On classical weight one forms in Hida families
Journal de Théorie des Nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 669-690.

We give precise estimates for the number of classical weight one specializations of a non-CM family of ordinary cuspidal eigenforms. We also provide examples to show how uniqueness fails with respect to membership of weight one forms in families.

Nous effectuons une estimation précise du nombre de spécialisations classiques en poids un d’une famille non-CM de formes modulaires propres ordinaires cuspidales. Nous donnons aussi des exemples où plusieurs familles se spécialisent sur la même forme de poids un.

Received:
Published online:
DOI: 10.5802/jtnb.816
Classification: 11F80,  11F33,  11R23
@article{JTNB_2012__24_3_669_0,
     author = {Mladen Dimitrov and Eknath Ghate},
     title = {On classical weight one forms in {Hida} families},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {669--690},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {24},
     number = {3},
     year = {2012},
     doi = {10.5802/jtnb.816},
     zbl = {1271.11060},
     mrnumber = {3010634},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.816/}
}
TY  - JOUR
TI  - On classical weight one forms in Hida families
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2012
DA  - 2012///
SP  - 669
EP  - 690
VL  - 24
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.816/
UR  - https://zbmath.org/?q=an%3A1271.11060
UR  - https://www.ams.org/mathscinet-getitem?mr=3010634
UR  - https://doi.org/10.5802/jtnb.816
DO  - 10.5802/jtnb.816
LA  - en
ID  - JTNB_2012__24_3_669_0
ER  - 
%0 Journal Article
%T On classical weight one forms in Hida families
%J Journal de Théorie des Nombres de Bordeaux
%D 2012
%P 669-690
%V 24
%N 3
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.816
%R 10.5802/jtnb.816
%G en
%F JTNB_2012__24_3_669_0
Mladen Dimitrov; Eknath Ghate. On classical weight one forms in Hida families. Journal de Théorie des Nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 669-690. doi : 10.5802/jtnb.816. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.816/

[B03] K. Buzzard, Analytic continuation of overconvergent eigenforms. J. Amer. Math. Soc., 16 (2003), 29–55. | MR: 1937198 | Zbl: 1076.11029

[BD12] J. Bellaïche and M. Dimitrov, On the Eigencurve at classical weight one points. Preprint (2012).

[BT99] K. Buzzard and R. Taylor, Companion forms and weight 1 forms. Ann. of Math., 149 (1999), 905–919. | MR: 1709306 | Zbl: 0965.11019

[CV03] S. Cho and V. Vatsal, Deformations of induced Galois representations. J. Reine Angew. Math., 556 (2003), 79–98. | MR: 1971139 | Zbl: 1041.11039

[EPW06] M. Emerton, R. Pollack and T. Weston, Variation of Iwasawa invariants in Hida families. Invent. Math., 163 (2006), 523–580. | MR: 2207234 | Zbl: 1093.11065

[F02] A. Fischman, On the image of Λ-adic Galois representations. Ann. Inst. Fourier, Grenoble, 52 (2002), no. 2, 351–378. | Numdam | MR: 1906479 | Zbl: 1020.11037

[GK12] E. Ghate and N. Kumar, Control theorems for ordinary 2-adic families of modular forms. In preparation.

[GV04] E. Ghate and V. Vatsal, On the local behaviour of ordinary Λ-adic representations. Ann. Inst. Fourier, Grenoble, 54 (2004), no. 7, 2143–2162. | Numdam | MR: 2139691 | Zbl: 1131.11341

[H85] H. Hida, Galois representations into GL 2 (Z p [[X]]) attached to ordinary cusp forms. Invent. Math., 85 (1986), 545–613. | MR: 848685 | Zbl: 0612.10021

[H86] Iwasawa modules attached to congruences of cusp forms. Ann. Sci. Ecole Norm. Sup. (4), 19 (1986), 231–273. | MR: 868300

[S77] J-P. Serre, Modular forms of weight one and Galois representations. Proc. Sympos. Univ. Durham, Durham (1975), Academic Press, London, 1977, 193–268. | MR: 450201 | Zbl: 0366.10022

[W88] A. Wiles, On ordinary λ-adic representations associated to modular forms. Invent. Math., 94 (1988), 529–573. | MR: 969243 | Zbl: 0664.10013

Cited by Sources: