Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture

Krzysztof Klosin[1]

  • [1] Cornell University Department of Mathematics 310 Malott Hall Ithaca, NY 14853-4201 (USA)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 1, page 81-166
  • ISSN: 0373-0956

Abstract

top
Let k be a positive integer divisible by 4, p > k a prime, f an elliptic cuspidal eigenform (ordinary at p ) of weight k - 1 , level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives ad 0 M ( - 1 ) and ad 0 M ( 2 ) , where M is the motif attached to f . More precisely, we prove that under certain conditions the p -adic valuation of the algebraic part of the symmetric square L -function of f evaluated at k provides a lower bound for the p -adic valuation of the order of the Pontryagin dual of the Selmer group for the adjoint of the p -adic Galois representation attached to f restricted to the Gaussian field and twisted by the inverse of the cyclotomic character. Our method uses an idea of Ribet, in that we introduce an intermediate step and produce congruences between CAP and non-CAP modular forms on the unitary group U ( 2 , 2 ) .

How to cite

top

Klosin, Krzysztof. "Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture." Annales de l’institut Fourier 59.1 (2009): 81-166. <http://eudml.org/doc/10399>.

@article{Klosin2009,
abstract = {Let $k$ be a positive integer divisible by 4, $p&gt;k$ a prime, $f$ an elliptic cuspidal eigenform (ordinary at $p$) of weight $k-1$, level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives $\text\{ad\}^0 M(-1)$ and $\text\{ad\}^0 M(2)$, where $M$ is the motif attached to $f$. More precisely, we prove that under certain conditions the $p$-adic valuation of the algebraic part of the symmetric square $L$-function of $f$ evaluated at $k$ provides a lower bound for the $p$-adic valuation of the order of the Pontryagin dual of the Selmer group for the adjoint of the $p$-adic Galois representation attached to $f$ restricted to the Gaussian field and twisted by the inverse of the cyclotomic character. Our method uses an idea of Ribet, in that we introduce an intermediate step and produce congruences between CAP and non-CAP modular forms on the unitary group $\text\{U\}(2,2)$.},
affiliation = {Cornell University Department of Mathematics 310 Malott Hall Ithaca, NY 14853-4201 (USA)},
author = {Klosin, Krzysztof},
journal = {Annales de l’institut Fourier},
keywords = {Automorphic forms on unitary groups; congruences; Selmer groups; Bloch-Kato conjecture; automorphic forms on unitary groups},
language = {eng},
number = {1},
pages = {81-166},
publisher = {Association des Annales de l’institut Fourier},
title = {Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture},
url = {http://eudml.org/doc/10399},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Klosin, Krzysztof
TI - Congruences among modular forms on U(2,2) and the Bloch-Kato conjecture
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 1
SP - 81
EP - 166
AB - Let $k$ be a positive integer divisible by 4, $p&gt;k$ a prime, $f$ an elliptic cuspidal eigenform (ordinary at $p$) of weight $k-1$, level 4 and non-trivial character. In this paper we provide evidence for the Bloch-Kato conjecture for the motives $\text{ad}^0 M(-1)$ and $\text{ad}^0 M(2)$, where $M$ is the motif attached to $f$. More precisely, we prove that under certain conditions the $p$-adic valuation of the algebraic part of the symmetric square $L$-function of $f$ evaluated at $k$ provides a lower bound for the $p$-adic valuation of the order of the Pontryagin dual of the Selmer group for the adjoint of the $p$-adic Galois representation attached to $f$ restricted to the Gaussian field and twisted by the inverse of the cyclotomic character. Our method uses an idea of Ribet, in that we introduce an intermediate step and produce congruences between CAP and non-CAP modular forms on the unitary group $\text{U}(2,2)$.
LA - eng
KW - Automorphic forms on unitary groups; congruences; Selmer groups; Bloch-Kato conjecture; automorphic forms on unitary groups
UR - http://eudml.org/doc/10399
ER -

