Non-abelian p -adic L -functions and Eisenstein series of unitary groups – The CM method

Thanasis Bouganis[1]

  • [1] Department of Mathematical Sciences Durham University Science Laboratories, South Rd. Durham DH1 3LE (U.K.)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 2, page 793-891
  • ISSN: 0373-0956

Abstract

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In this work we prove various cases of the so-called “torsion congruences” between abelian p -adic L -functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of n variables and we obtain more explicit results in the special cases of n = 1 and n = 2 . In both of these cases we also explain their implications for some particular “motives”, as for example elliptic curves with complex multiplication. Finally we also discuss a new kind of congruences, which we call “average torsion congruences”

How to cite

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Bouganis, Thanasis. "Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups – The CM method." Annales de l’institut Fourier 64.2 (2014): 793-891. <http://eudml.org/doc/275632>.

@article{Bouganis2014,
abstract = {In this work we prove various cases of the so-called “torsion congruences” between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results in the special cases of $n=1$ and $n=2$. In both of these cases we also explain their implications for some particular “motives”, as for example elliptic curves with complex multiplication. Finally we also discuss a new kind of congruences, which we call “average torsion congruences”},
affiliation = {Department of Mathematical Sciences Durham University Science Laboratories, South Rd. Durham DH1 3LE (U.K.)},
author = {Bouganis, Thanasis},
journal = {Annales de l’institut Fourier},
keywords = {($p$-adic) $L$-functions; Eisenstein Series; Unitary Groups; Congruences; ($p$-adic) $L$-functions; Eisenstein series; unitary groups; congruences},
language = {eng},
number = {2},
pages = {793-891},
publisher = {Association des Annales de l’institut Fourier},
title = {Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups – The CM method},
url = {http://eudml.org/doc/275632},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Bouganis, Thanasis
TI - Non-abelian $p$-adic $L$-functions and Eisenstein series of unitary groups – The CM method
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 2
SP - 793
EP - 891
AB - In this work we prove various cases of the so-called “torsion congruences” between abelian $p$-adic $L$-functions that are related to automorphic representations of definite unitary groups. These congruences play a central role in the non-commutative Iwasawa theory as it became clear in the works of Kakde, Ritter and Weiss on the non-abelian Main Conjecture for the Tate motive. We tackle these congruences for a general definite unitary group of $n$ variables and we obtain more explicit results in the special cases of $n=1$ and $n=2$. In both of these cases we also explain their implications for some particular “motives”, as for example elliptic curves with complex multiplication. Finally we also discuss a new kind of congruences, which we call “average torsion congruences”
LA - eng
KW - ($p$-adic) $L$-functions; Eisenstein Series; Unitary Groups; Congruences; ($p$-adic) $L$-functions; Eisenstein series; unitary groups; congruences
UR - http://eudml.org/doc/275632
ER -

