {11F66 (11F72 11L07)},
doi = {10.1007/s00039-015-0318-7},
journal = {Geom. Funct. Anal.},
zblnumber = {06438343},
volume = {25},
mrnumber = {3334233},
fjournal = {Geometric and Functional Analysis},
mrreviewer = {Ravi Raghunathan},
title = {The second moment of twisted modular {$L$}-functions},
year = {2015},
pages = {453--516},
} -
[BFG] D. Bump, S. Friedberg, and D. Goldfeld, "Poincaré series and Kloosterman sums for ${ SL}(3,{\bf Z})$," Acta Arith., vol. 50, iss. 1, pp. 31-89, 1988.
@ARTICLE{BFG, mrkey = {0945275},
number = {1},
issn = {0065-1036},
author = {Bump, Daniel and Friedberg, Solomon and Goldfeld, Dorian},
mrclass = {11F37 (11L05)},
journal = {Acta Arith.},
zblnumber = {0647.10020},
volume = {50},
mrnumber = {0945275},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
mrreviewer = {Shigeaki Tsuyumine},
title = {Poincaré series and {K}loosterman sums for {${\rm SL}(3,{\bf Z})$}},
year = {1988},
pages = {31--89},
} -
[sga4h] P. Deligne, Cohomologie Étale, New York: Springer-Verlag, 1977, vol. 569.
@BOOK{sga4h, mrkey = {0463174},
author = {Deligne, Pierre},
mrclass = {14F20},
series = {Lecture Notes in Math.},
address = {New York},
publisher = {Springer-Verlag},
volume = {569},
mrnumber = {0463174},
note = {Séminaire de Géométrie Algébrique du Bois-Marie SGA 4${\frac{1}{2}}$; avec la collaboration de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier},
mrreviewer = {J. S. Milne},
title = {Cohomologie Étale},
year = {1977},
pages = {iv+312pp},
doi = {10.1007/BFb0091516},
zblnumber = {0345.00010},
} -
[WeilII] P. Deligne, "La conjecture de Weil. II," Inst. Hautes Études Sci. Publ. Math., iss. 52, pp. 137-252, 1980.
@ARTICLE{WeilII, mrkey = {0601520},
number = {52},
issn = {0073-8301},
author = {Deligne, Pierre},
mrclass = {14G13 (10H10)},
url = {http://www.numdam.org/item?id=PMIHES_1980__52__137_0},
journal = {Inst. Hautes Études Sci. Publ. Math.},
zblnumber = {0456.14014},
mrnumber = {0601520},
fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
mrreviewer = {Spencer J. Bloch},
title = {La conjecture de {W}eil. {II}},
year = {1980},
pages = {137--252},
doi = {10.1007/BF02684780},
} -
[SGA7] P. Deligne and N. Katz, Groupes de Monodromie en Géométrie Algébrique. II, New York: Springer-Verlag, 1973, vol. 340.
@BOOK{SGA7, mrkey = {0354657},
author = {Deligne, Pierre and Katz, N.},
mrclass = {14-06},
series = {Lecture Notes in Math.},
address = {New York},
publisher = {Springer-Verlag},
volume = {340},
mrnumber = {0354657},
note = {Séminaire de Géométrie Algébrique du Bois-Marie 1967--1969 (SGA 7 II)},
title = {Groupes de Monodromie en Géométrie Algébrique. {II}},
year = {1973},
pages = {x+438},
doi = {10.1007/BFb0060505},
} -
[DesIw] J. -M. Deshouillers and H. Iwaniec, "Kloosterman sums and Fourier coefficients of cusp forms," Invent. Math., vol. 70, iss. 2, pp. 219-288, 1982/83.
