Skip to main content Accessibility help
×
Home

$p$ -ADIC $L$ -FUNCTIONS FOR UNITARY GROUPS

Abstract

This paper completes the construction of $p$ -adic $L$ -functions for unitary groups. More precisely, in Harris, Li and Skinner [‘ $p$ -adic $L$ -functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math.Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such $p$ -adic $L$ -functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and $p$ -adic differential operators [Eischen, ‘A $p$ -adic Eisenstein measure for unitary groups’, J. Reine Angew. Math.699 (2015), 111–142; ‘ $p$ -adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble)62(1) (2012), 177–243], Part II of the present paper provides the calculations of local $\unicode[STIX]{x1D701}$ -integrals occurring in the Euler product (including at $p$ ). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      $p$ -ADIC $L$ -FUNCTIONS FOR UNITARY GROUPS
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      $p$ -ADIC $L$ -FUNCTIONS FOR UNITARY GROUPS
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      $p$ -ADIC $L$ -FUNCTIONS FOR UNITARY GROUPS
      Available formats
      ×

Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

Hide All
[BHR94]Blasius, D., Harris, M. and Ramakrishnan, D., ‘Coherent cohomology, limits of discrete series, and Galois conjugation’, Duke Math. J. 73(3) (1994), 647685.
[CEF+16]Caraiani, A., Eischen, E., Fintzen, J., Mantovan, E. and Varma, I., ‘p-adic q-expansion principles on unitary shimura varieties’, inDirections in Number Theory, Vol. 3 (Springer, Cham, 2016), 197243.
[Cas95]Casselman, W., ‘Introduction to the theory of admissible representations of $p$-adic reductive groups’, Unpublished manuscript, 1995, https://www.math.ubc.ca/∼cass/research/pdf/p-adic-book.pdf.
[CCO14]Chai, C.-L., Conrad, B. and Oort, F., Complex Multiplication and Lifting Problems, Mathematical Surveys and Monographs, 195 (American Mathematical Society, Providence, RI, 2014).
[Che04]Chenevier, G., ‘Familles p-adiques de formes automorphes pour GLn’, J. Reine Angew. Math. 570 (2004), 143217.
[CHT08]Clozel, L., Harris, M. and Taylor, R., ‘Automorphy for some l-adic lifts of automorphic mod l Galois representations’, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1181. With Appendix A, summarizing unpublished work of Russ Mann, and Appendix B by Marie-France Vignéras.
[Coa89]Coates, J., ‘On p-adic L-functions attached to motives over Q . II’, Bol. Soc. Brasil. Mat. (N.S.) 20(1) (1989), 101112.
[CPR89]Coates, J. and Perrin-Riou, B., ‘On p-adic L-functions attached to motives over Q’, inAlgebraic Number Theory, Advanced Studies in Pure Mathematics, 17 (Academic Press, Boston, MA, 1989), 2354.
[Del79]Deligne, P., ‘Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques’, inAutomorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proceedings of Symposia in Pure Mathematics, XXXIII (American Mathematical Society, Providence, RI, 1979), 247289.
[EFMV18]Eischen, E., Fintzen, J., Mantovan, E. and Varma, I., ‘Differential operators and families of automorphic forms on unitary groups of arbitrary signature’, Doc. Math. 23 (2018), 445495.
[Eis12]Eischen, E. E., ‘p-adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble) 62(1) (2012), 177243.
[Eis14]Eischen, E., ‘A p-adic Eisenstein measure for vector-weight automorphic forms’, Algebra Number Theory 8(10) (2014), 24332469.
[Eis15]Eischen, E. E., ‘A p-adic Eisenstein measure for unitary groups’, J. Reine Angew. Math. 699 (2015), 111142.
[Eis16]Eischen, E. E., ‘Differential operators, pullbacks, and families of automorphic forms on unitary groups’, Ann. Math. Qué. 40(1) (2016), 5582.
