Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes
Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 673-707.

Soit f une forme primitive cuspidale à multiplication complexe (CM) et soit p un nombre premier impair tel que f soit non-ordinaire en p. Nous construisons des fonctions L p-adiques admissibles pour les puissances symétriques de f, vérifiant ainsi des cas particuliers de conjectures de Dabrowski et Panchishkin. À l’aide d’un résultat récent de Benois, nous prouvons la conjecture des zéros triviaux dans notre contexte. De plus, nous construisons des fonctions L p-adiques plus/moins “mixtes” et obtenons une décomposition des fonctions L p-adiques admissibles analogue à celle de Pollack. Du côté arithmétique, nous définissons les groupes de Selmer plus/moins mixtes correspondants et nous énonçons une Conjecture Principale.

Let f be a cuspidal newform with complex multiplication (CM) and let p be an odd prime at which f is non-ordinary. We construct admissible p-adic L-functions for the symmetric powers of f, thus verifying conjectures of Dabrowski and Panchishkin in this special case. We combine this with recent work of Benois to prove the trivial zero conjecture in this setting. We also construct “mixed” plus and minus p-adic L-functions and prove an analogue of Pollack’s decomposition of the admissible p-adic L-functions. On the arithmetic side, we define corresponding mixed plus and minus Selmer groups and formulate the Main Conjecture of Iwasawa Theory.

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DOI : https://doi.org/10.5802/jtnb.885
Classification : 11R23,  11F80,  11F67
@article{JTNB_2014__26_3_673_0,
     author = {Robert Harron and Antonio Lei},
     title = {Iwasawa theory for symmetric powers of {CM} modular forms at non-ordinary primes},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {673--707},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {26},
     number = {3},
     year = {2014},
     doi = {10.5802/jtnb.885},
     mrnumber = {3320498},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.885/}
}
Robert Harron; Antonio Lei. Iwasawa theory for symmetric powers of CM modular forms at non-ordinary primes. Journal de Théorie des Nombres de Bordeaux, Tome 26 (2014) no. 3, pp. 673-707. doi : 10.5802/jtnb.885. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.885/

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