学术活动

Functoriality, Congruence ideals and big image of Galois representations associated to Hida families - 陈欢 博士(法国巴黎第十三大学)

作者:   来源:  时间:2019-09-12

题目:Functoriality, Congruence ideals and big image of Galois representations associated to Hida families

报告人:陈欢 博士(法国巴黎第十三大学)

Abstract:

The study of images of Galois representations was initiated by J-P. Serre in 1968. He showed that if an elliptic curve defined over Q does not have CM, then the image of the associated p-adic Galois representation is open for any prime p. Furthermore, he showed that the p-adic Galois representation associated to such an elliptic curve is surjective (onto GL2(Zp)) for almost all primes p. The study of the image of the Galois representation attached to a modular form, and showing that it is large in the absence of CM, was first carried out by Serre and Swinnerton-Dyer in the early 1970s in the case of modular forms of level one with integral coefficients and generalized to classical modular forms by Ribet and Momose in 1980s.

In the 1980s, Hida developed his theory of p-adic families of ordinary modular forms over GL2 and the Galois representations attached to them. Roughly speaking, the Galois representation  arising from a Hida family F take values in GL2(Zp), where I is an integral domain that is finite flat over Zp[[T]]. Hida has shown under some technical hypotheses that, if F does not have CM, then Im( ) is "big" with respect to the ring, which means that it contains acongruence subgroup of SL2( ).

Following the work of Hida-Tilouine, J. Lang and A. Conti, I generalized the results of Hida to the case of Hida families over general reductive groups. Assuming the existence of associated Galois representations, under technical hypotheses, I proved that the images of attached Galois representations are also "big". In my talk, I will begin with some preliminaries on Hida families of modular forms, then explain the details of my work and the precise result, give the main steps of the proof. Finally, I will talk about some possible directions of further research.

时间:

2019年9月12日(周四)上午9:00-10:30,813教室

2019年9月19日(周四)下午15:00-16:30,627教室

2019年9月26日(周四)下午14:00-15:30,627教室

地点:首都师范大学本部教二楼

联系人:童纪龙

 

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