[EM19]Eischen, E. and Mantovan, E., ‘p-adic families of automorphic forms in the 𝜇-ordinary setting’, Amer. J. Math. (2019), Accepted for publication.
[Gar84]Garrett, P. B., ‘Pullbacks of Eisenstein series; applications’, inAutomorphic Forms of Several Variables (Katata, 1983), Progress in Mathematics, 46 (Birkhäuser Boston, Boston, MA, 1984), 114137.
[Gar08]Garrett, P., ‘Values of Archimedean zeta integrals for unitary groups’, inEisenstein Series and Applications, Progress in Mathematics, 258 (Birkhäuser, Boston, Boston, MA, 2008), 125148.
[GPSR87]Gelbart, S., Piatetski-Shapiro, I. and Rallis, S., Explicit Constructions of Automorphic L-functions, Lecture Notes in Mathematics, 1254 (Springer, Berlin, 1987).
[GW09]Goodman, R. and Wallach, N. R., Symmetry, Representations, and Invariants, Graduate Texts in Mathematics, 255 (Springer, Dordrecht, 2009).
[Har86]Harris, M., ‘Arithmetic vector bundles and automorphic forms on Shimura varieties. II’, Compositio Math. 60(3) (1986), 323378.
[Har89]Harris, M., ‘Functorial properties of toroidal compactifications of locally symmetric varieties’, Proc. Lond. Math. Soc. (3) 59(1) (1989), 122.
[Har90]Harris, M., ‘Automorphic forms of -cohomology type as coherent cohomology classes’, J. Differential Geom. 32(1) (1990), 163.
[Har97]Harris, M., ‘L-functions and periods of polarized regular motives’, J. Reine Angew. Math. 483 (1997), 75161.
[Har08]Harris, M., ‘A simple proof of rationality of Siegel–Weil Eisenstein series’, inEisenstein Series and Applications, Progress in Mathematics, 258 (Birkhäuser, Boston, MA, 2008), 149185.
[Har13a]Harris, M., ‘Beilinson–Bernstein localization over ℚ and periods of automorphic forms’, Int. Math. Res. Not. IMRN 9 (2013), 20002053.
[Har13b]Harris, M., ‘The Taylor–Wiles method for coherent cohomology’, J. Reine Angew. Math. 679 (2013), 125153.
[HKS96]Harris, M., Kudla, S. S. and Sweet, W. J., ‘Theta dichotomy for unitary groups’, J. Amer. Math. Soc. 9(4) (1996), 9411004.
[HLS05]Harris, M., Li, J.-S. and Skinner, C. M., ‘The Rallis inner product formula and p-adic L-functions’, inAutomorphic Representations, L-functions and Applications: Progress and Prospects, Ohio State Univ. Math. Res. Inst. Publ., 11 (de Gruyter, Berlin, 2005), 225255.
[HLS06]Harris, M., Li, J.-S. and Skinner, C. M., ‘p-adic L-functions for unitary Shimura varieties. I. Construction of the Eisenstein Measure’, Doc. Math. Extra Vol. (2006), 393464 (electronic).
[Hid88]Hida, H., ‘A p-adic measure attached to the zeta functions associated with two elliptic modular forms. II’, Ann. Inst. Fourier (Grenoble) 38(3) (1988), 183.
[Hid96]Hida, H., ‘On the search of genuine p-adic modular L-functions for GL(n)’, Mém. Soc. Math. Fr. (N.S.) 67 (1996), vi+110, With a correction to: ‘On $p$-adic $L$-functions of $\text{GL}(2)\times \text{GL}(2)$ over totally real fields’ Ann. Inst. Fourier (Grenoble) 41(2) (1991), 311–391.
[Hid98]Hida, H., ‘Automorphic induction and Leopoldt type conjectures for GL(n)’, Asian J. Math. 2(4) (1998), 667710. Mikio Sato: a great Japanese mathematician of the twentieth century.
[Hid02]Hida, H., ‘Control theorems of coherent sheaves on Shimura varieties of PEL type’, J. Inst. Math. Jussieu 1(1) (2002), 176.
[Hid04]Hida, H., p-adic Automorphic Forms on Shimura Varieties, Springer Monographs in Mathematics (Springer, New York, 2004).
[Jac79]Jacquet, H., ‘Principal L-functions of the linear group’, inAutomorphic Forms, Representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proceedings of Symposia in Applied Mathematics, XXXIII (American Mathematical Society, Providence, RI, 1979), 6386.