@ARTICLE{DesIw, mrkey = {0684172},
number = {2},
issn = {0020-9910},
author = {Deshouillers, J.-M. and Iwaniec, H.},
mrclass = {10D12 (10G10)},
doi = {10.1007/BF01390728},
journal = {Invent. Math.},
zblnumber = {0502.10021},
volume = {70},
mrnumber = {0684172},
fjournal = {Inventiones Mathematicae},
mrreviewer = {Audrey A. Terras},
title = {Kloosterman sums and {F}ourier coefficients of cusp forms},
year = {1982/83},
pages = {219--288},
} -
[FGKM] &. Fouvry, S. Ganguly, E. Kowalski, and P. Michel, "Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions," Comment. Math. Helv., vol. 89, iss. 4, pp. 979-1014, 2014.
@ARTICLE{FGKM, mrkey = {3284303},
number = {4},
issn = {0010-2571},
author = {Fouvry, Étienne and Ganguly, Satadal and Kowalski, Emmanuel and Michel, Philippe},
mrclass = {11F11 (11F30 11T23 60F05)},
doi = {10.4171/CMH/342},
journal = {Comment. Math. Helv.},
zblnumber = {1306.11079},
volume = {89},
mrnumber = {3284303},
fjournal = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society},
title = {Gaussian distribution for the divisor function and {H}ecke eigenvalues in arithmetic progressions},
year = {2014},
pages = {979--1014},
} -
[FKM2] &. Fouvry, E. Kowalski, and P. Michel, "Algebraic trace functions over the primes," Duke Math. J., vol. 163, iss. 9, pp. 1683-1736, 2014.
@ARTICLE{FKM2, mrkey = {3217765},
number = {9},
issn = {0012-7094},
author = {Fouvry, Étienne and Kowalski, Emmanuel and Michel, Philippe},
mrclass = {11N05 (11N32 11N35)},
doi = {10.1215/00127094-2690587},
journal = {Duke Math. J.},
zblnumber = {1318.11103},
volume = {163},
mrnumber = {3217765},
fjournal = {Duke Mathematical Journal},
mrreviewer = {Ben Joseph Green},
title = {Algebraic trace functions over the primes},
year = {2014},
pages = {1683--1736},
} -
[FKM1] &. Fouvry, E. Kowalski, and P. Michel, "Algebraic twists of modular forms and Hecke orbits," Geom. Funct. Anal., vol. 25, iss. 2, pp. 580-657, 2015.
@ARTICLE{FKM1, mrkey = {3334236},
number = {2},
issn = {1016-443X},
author = {Fouvry, Étienne and Kowalski, Emmanuel and Michel, Philippe},
mrclass = {11F11 (11F32 11F37 11L05 11T23)},
doi = {10.1007/s00039-015-0310-2},
journal = {Geom. Funct. Anal.},
zblnumber = {1344.11036},
volume = {25},
mrnumber = {3334236},
fjournal = {Geometric and Functional Analysis},
mrreviewer = {Guangshi LÂ${}^1\!/\!_4$},
title = {Algebraic twists of modular forms and {H}ecke orbits},
year = {2015},
pages = {580--657},
} -
[FKM3] &. Fouvry, E. Kowalski, and P. Michel, "On the exponent of distribution of the ternary divisor function," Mathematika, vol. 61, iss. 1, pp. 121-144, 2015.
@ARTICLE{FKM3, mrkey = {3333965},
number = {1},
issn = {0025-5793},
author = {Fouvry, Étienne and Kowalski, Emmanuel and Michel, Philippe},
mrclass = {11N37 (11L07 11N25 11T23)},
doi = {10.1112/S0025579314000096},
journal = {Mathematika},
zblnumber = {1317.11080},
volume = {61},
mrnumber = {3333965},
fjournal = {Mathematika. A Journal of Pure and Applied Mathematics},
mrreviewer = {D. R. Heath-Brown},
title = {On the exponent of distribution of the ternary divisor function},
year = {2015},
pages = {121--144},
} -
[FKMSP] &. Fouvry, E. Kowalski, and P. Michel, "A study in sums of products," Philos. Trans. Roy. Soc. A, vol. 373, iss. 2040, p. 20140309, 2015.
