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IWASAWA THEORY FOR THE SYMMETRIC SQUARE OF A CM MODULAR FORM AT INERT PRIMES

Published online by Cambridge University Press:  12 December 2011

ANTONIO LEI
ANTONIO LEI*
Affiliation:
School of Mathematical Sciences, Monash University, Clayton, VIC 3800, Australia e-mail: antonio.lei@monash.edu
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Abstract

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Let f be a modular form with complex multiplication (CM) and p an odd prime that is inert in the CM field. We construct two p-adic L-functions for the symmetric square of f, one of which has the same interpolating properties as the one constructed by Delbourgo and Dabrowski (A. Dabrowski and D. Delbourgo, S-adic L-functions attached to the symmetric square of a newform, Proc. Lond. Math. Soc. 74(3) (1997), 559–611), whereas the other one has a similar interpolating properties but corresponds to a different eigenvalue of the Frobenius. The symmetry between these two p-adic L-functions allows us to define the plus and minus p-adic L-functions à la Pollack (R. Pollack, on the p-adic L-function of a modular form at a supersingular prime, Duke Math. J. 118(3) (2003), 523–558). We also define the plus and minus p-Selmer groups analogous to the ones defined by Kobayashi (S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152(1) (2003), 1–36). We explain how to relate these two sets of objects via a main conjecture.

Keywords

11R2311F80
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 54 , Issue 2 , May 2012 , pp. 241 - 259
Copyright
Copyright © Glasgow Mathematical Journal Trust 2011

References

REFERENCES

1.Coates, J. and Schmidt, C.-G., Iwasawa theory for the symmetric square of an elliptic curve, J. Reine Angew. Math. 375/376 (1987), 104–156.Google Scholar
2.Dabrowski, A., Bounded p-adic L-functions of motives at supersingular primes, C. R. Math. Acad. Sci. Paris 349 (7–8) (2011), 365–368.CrossRefGoogle Scholar
3.Dabrowski, A. and Delbourgo, D., S-adic L-functions attached to the symmetric square of a new form, Proc. Lond. Math. Soc. 74 (3) (1997), 559–611.CrossRefGoogle Scholar
4.Deligne, P., Formes modulaires et représentations l-adiques, Séminaire Bourbaki 21, Exp. No. 355 (1968–1969), 139–172.Google Scholar
5.Kato, K., p-adic Hodge theory and values of zeta functions of modular forms, Astérisque (295) (2004), ix, 117–290, Cohomologies p-adiques et applications arithmétiques. III.Google Scholar
6.Lei, A., Iwasawa theory for modular forms at supersingular primes, Compos. Math. 147 (3) (2011), 803–838.CrossRefGoogle Scholar
7.Lei, A., Loeffler, D. and Zerbes, S. L., Wach modules and Iwasawa theory for modular forms, Asian J. Math. 14 (4) (2010), 475–528.CrossRefGoogle Scholar
8.Mazur, B. and Wiles, A., Class fields of Abelian extensions of ℚ, Invent. Math. 76 (2) (1984), 179–330.CrossRefGoogle Scholar
9.Perrin-Riou, B., Fonctions L p-adiques d'une courbe elliptique et points rationnels, Ann. Inst. Fourier (Grenoble) 43 (4) (1993), 945–995.CrossRefGoogle Scholar
10.Perrin-Riou, B., Zéros triviaux des fonctions L p-adiques, un cas particulier, Compos. Math. 114 (1) (1998), 37–76.Google Scholar
11.Pollack, R., On the p-adic L-function of a modular form at a supersingular prime, Duke Math. J. 118 (3) (2003), 523–558.CrossRefGoogle Scholar
12.Pollack, R. and Rubin, K., The main conjecture for CM elliptic curves at supersingular primes, Ann. Math. (2) 159 (1) (2004), 447–464.CrossRefGoogle Scholar
13.Rubin, K., The “main conjecture” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1) (1991), 25–68.CrossRefGoogle Scholar