[KMSW14]Kaletha, T., Minguez, A., Shin, S. W. and White, P.-J., ‘Endoscopic classification of representations: Inner forms of unitary groups’, Preprint, 2014,arXiv:1409.3731.pdf.
[Kat78]Katz, N. M., ‘p-adic L-functions for CM fields’, Invent. Math. 49(3) (1978), 199297.
[Kot92]Kottwitz, R. E., ‘Points on some Shimura varieties over finite fields’, J. Amer. Math. Soc. 5(2) (1992), 373444.
[Lab11]Labesse, J.-P., ‘Changement de base CM et séries discrètes’, inOn the Stabilization of the Trace Formula, Stab. Trace Formula Shimura Var. Arith. Appl., 1 (Int. Press, Somerville, MA, 2011), 429470.
[Lan12]Lan, K.-W., ‘Comparison between analytic and algebraic constructions of toroidal compactifications of PEL-type Shimura varieties’, J. Reine Angew. Math. 664 (2012), 163228.
[Lan13]Lan, K.-W., Arithmetic Compactifications of PEL-type Shimura Varieties, London Mathematical Society Monographs, 36 (Princeton University Press, Princeton, NJ, 2013).
[Lan16]Lan, K.-W., ‘Higher Koecher’s principle’, Math. Res. Lett. 23(1) (2016), 163199.
[Lan17]Lan, K.-W., ‘Integral models of toroidal compactifications with projective cone decompositions’, Int. Math. Res. Not. IMRN 11 (2017), 32373280.
[Lan18]Lan, K.-W., Compactifications of PEL-type Shimura Varieties and Kuga Families with Ordinary Loci, (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2018).
[LS13]Lan, K.-W. and Suh, J., ‘Vanishing theorems for torsion automorphic sheaves on general PEL-type Shimura varieties’, Adv. Math. 242 (2013), 228286.
[Li92]Li, J.-S., ‘Nonvanishing theorems for the cohomology of certain arithmetic quotients’, J. Reine Angew. Math. 428 (1992), 177217.
[Liu19a]Liu, Z., ‘The doubling Archimedean zeta integrals for p-adic interpolation’, Math. Res. Lett. (2019), Accepted for publication. Preprint available at arXiv:1904.07121.
[Liu19b]Liu, Z., ‘p-adic L-functions for ordinary families on symplectic groups’, J. Inst. Math. Jussieu (2019), 161.
[MVW87]Mœglin, C., Vignéras, M.-F. and Waldspurger, J.-L., Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics, 1291 (Springer, Berlin, 1987).
[Mok15]Mok, C. P., ‘Endoscopic classification of representations of quasi-split unitary groups’, Mem. Amer. Math. Soc. 235(1108) (2015), vi+248.
[Moo04]Moonen, B., ‘Serre–Tate theory for moduli spaces of PEL type’, Ann. Sci. Éc. Norm. Supér. (4) 37(2) (2004), 223269.
[Pan94]Panchishkin, A. A., ‘Motives over totally real fields and p-adic L-functions’, Ann. Inst. Fourier (Grenoble) 44(4) (1994), 9891023.
[Pil11]Pilloni, V., ‘Prolongement analytique sur les variétés de Siegel’, Duke Math. J. 157(1) (2011), 167222.
[Shi97]Shimura, G., Euler Products and Eisenstein Series, CBMS Regional Conference Series in Mathematics, 93 (Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997).
[Shi00]Shimura, G., Arithmeticity in the Theory of Automorphic Forms, Mathematical Surveys and Monographs, 82 (American Mathematical Society, Providence, RI, 2000).
[SU02]Skinner, C. and Urban, E., ‘Sur les déformations p-adiques des formes de Saito–Kurokawa’, C. R. Math. Acad. Sci. Paris 335(7) (2002), 581586.
[SU14]Skinner, C. and Urban, E., ‘The Iwasawa main conjectures for GL2’, Invent. Math. 195(1) (2014), 1277.
[Wan15]Wan, X., ‘Families of nearly ordinary Eisenstein series on unitary groups’, Algebra Number Theory 9(9) (2015), 19552054. With an appendix by Kai-Wen Lan.
[Wed99]Wedhorn, T., ‘Ordinariness in good reductions of Shimura varieties of PEL-type’, Ann. Sci. Éc. Norm. Supér. (4) 32(5) (1999), 575618.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

$p$ -ADIC $L$ -FUNCTIONS FOR UNITARY GROUPS

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.