@ARTICLE{FKMSP, mrkey = {3338119},
number = {2040},
issn = {1364-503X},
author = {Fouvry, Étienne and Kowalski, Emmanuel and Michel, Philippe},
mrclass = {11L05},
doi = {10.1098/rsta.2014.0309},
journal = {Philos. Trans. Roy. Soc. A},
zblnumber = {06674711},
volume = {373},
mrnumber = {3338119},
fjournal = {Philosophical Transactions of the Royal Society A. Mathematical, Physical and Engineering Sciences},
mrreviewer = {S. W. Graham},
title = {A study in sums of products},
year = {2015},
pages = {20140309, 26},
} -
[FoMi] &. Fouvry and P. Michel, "Sur certaines sommes d’exponentielles sur les nombres premiers," Ann. Sci. École Norm. Sup., vol. 31, iss. 1, pp. 93-130, 1998.
@ARTICLE{FoMi, mrkey = {1604298},
number = {1},
issn = {0012-9593},
author = {Fouvry, Étienne and Michel, Philippe},
mrclass = {11L07 (11G25)},
doi = {10.1016/S0012-9593(98)80019-0},
journal = {Ann. Sci. École Norm. Sup.},
volume = {31},
mrnumber = {1604298},
fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
mrreviewer = {John B. Friedlander},
title = {Sur certaines sommes d'exponentielles sur les nombres premiers},
year = {1998},
pages = {93--130},
zblnumber = {0915.11045},
} -
[FrIw] J. B. Friedlander and H. Iwaniec, "Incomplete Kloosterman sums and a divisor problem," Ann. of Math., vol. 121, iss. 2, pp. 319-350, 1985.
@ARTICLE{FrIw, mrkey = {0786351},
number = {2},
issn = {0003-486X},
author = {Friedlander, John B. and Iwaniec, Henryk},
mrclass = {11N25 (11L40 11N69)},
doi = {10.2307/1971175},
journal = {Ann. of Math.},
zblnumber = {0572.10029},
volume = {121},
mrnumber = {0786351},
note = {with an appendix by {B. J. B}irch and {E.} Bombieri},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {D. R. Heath-Brown},
title = {Incomplete {K}loosterman sums and a divisor problem},
year = {1985},
pages = {319--350},
} -
[Fu] L. Fu, "Calculation of $\ell$-adic local Fourier transformations," Manuscripta Math., vol. 133, iss. 3-4, pp. 409-464, 2010.
@ARTICLE{Fu, mrkey = {2729262},
number = {3-4},
issn = {0025-2611},
author = {Fu, Lei},
mrclass = {14F20 (11S15 11S20 11T23)},
doi = {10.1007/s00229-010-0377-x},
journal = {Manuscripta Math.},
zblnumber = {1206.14035},
volume = {133},
mrnumber = {2729262},
fjournal = {Manuscripta Mathematica},
mrreviewer = {Adolfo QuirÂ${}^3$s},
title = {Calculation of {$\ell$}-adic local {F}ourier transformations},
year = {2010},
pages = {409--464},
} -
[fu-book] L. Fu, Etale Cohomology Theory, Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2011, vol. 13.
@BOOK{fu-book, mrkey = {2791606},
author = {Fu, Lei},
mrclass = {14F20 (11G25 14F05 14G10 14G15 18F10)},
series = {Nankai Tracts in Math.},
isbn = {978-981-4307-72-7; 981-4307-72-6},
address = {Hackensack, NJ},
publisher = {World Scientific Publishing Co. Pte. Ltd.},
doi = {10.1142/9789814307734},
zblnumber = {1228.14001},
volume = {13},
mrnumber = {2791606},
mrreviewer = {Adolfo QuirÂ${}^3$s},
title = {Etale Cohomology Theory},
year = {2011},
pages = {x+611},
} -
[Fu2] L. Fu, "$\ell$-adic GKZ hypergeometric sheaves and exponential sums," Adv. Math., vol. 298, pp. 51-88, 2016.