References

top
  1. J. Bellaïche, G. Chenevier, p -adic families of Galois representations and higher rank Selmer groups Zbl1192.11035
  2. J. Bellaïche, G. Chenevier, Formes non tempérées pour U ( 3 ) et conjectures de Bloch-Kato, Ann. Sci. École Norm. Sup. (4) 37 (2004), 611-662 Zbl1201.11051MR2097894
  3. T. Berger, An Eisenstein ideal for imaginary quadratic fields and the Bloch-Kato conjecture for Hecke characters, (2007) 
  4. D. Blasius, J. D. Rogawski, Zeta functions of Shimura varieties, Motives (Seattle, WA, 1991) 55 (1994), 525-571, Amer. Math. Soc., Providence, RI Zbl0827.11033MR1265563
  5. S. Bloch, K. Kato, L -functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. I 86 (1990), 333-400, Birkhäuser Boston, Boston, MA Zbl0768.14001MR1086888
  6. H. Braun, Hermitian modular functions, Ann. of Math. (2) 50 (1949), 827-855 Zbl0038.23803MR32699
  7. H. Braun, Hermitian modular functions. II. Genus invariants of Hermitian forms, Ann. of Math. (2) 51 (1950), 92-104 Zbl0038.23901MR32700
  8. H. Braun, Hermitian modular functions. III, Ann. of Math. (2) 53 (1951), 143-160 Zbl0041.41603MR39005
  9. J. Brown, Saito-Kurokawa lifts and applications to the Bloch-Kato conjecture, Compos. Math. 143 (2007), 290-322 Zbl1172.11015MR2309988
  10. D. Bump, Automorphic forms and representations, 55 (1997), Cambridge University Press, Cambridge Zbl0868.11022MR1431508
  11. H. Darmon, F. Diamond, R. Taylor, Fermat’s last theorem, Elliptic curves, modular forms & Fermat’s last theorem (Hong Kong, 1993) (1997), 2-140, Internat. Press, Cambridge, MA Zbl0877.11035MR1605752
  12. P. Deligne, Valeurs de fonctions L et périodes d’intégrales, Automorphic forms, representations and -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 (1979), 313-346, Amer. Math. Soc., Providence, R.I. Zbl0449.10022
  13. F. Diamond, J. Im, Modular forms and modular curves, Seminar on Fermat’s Last Theorem (Toronto, ON, 1993–1994) 17 (1995), 39-133, Amer. Math. Soc., Providence, RI Zbl0853.11032MR1357209
  14. Fred Diamond, Matthias Flach, Li Guo, The Tamagawa number conjecture of adjoint motives of modular forms, Ann. Sci. École Norm. Sup. (4) 37 (2004), 663-727 Zbl1121.11045MR2103471
  15. N. Dummigan, Period ratios of modular forms, Math. Ann. 318 (2000), 621-636 Zbl1041.11037MR1800772
  16. N. Dummigan, Symmetric square L -functions and Shafarevich-Tate groups, Experiment. Math. 10 (2001), 383-400 Zbl1039.11029MR1917426
  17. N. Dummigan, Symmetric square L-functions and Shafarevich-Tate groups, II, (2008) Zbl1039.11029
  18. D. Eisenbud, Commutative algebra, 150 (1995), Springer-Verlag, New York Zbl0819.13001MR1322960
  19. J.-M. Fontaine, Sur certains types de représentations p -adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti-Tate, Ann. of Math. (2) 115 (1982), 529-577 Zbl0544.14016MR657238
  20. J.-M. Fontaine, B. Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L , Motives (Seattle, WA, 1991) 55 (1994), 599-706, Amer. Math. Soc., Providence, RI MR1265546
  21. E. Freitag, Siegelsche Modulfunktionen, 254 (1983), Springer-Verlag, Berlin Zbl0498.10016MR871067
  22. S. Gelbart, Automorphic forms on adèle groups, (1975), Princeton University Press, Princeton, N.J. Zbl0329.10018MR379375
  23. V. A. Gritsenko, The Maass space for SU ( 2 , 2 ) . The Hecke ring, and zeta functions, Trudy Mat. Inst. Steklov. 183 (1990), 68-78, 223–225 Zbl0764.11026MR1092016
  24. V. A. Gritsenko, Parabolic extensions of the Hecke ring of the general linear group. II, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 183 (1990), 56-76, 165, 167 Zbl0748.11026MR1075006
  25. H. Hida, Galois representations and the theory of p -adic Hecke algebras, Sūgaku 39 (1987), 124-139 Zbl0641.10025MR904860
  26. H. Hida, Elementary theory of L -functions and Eisenstein series, 26 (1993), Cambridge University Press, Cambridge Zbl0942.11024MR1216135
  27. H. Hida, Modular forms and Galois cohomology, 69 (2000), Cambridge University Press, Cambridge Zbl0952.11014MR1779182
  28. H. Hida, p -adic automorphic forms on Shimura varieties, (2004), Springer-Verlag, New York Zbl1055.11032MR2055355
  29. T. Hina, T. Sugano, On the local Hecke series of some classical groups over 𝔭 -adic fields, J. Math. Soc. Japan 35 (1983), 133-152 Zbl0496.14014MR679080
  30. T. Ikeda, On the lifting of hermitian modular forms, (2007) Zbl1155.11025MR2457521
  31. H. Iwaniec, Topics in classical automorphic forms, 17 (1997), American Mathematical Society, Providence, RI Zbl0905.11023MR1474964
  32. H. Kim, Automorphic L -functions, Lectures on automorphic -functions 20 (2004), 97-201, Amer. Math. Soc., Providence, RI MR2071507
  33. G. Kings, The Bloch-Kato conjecture on special values of L -functions. A survey of known results, J. Théor. Nombres Bordeaux 15 (2003), 179-198 Zbl1050.11063MR2019010
  34. K. Klosin, Congruences among automorphic forms on the unitary group U ( 2 , 2 ) , (2006), University of Michigan, Ann Arbor 
  35. K. Klosin, Adelic Maass spaces on U ( 2 , 2 ) , (2007) 
  36. H. Kojima, An arithmetic of Hermitian modular forms of degree two, Invent. Math. 69 (1982), 217-227 Zbl0502.10011MR674402
  37. A. Krieg, The Maaß spaces on the Hermitian half-space of degree 2 , Math. Ann. 289 (1991), 663-681 Zbl0713.11033MR1103042
  38. R. P. Langlands, On the functional equations satisfied by Eisenstein series, (1976), Springer-Verlag, Berlin Zbl0332.10018MR579181
  39. B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. (1977), 33-186 (1978) Zbl0394.14008MR488287
  40. B. Mazur, An introduction to the deformation theory of Galois representations, Modular forms and Fermat’s last theorem (Boston, MA, 1995) (1997), 243-311, Springer, New York Zbl0901.11015MR1638481
  41. B. Mazur, A. Wiles, Class fields of abelian extensions of Q , Invent. Math. 76 (1984), 179-330 Zbl0545.12005MR742853
  42. T. Miyake, Modular forms, (1989), Springer-Verlag, Berlin Zbl0701.11014MR1021004
  43. C. Mœglin, J.-L. Waldspurger, Le spectre résiduel de GL ( n ) , Ann. Sci. École Norm. Sup. (4) 22 (1989), 605-674 Zbl0696.10023MR1026752
  44. S. Raghavan, J. Sengupta, A Dirichlet series for Hermitian modular forms of degree 2 , Acta Arith. 58 (1991), 181-201 Zbl0685.10022MR1121080
  45. K. A. Ribet, A modular construction of unramified p -extensions of Q ( μ p ) , Invent. Math. 34 (1976), 151-162 Zbl0338.12003MR419403
  46. K. Rubin, Euler systems, 147 (2000), Princeton University Press, Princeton, NJ Zbl0977.11001MR1749177
  47. C.-G. Schmidt, p -adic measures attached to automorphic representations of GL ( 3 ) , Invent. Math. 92 (1988), 597-631 Zbl0656.10023MR939477
  48. J.-P. Serre, Groupes de Lie l -adiques attachés aux courbes elliptiques, Les Tendances Géom. en Algébre et Théorie des Nombres (1966), 239-256, Éditions du Centre National de la Recherche Scientifique, Paris Zbl0148.41502MR218366
  49. G. Shimura, Confluent hypergeometric functions on tube domains, Math. Ann. 260 (1982), 269-302 Zbl0502.10013MR669297
  50. G. Shimura, Euler products and Eisenstein series, 93 (1997), Published for the Conference Board of the Mathematical Sciences, Washington, DC Zbl0906.11020MR1450866
  51. G. Shimura, Arithmeticity in the theory of automorphic forms, 82 (2000), American Mathematical Society, Providence, RI Zbl0967.11001MR1780262
  52. C. M. Skinner, Selmer groups, (2004) 
  53. J. Sturm, Special values of zeta functions, and Eisenstein series of half integral weight, Amer. J. Math. 102 (1980), 219-240 Zbl0433.10015MR564472
  54. R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553-572 Zbl0823.11030MR1333036
  55. E. Urban, Selmer groups and the Eisenstein-Klingen ideal, Duke Math. J. 106 (2001), 485-525 Zbl1061.11027MR1813234
  56. V. Vatsal, Canonical periods and congruence formulae, Duke Math. J. 98 (1999), 397-419 Zbl0979.11027MR1695203
  57. V. Vatsal, Special values of anticyclotomic L -functions, Duke Math. J. 116 (2003), 219-261 Zbl1065.11048MR1953292
  58. L. C. Washington, Galois cohomology, Modular forms and Fermat’s last theorem (Boston, MA, 1995) (1997), 101-120, Springer, New York Zbl0928.12003MR1638477
  59. L. C. Washington, Introduction to cyclotomic fields, 83 (1997), Springer-Verlag, New York Zbl0966.11047MR1421575
  60. A. Wiles, The Iwasawa conjecture for totally real fields, Ann. of Math. (2) 131 (1990), 493-540 Zbl0719.11071MR1053488
  61. A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443-551 Zbl0823.11029MR1333035

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.