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**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).

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).

Reviewer: Andrzej Dąbrowski (Szczecin)

### MSC:

11F80 | Galois representations |

11F33 | Congruences for modular and \(p\)-adic modular forms |

11R23 | Iwasawa theory |

### Citations:

Zbl 1131.11341
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\textit{M. Dimitrov} and \textit{E. Ghate}, J. Théor. Nombres Bordx. 24, No. 3, 669--690 (2012; Zbl 1271.11060)

### References:

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