@ARTICLE{Fu2, mrkey = {3505737},
issn = {0001-8708},
author = {Fu, Lei},
mrclass = {14F20 (11T23 14G15 33C70)},
doi = {10.1016/j.aim.2016.04.021},
journal = {Adv. Math.},
zblnumber = {06591692},
volume = {298},
mrnumber = {3505737},
fjournal = {Advances in Mathematics},
mrreviewer = {Edwin LeÂ${}^3$n-Cardenal},
title = {{$\ell$}-adic {GKZ} hypergeometric sheaves and exponential sums},
year = {2016},
pages = {51--88},
} -
[GKR] P. Gao, R. Khan, and G. Ricotta, "The second moment of Dirichlet twists of Hecke $L$-functions," Acta Arith., vol. 140, iss. 1, pp. 57-65, 2009.
@ARTICLE{GKR, mrkey = {2557853},
number = {1},
issn = {0065-1036},
author = {Gao, Peng and Khan, Rizwanur and Ricotta, Guillaume},
mrclass = {11F66 (11M41)},
doi = {10.4064/aa140-1-4},
journal = {Acta Arith.},
zblnumber = {1242.11035},
volume = {140},
mrnumber = {2557853},
fjournal = {Acta Arithmetica},
mrreviewer = {Arnaud Chadozeau},
title = {The second moment of {D}irichlet twists of {H}ecke {$L$}-functions},
year = {2009},
pages = {57--65},
} -
[sga1] A. Grothendieck and M. Raynaud, Revêtements Étales et Groupe Fondamental, New York: Springer-Verlag, 1971, vol. 224.
@BOOK{sga1, mrkey = {0354651},
author = {Grothendieck, A. and Raynaud, M.},
mrclass = {14-06 (14E20)},
series = {Lecture Notes in Math.},
address = {New York},
publisher = {Springer-Verlag},
volume = {224},
mrnumber = {0354651},
note = {Séminaire de Géométrie Algébrique du Bois Marie 1960--1961 (SGA~1); dirigé par Alexandre Grothendieck.; augmenté de deux exposés de M. Raynaud},
title = {Revêtements Étales et Groupe Fondamental},
year = {1971},
pages = {xxii+447},
doi = {10.1007/BFb0058656},
} -
[hartshorne] R. Hartshorne, Algebraic Geometry, New York: Springer-Verlag, 1977, vol. 52.
@BOOK{hartshorne, mrkey = {0463157},
author = {Hartshorne, Robin},
mrclass = {14-01},
series = {Graduate Texts in Math.},
address = {New York},
isbn = {0-387-90244-9},
publisher = {Springer-Verlag},
volume = {52},
mrnumber = {0463157},
mrreviewer = {Robert Speiser},
title = {Algebraic Geometry},
year = {1977},
pages = {xvi+496},
zblnumber = {0367.14001},
doi = {10.1007/978-1-4757-3849-0},
} -
[HBfourth] D. R. Heath-Brown, "The fourth power mean of Dirichlet’s $L$-functions," Analysis, vol. 1, iss. 1, pp. 25-32, 1981.
@ARTICLE{HBfourth, mrkey = {0623640},
number = {1},
issn = {0174-4747},
author = {Heath-Brown, D. R.},
mrclass = {10H08},
doi = {10.1524/anly.1981.1.1.25},
journal = {Analysis},
zblnumber = {0479.10027},
volume = {1},
mrnumber = {0623640},
fjournal = {Analysis. International Journal of Analysis and its Application},
mrreviewer = {Matti Jutila},
title = {The fourth power mean of {D}irichlet's {$L$}-functions},
year = {1981},
pages = {25--32},
} -
[HB] D. R. Heath-Brown, "The divisor function $d_3(n)$ in arithmetic progressions," Acta Arith., vol. 47, iss. 1, pp. 29-56, 1986.
@ARTICLE{HB, mrkey = {0866901},
number = {1},
issn = {0065-1036},
author = {Heath-Brown, D. R.},
mrclass = {11N25 (11N37)},
journal = {Acta Arith.},
zblnumber = {0549.10034},
volume = {47},
mrnumber = {0866901},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
mrreviewer = {A. J. Hildebrand},
title = {The divisor function {$d_3(n)$} in arithmetic progressions},
year = {1986},
pages = {29--56},
} -
[HL] J. Hoffstein and M. Lee, Second Moments and simultaneous non-vanishing of $\mathrm{GL}(2)$ automorphic $L$-series.
@MISC{HL,
author = {Hoffstein, J. and Lee, M.},
arxiv = {1308.5980},
title = {Second Moments and simultaneous non-vanishing of {$\mathrm{GL}(2)$} automorphic {$L$}-series},
} -
[IwKo] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53.
@BOOK{IwKo, mrkey = {2061214},
author = {Iwaniec, Henryk and Kowalski, Emmanuel},
mrclass = {11-02 (11Fxx 11Lxx 11Mxx 11Nxx)},
series = {Amer. Math. Soc. Colloq. Publ.},
isbn = {0-8218-3633-1},
address = {Providence, RI},
publisher = {Amer. Math. Soc.},
doi = {10.1090/coll/053},
zblnumber = {1059.11001},
volume = {53},
mrnumber = {2061214},
mrreviewer = {K. Soundararajan},
title = {Analytic Number Theory},
year = {2004},
pages = {xii+615},
} -
[Sommes] N. M. Katz, Sommes Sxponentielles, Paris: Soc. Math. France, 1980, vol. 79.
@BOOK{Sommes, mrkey = {0617009},
author = {Katz, Nicholas M.},
mrclass = {10G05},
series = {Astérisque},
address = {Paris},
publisher = {Soc. Math. France},
volume = {79},
mrnumber = {0617009},
mrreviewer = {J. S. Milne},
title = {Sommes Sxponentielles},
year = {1980},
pages = {209},
zblnumber = {0469.12007},
} -
[GKM] N. M. Katz, Gauss sums, Kloosterman Sums, and Monodromy Groups, Princeton, NJ: Princeton Univ. Press, 1988, vol. 116.
@BOOK{GKM, mrkey = {0955052},
author = {Katz, Nicholas M.},
mrclass = {11G40 (11L05 11T24 14G10 14G15)},
series = {Ann. of Math. Stud.},
isbn = {0-691-08432-7; 0-691-08433-5},
address = {Princeton, NJ},
publisher = {Princeton Univ. Press},
doi = {10.1515/9781400882120},
zblnumber = {0675.14004},
volume = {116},
mrnumber = {0955052},
mrreviewer = {P. Bayer},
title = {Gauss sums, {K}loosterman Sums, and Monodromy Groups},
year = {1988},
pages = {x+246},
} -
[ESDE] N. M. Katz, Exponential Sums and Differential Equations, Princeton, NJ: Princeton Univ. Press, 1990, vol. 124.
@BOOK{ESDE, mrkey = {1081536},
author = {Katz, Nicholas M.},
mrclass = {14D10 (11L03 11T23 14G15)},
series = {Ann. of Math. Stud.},
isbn = {0-691-08598-6; 0-691-08599-4},
address = {Princeton, NJ},
publisher = {Princeton Univ. Press},
doi = {10.1515/9781400882434},
zblnumber = {0731.14008},
volume = {124},
mrnumber = {1081536},
mrreviewer = {Hernando Enrique Sierra-Morales},
title = {Exponential Sums and Differential Equations},
year = {1990},
pages = {xii+430},
} -
[rigid] N. M. Katz, Rigid Local Systems, Princeton, NJ: Princeton Univ. Press, 1996, vol. 139.
@BOOK{rigid, mrkey = {1366651},
author = {Katz, Nicholas M.},
mrclass = {14F20 (14F05)},
series = {Ann. of Math. Stud.},
isbn = {0-691-01118-4},
address = {Princeton, NJ},
publisher = {Princeton Univ. Press},
doi = {10.1515/9781400882595},
zblnumber = {0864.14013},
volume = {139},
mrnumber = {1366651},
mrreviewer = {Abdellah Mokrane},
title = {Rigid Local Systems},
year = {1996},
pages = {viii+223},
} -
[KatzBetti] N. M. Katz, "Sums of Betti numbers in arbitrary characteristic," Finite Fields Appl., vol. 7, iss. 1, pp. 29-44, 2001.
