On classical weight one forms in Hida families. (English. French summary) Zbl 1271.11060

Let \(p\) be an odd prime and let \(F\) be a primitive \(p\)-adic Hida family of ordinary, cuspidal eigenforms. Then \(F\) admits infinitely many classical specializations of any given weight \(\geq 2\). Moreover, a CM family contains infinitely many CM classical specializations of weight one. It is also shown that a non-CM family admits only finitely many classical weight one specializations [E. Ghate and V. Vatsal, Ann. Inst. Fourier 54, No. 7, 2143–2162 (2004; Zbl 1131.11341)].
The authors give precise estimates for the number of such specializations. The results depend on the residual type of the family (see §4 for precise definitions). For instance, in the exceptional case and under some mild assumptions, there is exactly one classical weight one specialization (Theorem 5.1 and Prop. 5.2). But in the RM (real multiplication) case, they provide examples to show how the uniqueness fails (§6.2). The authors also provide an example of two non-Galois conjugate Hida families specializing to the same classical weight one form (§7.4).


11F80 Galois representations
11F33 Congruences for modular and \(p\)-adic modular forms
11R23 Iwasawa theory


Zbl 1131.11341
Full Text: DOI Numdam


[1] J. Bellaïche and M. Dimitrov, On the Eigencurve at classical weight one points. Preprint (2012). · Zbl 1404.11047
[2] K. Buzzard and R. Taylor, Companion forms and weight \(1\) forms. Ann. of Math., 149 (1999), 905-919. · Zbl 0965.11019
[3] K. Buzzard, Analytic continuation of overconvergent eigenforms. J. Amer. Math. Soc., 16 (2003), 29-55. · Zbl 1076.11029
[4] S. Cho and V. Vatsal, Deformations of induced Galois representations. J. Reine Angew. Math., 556 (2003), 79-98. · Zbl 1041.11039
[5] M. Emerton, R. Pollack and T. Weston, Variation of Iwasawa invariants in Hida families. Invent. Math., 163 (2006), 523-580. · Zbl 1093.11065
[6] A. Fischman, On the image of \(\Lambda \)-adic Galois representations. Ann. Inst. Fourier, Grenoble, 52 (2002), no. 2, 351-378. · Zbl 1020.11037
[7] E. Ghate and N. Kumar, Control theorems for ordinary \(2\)-adic families of modular forms. In preparation. · Zbl 1307.11068
[8] E. Ghate and V. Vatsal, On the local behaviour of ordinary \({\Lambda }\)-adic representations. Ann. Inst. Fourier, Grenoble, 54 (2004), no. 7, 2143-2162. · Zbl 1131.11341
[9] H. Hida, Galois representations into \({\rm GL}_2({\bf Z}_p[[X]])\) attached to ordinary cusp forms. Invent. Math., 85 (1986), 545-613. · Zbl 0612.10021
[10] Iwasawa modules attached to congruences of cusp forms. Ann. Sci. Ecole Norm. Sup. (4), 19 (1986), 231-273. · Zbl 0607.10022
[11] J-P. Serre, Modular forms of weight one and Galois representations. Proc. Sympos. Univ. Durham, Durham (1975), Academic Press, London, 1977, 193-268. · Zbl 0366.10022
[12] A. Wiles, On ordinary \(\lambda \)-adic representations associated to modular forms. Invent. Math., 94 (1988), 529-573. · Zbl 0664.10013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.