Hida families, p -adic heights, and derivatives

Trevor Arnold[1]

  • [1] McMaster University Department of Mathematics & Statistics 1280 Main Street West Hamilton, ON L8S 4K1 (Canada)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 6, page 2275-2299
  • ISSN: 0373-0956

Abstract

top
This paper concerns the arithmetic of certain p -adic families of elliptic modular forms. We relate, using a formula of Rubin, some Iwasawa-theoretic aspects of the three items in the title of this paper. In particular, we examine several conjectures, three of which assert the non-triviality of an Euler system, a p -adic regulator, and the derivative of a p -adic L -function. We investigate sufficient conditions for the first conjecture to hold and show that, under additional assumptions, the first conjecture implies the equivalence of the last two.

How to cite

top

Arnold, Trevor. "Hida families, $p$-adic heights, and derivatives." Annales de l’institut Fourier 60.6 (2010): 2275-2299. <http://eudml.org/doc/116333>.

@article{Arnold2010,
abstract = {This paper concerns the arithmetic of certain $p$-adic families of elliptic modular forms. We relate, using a formula of Rubin, some Iwasawa-theoretic aspects of the three items in the title of this paper. In particular, we examine several conjectures, three of which assert the non-triviality of an Euler system, a $p$-adic regulator, and the derivative of a $p$-adic $L$-function. We investigate sufficient conditions for the first conjecture to hold and show that, under additional assumptions, the first conjecture implies the equivalence of the last two.},
affiliation = {McMaster University Department of Mathematics & Statistics 1280 Main Street West Hamilton, ON L8S 4K1 (Canada)},
author = {Arnold, Trevor},
journal = {Annales de l’institut Fourier},
keywords = {Iwasawa theory; Hida family; $p$-adic height; $p$-adic $L$-function; -adic height; -adic -function; Euler system},
language = {eng},
number = {6},
pages = {2275-2299},
publisher = {Association des Annales de l’institut Fourier},
title = {Hida families, $p$-adic heights, and derivatives},
url = {http://eudml.org/doc/116333},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Arnold, Trevor
TI - Hida families, $p$-adic heights, and derivatives
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 6
SP - 2275
EP - 2299
AB - This paper concerns the arithmetic of certain $p$-adic families of elliptic modular forms. We relate, using a formula of Rubin, some Iwasawa-theoretic aspects of the three items in the title of this paper. In particular, we examine several conjectures, three of which assert the non-triviality of an Euler system, a $p$-adic regulator, and the derivative of a $p$-adic $L$-function. We investigate sufficient conditions for the first conjecture to hold and show that, under additional assumptions, the first conjecture implies the equivalence of the last two.
LA - eng
KW - Iwasawa theory; Hida family; $p$-adic height; $p$-adic $L$-function; -adic height; -adic -function; Euler system
UR - http://eudml.org/doc/116333
ER -

References

top
  1. P. Deligne, Formes modulaires et représentations ℓ -adiques, Séminaire Bourbaki 1968/1969, exp. 355 179 (1971), 139-172, Springer, Berlin Zbl0206.49901
  2. R. Greenberg, Iwasawa theory for motives, -functions and arithmetic (Durham, 1989) 153 (1991), 211-233, Cambridge Univ. Press Zbl0727.11043MR1110394
  3. R. Greenberg, Elliptic curves and p -adic deformations, Elliptic curves and related topics 4 (1994), 101-110, AMS Zbl0821.14021MR1260957
  4. R. Greenberg, On the structure of certain Galois cohomology groups, Doc. Math. (2006) Zbl1138.11048MR2290593
  5. H. Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm. Sup. (4) 19 (1986), 231-273 Zbl0607.10022MR868300
  6. K. Kato, p -adic Hodge theory and values of zeta functions of modular forms, Astérisque (2004), 117-290 Zbl1142.11336MR2104361
  7. K. Kitagawa, On standard p -adic L -functions of families of elliptic cusp forms, -adic monodromy and the Birch and Swinnerton-Dyer conjecture 165 (1994), 81-110, AMS Zbl0841.11028MR1279604
  8. B. Mazur, J. Tilouine, Représentations galoisiennes, différentielles de Kähler et ‘conjectures principales’, Inst. Hautes Études Sci. Publ. Math. (1990), 65-103 Zbl0744.11053MR1079644
  9. J. Milne, Arithmetic duality theorems, 1 (1986), Academic Press Zbl0613.14019MR881804
  10. J. Nekovář, Selmer complexes, Astérisque (2006) Zbl1211.11120MR2333680
  11. T. Ochiai, A generalization of the Coleman map for Hida deformations, Amer. J. Math. 125 (2003), 849-892 Zbl1057.11048MR1993743
  12. T. Ochiai, Euler system for Galois deformation, Ann. Inst. Fourier (Grenoble) 55 (2005), 113-146 Zbl1112.11031MR2141691
  13. T. Ochiai, On the two-variable Iwasawa main conjecture, Compositio Math. 142 (2006), 1157-1200 Zbl1112.11051MR2264660
  14. B. Perrin-Riou, Théorie d’Iwasawa et hauteurs p -adiques, Invent. Math. 109 (1992), 137-185 Zbl0781.14013MR1168369
  15. A. Plater, Height pairings in families of deformations, J. reine angew. Math. 486 (1997), 97-127 Zbl0872.14018MR1450752
  16. K. Rubin, Abelian varieties, p -adic heights and derivatives, Algebra and number theory (1994), 247-266, Walter de Gruyter and Co. Zbl0829.11034MR1285370
  17. K. Rubin, Euler Systems, (2000), Princeton University Press Zbl0977.11001MR1749177
  18. A. Wiles, On λ -adic representations associated to modular forms, Invent. Math. 94 (1988), 529-573 Zbl0664.10013MR969243

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.