## The Density of Ramified Primes

##### Doc. Math. 24, 2423-2429 (2019)
DOI: 10.25537/dm.2019v24.2423-2429

### Summary

Let $F$ be a number field, $\mathcal{O}$ be a domain with fraction field $\mathcal{K}$ of characteristic zero and $\rho:\text{Gal}(\overline F/F)\to\text{GL}_n(\mathcal{O})$ be a representation such that $\rho\otimes\overline{\mathcal{K}}$ is semisimple. If $\mathcal{O}$ admits a finite monomorphism from a power series ring with coefficients in a $p$-adic integer ring (resp. $\mathcal{O}$ is an affinoid algebra over a $p$-adic number field) and $\rho$ is continuous with respect to the maximal ideal adic topology (resp. the Banach algebra topology), then we prove that the set of ramified primes of $\rho$ is of density zero. If $\mathcal{O}$ is a complete local Noetherian ring over $\mathbb{Z}_p$ with finite residue field of characteristic $p,\rho$ is continuous with respect to the maximal ideal adic topology and the kernels of pure specializations of $\rho$ form a Zariski-dense subset of $\text{Spec}\mathcal{O}$, then we show that the set of ramified primes of $\rho$ is of density zero. These results are analogues, in the context of big Galois representations, of a result of Khare and Rajan, and are proved relying on their result.

11F80

### Keywords/Phrases

Galois representations, ramification

### References

• [BC09]. Joël Bellaïche and Gaëtan Chenevier. Families of Galois representations and Selmer groups. Astérisque, 324:xii+314, 2009. zbl 1192.11035; MR2656025.
• [Bos14]. Siegfried Bosch. Lectures on formal and rigid geometry, volume 2105 of Lecture Notes in Mathematics. Springer, Cham, 2014. DOI 10.1007/978-3-319-04417-0; zbl 1314.14002; MR3309387.
• [Che04]. Gaëtan Chenevier. Familles $p$-adiques de formes automorphes pour $GL_n$. J. Reine Angew. Math., 570:143-217, 2004. DOI 10.1515/crll.2004.031; zbl 1093.11036; MR2075765.
• [Del71]. Pierre Deligne. Formes modulaires et représentations $l$-adiques. In Séminaire Bourbaki. Vol. 1968/69: Exposés 347-363, volume 175 of Lecture Notes in Math., pages Exp. No. 355, 139-172. Springer, Berlin, 1971. zbl 0206.49901; MR3077124.
• [Eic54]. Martin Eichler. Quaternäre quadratische Formen und die Riemannsche Vermutung für die Kongruenzzetafunktion. Arch. Math., 5:355-366, 1954. DOI 10.1007/BF01898377; zbl 0059.03804; MR0063406.
• [FvdP04]. Jean Fresnel and Marius van der Put. Rigid analytic geometry and its applications, volume 218 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 2004. zbl 1096.14014; MR2014891.
• [Hid86a]. Haruzo Hida. Galois representations into $GL_2({\mathbf Z}_p[[X]])$ attached to ordinary cusp forms. Invent. Math., 85(3):545-613, 1986. DOI 10.1007/BF01390329; zbl 0612.10021; MR0848685.
• [Hid86b]. Haruzo Hida. Iwasawa modules attached to congruences of cusp forms. Ann. Sci. École Norm. Sup. (4), 19(2):231-273, 1986. DOI 10.24033/asens.1507; zbl 0607.10022; MR0868300.
• [KR01]. Chandrashekhar Khare and C.S. Rajan. The density of ramified primes in semisimple $p$-adic Galois representations. Internat. Math. Res. Notices, (12):601-607, 2001. DOI 10.1155/S1073792801000319; zbl 1043.11049; MR1836789; arxiv math/0011272.
• [Mat80]. Hideyuki Matsumura. Commutative algebra, volume 56 of Mathematics Lecture Note Series. Benjamin/Cummings Publishing Co., Inc., Reading, Mass., second edition, 1980. zbl 0441.13001; MR0575344.
• [Ram00]. Ravi Ramakrishna. Infinitely ramified Galois representations. Ann. of Math. (2), 151(2):793-815, 2000. DOI 10.2307/121048; zbl 1078.11510; MR1765710; arxiv math/0003241.
• [Sah17]. Jyoti Prakash Saha. Purity for families of Galois representations. Ann. Inst. Fourier (Grenoble), 67(2):879-910, 2017. DOI 10.5802/aif.3099; zbl 06821963; MR3669514; arxiv 1410.3843.
• [Ser98]. Jean-Pierre Serre. Abelian $l$-adic representations and elliptic curves, volume 7 of Research Notes in Mathematics. A K Peters, Ltd., Wellesley, MA, 1998. With the collaboration of Willem Kuyk and John Labute, Revised reprint of the 1968 original. zbl 0902.14016; MR1484415.
• [Shi58]. Goro Shimura. Correspondances modulaires et les fonctions $\zeta$ de courbes algébriques. J. Math. Soc. Japan, 10:1-28, 1958. DOI 10.2969/jmsj/01010001; zbl 0081.07603; MR0095173.
• [ST68]. Jean-Pierre Serre and John Tate. Good reduction of abelian varieties. Ann. of Math. (2), 88:492-517, 1968. DOI 10.2307/1970722; zbl 0172.46101; MR0236190.
• [{Sta}18]. The Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu, 2018.

### Affiliation

Saha, Jyoti Prakash
Department of Mathematics, Indian Institute of Science Education and Research Bhopal, Bhopal Bypass Road, Bhauri, Bhopal 462066, Madhya Pradesh, India