@ARTICLE{KatzBetti, mrkey = {1803934},
number = {1},
issn = {1071-5797},
author = {Katz, Nicholas M.},
mrclass = {14F25 (14F20 14R05)},
doi = {10.1006/ffta.2000.0303},
journal = {Finite Fields Appl.},
zblnumber = {1068.14501},
volume = {7},
mrnumber = {1803934},
fjournal = {Finite Fields and their Applications},
mrreviewer = {Jean-Yves ‰tesse},
title = {Sums of {B}etti numbers in arbitrary characteristic},
year = {2001},
pages = {29--44},
} -
[katz-mellin] N. M. Katz, Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms, Princeton, NJ: Princeton Univ. Press, 2012, vol. 180.
@BOOK{katz-mellin, mrkey = {2850079},
author = {Katz, Nicholas M.},
mrclass = {11T24 (11G25 14F05 14F20)},
series = {Ann. Math. Stud.},
isbn = {978-0-691-15331-5},
address = {Princeton, NJ},
publisher = {Princeton Univ. Press},
zblnumber = {1261.11084},
volume = {180},
mrnumber = {2850079},
mrreviewer = {Florent Jouve},
title = {Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms},
year = {2012},
pages = {viii+203},
} -
[katz-laumon] N. M. Katz and G. Laumon, "Transformation de Fourier et majoration de sommes exponentielles," Inst. Hautes Études Sci. Publ. Math., iss. 62, pp. 361-418, 1985.
@ARTICLE{katz-laumon, mrkey = {0823177},
issn = {0073-8301},
author = {Katz, Nicholas M. and Laumon, Gérard},
mrclass = {14G15 (11G25 11T21 14F12)},
url = {http://www.numdam.org/item?id=PMIHES_1985__62__145_0},
journal = {Inst. Hautes Études Sci. Publ. Math.},
zblnumber = {0603.14015},
number = {62},
mrnumber = {0823177},
fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
mrreviewer = {G. Horrocks},
title = {Transformation de {F}ourier et majoration de sommes exponentielles},
year = {1985},
pages = {361--418},
} -
[k-repr] E. Kowalski, An Introduction to the Representation Theory of Groups, Amer. Math. Soc., 2014, vol. 155.
@BOOK{k-repr, mrkey = {3236265},
author = {Kowalski, Emmanuel},
mrclass = {20C15 (22-01 22A25 43A10)},
series = {Grad. Stud. in Math.},
isbn = {978-1-4704-0966-1},
publisher = {Amer. Math. Soc.},
zblnumber = {1320.20008},
volume = {155},
mrnumber = {3236265},
mrreviewer = {Pierre Clare},
title = {An Introduction to the Representation Theory of Groups},
year = {2014},
pages = {vi+432},
} -
[KMV] E. Kowalski, P. Michel, and J. VanderKam, "Rankin-Selberg $L$-functions in the level aspect," Duke Math. J., vol. 114, iss. 1, pp. 123-191, 2002.
@ARTICLE{KMV, mrkey = {1915038},
number = {1},
issn = {0012-7094},
author = {Kowalski, Emmanuel and Michel, P. and VanderKam, J.},
mrclass = {11F66 (11G40)},
doi = {10.1215/S0012-7094-02-11416-1},
journal = {Duke Math. J.},
zblnumber = {1035.11018},
volume = {114},
mrnumber = {1915038},
fjournal = {Duke Mathematical Journal},
mrreviewer = {K. Soundararajan},
title = {Rankin-{S}elberg {$L$}-functions in the level aspect},
year = {2002},
pages = {123--191},
} -
[LaumonSMF] G. Laumon, "Semi-continuité du conducteur de Swan (d’après P. Deligne)," in The Euler-Poincaré Characteristic (French), Paris: Soc. Math. France, 1981, vol. 83, pp. 173-219.