References

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  1. Thanasis Bouganis, Non abelian p -adic L -functions and Eisenstein series of unitary groups II; the CM-method Zbl06387293
  2. Thanasis Bouganis, Non abelian p -adic L -functions and Eisenstein series of unitary groups; the Constant Term Method Zbl06387293
  3. Thanasis Bouganis, Special values of L -functions and false Tate curve extensions, J. Lond. Math. Soc. (2) 82 (2010), 596-620 Zbl1210.11115MR2739058
  4. Thanasis Bouganis, Non-abelian congruences between special values of L -functions of elliptic curves: the CM case, Int. J. Number Theory 7 (2011), 1883-1934 Zbl1279.11107MR2854221
  5. Thanasis Bouganis, F. Nuccio, Kongruenzen zwischen abelschen pseudo-Maßen und die Shintani Zerlegung 
  6. Thanasis Bouganis, Otmar Venjakob, On the non-commutative main conjecture for elliptic curves with complex multiplication, Asian J. Math. 14 (2010), 385-416 Zbl1214.11122MR2755723
  7. John Coates, Motivic p -adic L -functions, -functions and arithmetic (Durham, 1989) 153 (1991), 141-172, Cambridge Univ. Press, Cambridge Zbl0725.11029MR1110392
  8. John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha, Otmar Venjakob, The GL 2 main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Études Sci. (2005), 163-208 Zbl1108.11081MR2217048
  9. Daniel Delbourgo, Thomas Ward, Non-abelian congruences between L -values of elliptic curves, Ann. Inst. Fourier (Grenoble) 58 (2008), 1023-1055 Zbl1165.11077MR2427518
  10. Daniel Delbourgo, Thomas Ward, The growth of CM periods over false Tate extensions, Experiment. Math. 19 (2010), 195-210 Zbl1200.11081MR2676748
  11. T. Dokchitser, V. Dokchitser, Computations in non-commutative Iwasawa theory, Proc. Lond. Math. Soc. (3) 94 (2007), 211-272 Zbl1206.11083MR2294995
  12. Ellen E. Eischen, A p -adic Eisenstein Measure for Unitary Groups, (2011) Zbl1322.11040
  13. Ellen E. Eischen, p -adic differential operators on automorphic forms on unitary groups, Ann. Inst. Fourier (Grenoble) 62 (2012), 177-243 Zbl1257.11054MR2986270
  14. Ellen E. Eischen, M. Harris, J.-S. Li, C. Skinner, p -adic L -functions for unitary Shimura varieties, II 
  15. Takako Fukaya, Kazuya Kato, A formulation of conjectures on p -adic zeta functions in noncommutative Iwasawa theory, Proceedings of the St. Petersburg Mathematical Society. Vol. XII 219 (2006), 1-85, Amer. Math. Soc., Providence, RI Zbl1238.11105MR2276851
  16. Paul Garrett, Values of Archimedean zeta integrals for unitary groups, Eisenstein series and applications 258 (2008), 125-148, Birkhäuser Boston, Boston, MA Zbl1225.11065MR2402682
  17. Stephen Gelbart, Ilya Piatetski-Shapiro, Stephen Rallis, Explicit constructions of automorphic L -functions, 1254 (1987), Springer-Verlag, Berlin Zbl0612.10022MR892097
  18. Roger Godement, Hervé Jacquet, Zeta functions of simple algebras, (1972), Springer-Verlag, Berlin-New York Zbl0244.12011MR342495
  19. Takashi Hara, Iwasawa theory of totally real fields for certain non-commutative p -extensions, J. Number Theory 130 (2010), 1068-1097 Zbl1196.11148MR2600423
  20. Michael Harris, Unitary groups and Base Change Zbl1071.22025
  21. Michael Harris, L -functions of 2 × 2 unitary groups and factorization of periods of Hilbert modular forms, J. Amer. Math. Soc. 6 (1993), 637-719 Zbl0779.11023MR1186960
  22. Michael Harris, L -functions and periods of polarized regular motives, J. Reine Angew. Math. 483 (1997), 75-161 Zbl0859.11032MR1431843
  23. Michael Harris, A simple proof of rationality of Siegel-Weil Eisenstein series, Eisenstein series and applications 258 (2008), 149-185, Birkhäuser Boston, Boston, MA Zbl1225.11069MR2402683
  24. Michael Harris, Jian-Shu Li, Christopher M. Skinner, The Rallis inner product formula and p -adic L -functions, Automorphic representations, -functions and applications: progress and prospects 11 (2005), 225-255, de Gruyter, Berlin Zbl1103.11017MR2192825
  25. Michael Harris, Jian-Shu Li, Christopher M. Skinner, p -adic L -functions for unitary Shimura varieties. I. Construction of the Eisenstein measure, Doc. Math. (2006), 393-464 (electronic) Zbl1143.11019MR2290594
  26. H. Hida, J. Tilouine, Anti-cyclotomic Katz p -adic L -functions and congruence modules, Ann. Sci. École Norm. Sup. (4) 26 (1993), 189-259 Zbl0778.11061MR1209708
  27. Haruzo Hida, p -adic automorphic forms on Shimura varieties, (2004), Springer-Verlag, New York Zbl1055.11032MR2055355
  28. Haruzo Hida, Serre’s conjecture and base change for GL ( 2 ) , Pure Appl. Math. Q. 5 (2009), 81-125 Zbl1252.11089MR2520456
  29. Ming-Lun Hsieh, Ordinary p -adic Eisenstein series and p -adic L -functions for unitary groups, Ann. Inst. Fourier (Grenoble) 61 (2011), 987-1059 Zbl1271.11051MR2918724
  30. Ming-Lun Hsieh, Eisenstein congruence on unitary groups and Iwasawa main conjectures for CM fields, J. Amer. Math. Soc. 27 (2014), 753-862 Zbl1317.11108MR3194494
  31. Mahesh Kakde, Proof of the main conjecture of noncommutative Iwasawa theory for totally real number fields in certain cases, J. Algebraic Geom. 20 (2011), 631-683 Zbl1242.11084MR2819672
  32. Mahesh Kakde, From the classical to the noncommutative Iwasawa theory (for totally real number fields), Non-abelian fundamental groups and Iwasawa theory 393 (2012), 107-131, Cambridge Univ. Press, Cambridge Zbl1278.11100MR2905531
  33. K. Kato, Iwasawa theory of totally real fields for Galois extensions of Heisenberg type 
  34. Nicholas M. Katz, p -adic interpolation of real analytic Eisenstein series, Ann. of Math. (2) 104 (1976), 459-571 Zbl0354.14007MR506271
  35. Nicholas M. Katz, p -adic L -functions for CM fields, Invent. Math. 49 (1978), 199-297 Zbl0417.12003MR513095
  36. Jian-Shu Li, Nonvanishing theorems for the cohomology of certain arithmetic quotients, J. Reine Angew. Math. 428 (1992), 177-217 Zbl0749.11032MR1166512
  37. Jürgen Ritter, Alfred Weiss, Congruences between abelian pseudomeasures, Math. Res. Lett. 15 (2008), 715-725 Zbl1158.11047MR2424908
  38. Jürgen Ritter, Alfred Weiss, Congruences between abelian pseudomeasures, II, (2010) Zbl1158.11047
  39. Ehud de Shalit, On monomial relations between p -adic periods, J. Reine Angew. Math. 374 (1987), 193-207 Zbl0597.14038MR876224
  40. Goro Shimura, Euler products and Eisenstein series, 93 (1997), Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI Zbl0906.11020MR1450866
  41. Goro Shimura, Abelian varieties with complex multiplication and modular functions, 46 (1998), Princeton University Press, Princeton, NJ Zbl0908.11023MR1492449
  42. Goro Shimura, Arithmeticity in the theory of automorphic forms, 82 (2000), American Mathematical Society, Providence, RI Zbl0967.11001MR1780262
  43. Christopher Skinner, Eric Urban, The Iwasawa main conjectures for GL 2 , Invent. Math. 195 (2014), 1-277 Zbl1301.11074MR3148103

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