@INCOLLECTION{LaumonSMF, mrkey = {0629128},
author = {Laumon, G.},
mrclass = {14F20 (14H99)},
series = {Astérisque},
address = {Paris},
publisher = {Soc. Math. France},
zblnumber = {0504.14013},
volume = {83},
mrnumber = {0629128},
booktitle = {The {E}uler-{P}oincaré Characteristic ({F}rench)},
mrreviewer = {Gerd Faltings},
title = {Semi-continuité du conducteur de {S}wan (d'après {P}. {D}eligne)},
pages = {173--219},
year = {1981},
} -
[laumon87] G. Laumon, "Transformation de Fourier, constantes d’équations fonctionnelles et conjecture de Weil," Inst. Hautes Études Sci. Publ. Math., iss. 65, pp. 131-210, 1987.
@ARTICLE{laumon87, mrkey = {0908218},
number = {65},
issn = {0073-8301},
author = {Laumon, G.},
mrclass = {14G15 (11G40 11M41 11R39 11S37 14G13)},
url = {http://www.numdam.org/item?id=PMIHES_1987__65__131_0},
journal = {Inst. Hautes Études Sci. Publ. Math.},
zblnumber = {0641.14009},
mrnumber = {0908218},
fjournal = {Institut des Hautes Études Scientifiques. Publications Mathématiques},
mrreviewer = {Jean-Luc Brylinski},
title = {Transformation de {F}ourier, constantes d'équations fonctionnelles et conjecture de {W}eil},
year = {1987},
pages = {131--210},
} -
[laumonhomog] G. Laumon, "Transformation de Fourier homogène," Bull. Soc. Math. France, vol. 131, iss. 4, pp. 527-551, 2003.
@ARTICLE{laumonhomog, mrkey = {2044494},
number = {4},
issn = {0037-9484},
author = {Laumon, Gérard},
mrclass = {11G09 (11G45 11R39 11R58 14D20 14F05)},
journal = {Bull. Soc. Math. France},
zblnumber = {1088.11044},
volume = {131},
mrnumber = {2044494},
fjournal = {Bulletin de la Société Mathématique de France},
mrreviewer = {Alexey A. Panchishkin},
title = {Transformation de {F}ourier homogène},
year = {2003},
pages = {527--551},
} -
[LRS] W. Luo, Z. Rudnick, and P. Sarnak, "On Selberg’s eigenvalue conjecture," Geom. Funct. Anal., vol. 5, iss. 2, pp. 387-401, 1995.
@ARTICLE{LRS, mrkey = {1334872},
number = {2},
issn = {1016-443X},
author = {Luo, W. and Rudnick, Z. and Sarnak, P.},
mrclass = {11F72 (11F70)},
doi = {10.1007/BF01895672},
journal = {Geom. Funct. Anal.},
zblnumber = {0844.11038},
volume = {5},
mrnumber = {1334872},
fjournal = {Geometric and Functional Analysis},
mrreviewer = {Jens Bolte},
title = {On {S}elberg's eigenvalue conjecture},
year = {1995},
pages = {387--401},
} -
[milne] J. S. Milne, Étale Cohomology, Princeton, N.J.: Princeton Univ. Press, 1980, vol. 33.
@BOOK{milne, mrkey = {0559531},
author = {Milne, James S.},
mrclass = {14-02 (14F20 18F99)},
series = {Princeton Math. Ser.},
isbn = {0-691-08238-3},
address = {Princeton, N.J.},
publisher = {Princeton Univ. Press},
volume = {33},
mrnumber = {0559531},
mrreviewer = {G. Horrocks},
title = {Étale Cohomology},
year = {1980},
pages = {xiii+323},
zblnumber = {0433.14012},
} -
[Mu1] R. Munshi, "Shifted convolution of divisor function $d_3$ and Ramanujan $\tau$ function," in The Legacy of Srinivasa Ramanujan, Mysore: Ramanujan Math. Soc., 2013, vol. 20, pp. 251-260.
@INCOLLECTION{Mu1, mrkey = {3221314},
author = {Munshi, Ritabrata},
mrclass = {11F66 (11M41)},
series = {Ramanujan Math. Soc. Lect. Notes Ser.},
address = {Mysore},
publisher = {Ramanujan Math. Soc.},
zblnumber = {06389649},
volume = {20},
mrnumber = {3221314},
booktitle = {The Legacy of {S}rinivasa {R}amanujan},
mrreviewer = {A. Perelli},
title = {Shifted convolution of divisor function {$d_3$} and {R}amanujan {$\tau$} function},
pages = {251--260},
year = {2013},
} -
[Mu2] R. Munshi, "Shifted convolution sums for $\mathrm{GL}(3)\times\mathrm{GL}(2)$," Duke Math. J., vol. 162, iss. 13, pp. 2345-2362, 2013.
@ARTICLE{Mu2, mrkey = {3127803},
number = {13},
issn = {0012-7094},
author = {Munshi, Ritabrata},
mrclass = {11F66 (11M41)},
doi = {10.1215/00127094-2371416},
journal = {Duke Math. J.},
zblnumber = {1330.11033},
volume = {162},
mrnumber = {3127803},
fjournal = {Duke Mathematical Journal},
mrreviewer = {M. Ram Murty},
title = {Shifted convolution sums for {$\mathrm{GL}(3)\times\mathrm{GL}(2)$}},
year = {2013},
pages = {2345--2362},
} -
[Nunes] R. M. Nunes, Squarefree integers in large arithmetic progressions.
@MISC{Nunes,
author = {Nunes, R. M.},
arxiv = {1602.00311},
title = {Squarefree integers in large arithmetic progressions},
} -
[orgogozo] F. Orgogozo, "Altérations et groupe fondamental premier à $p$," Bull. Soc. Math. France, vol. 131, iss. 1, pp. 123-147, 2003.
@ARTICLE{orgogozo, mrkey = {1975807},
number = {1},
issn = {0037-9484},
author = {Orgogozo, Fabrice},
mrclass = {14F35},
journal = {Bull. Soc. Math. France},
zblnumber = {1083.14506},
volume = {131},
mrnumber = {1975807},
fjournal = {Bulletin de la Société Mathématique de France},
mrreviewer = {Daniel Matei},
title = {Altérations et groupe fondamental premier à {$p$}},
year = {2003},
pages = {123--147},
} -
[Sound] K. Soundararajan, "The fourth moment of Dirichlet $L$-functions," in Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2007, vol. 7, pp. 239-246.
@INCOLLECTION{Sound, mrkey = {2362204},
author = {Soundararajan, K.},
mrclass = {11M06},
series = {Clay Math. Proc.},
address = {Providence, RI},
publisher = {Amer. Math. Soc.},
zblnumber = {1208.11102},
volume = {7},
mrnumber = {2362204},
booktitle = {Analytic Number Theory},
mrreviewer = {Arnaud Chadozeau},
title = {The fourth moment of {D}irichlet {$L$}-functions},
pages = {239--246},
year = {2007},
} -
[Topa] B. Topacogullari, "The shifted convolution of divisor functions," Q. J. Math., vol. 67, iss. 2, pp. 331-363, 2016.
@ARTICLE{Topa,
author = {Topacogullari, Berke},
TITLE = {The shifted convolution of divisor functions},
JOURNAL = {Q. J. Math.},
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YEAR = {2016},
NUMBER = {2},
PAGES = {331--363},
ISSN = {0033-5606},
MRCLASS = {11N37 (11F30)},
MRNUMBER = {3509996},
MRREVIEWER = {Emre Alkan},
DOI = {10.1093/qmath/haw010},
ZBLNUMBER = {1356.11070},
} -
[Young] M. P. Young, "The fourth moment of Dirichlet $L$-functions," Ann. of Math., vol. 173, iss. 1, pp. 1-50, 2011.
@ARTICLE{Young, mrkey = {2753598},
number = {1},
issn = {0003-486X},
author = {Young, Matthew P.},
mrclass = {11M06 (11F72 11M50)},
doi = {10.4007/annals.2011.173.1.1},
journal = {Ann. of Math.},
zblnumber = {1296.11112},
volume = {173},
mrnumber = {2753598},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {Arnaud Chadozeau},
title = {The fourth moment of {D}irichlet {$L$}-functions},
year = {2011},
pages = {1--50},
}
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