## On Sharifi’s conjecture: Exceptional case

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- by Sheng-Chi Shih and Jun Wang PDF
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## Abstract:

In the present article, we study the conjecture of Sharifi on the surjectivity of the map $\varpi _{\theta }$. Here $\theta$ is a primitive even Dirichlet character of conductor $Np$, which is exceptional in the sense of Ohta. After localizing at the prime ideal $\mathfrak {p}$ of the Iwasawa algebra related to the trivial zero of the Kubota–Leopoldt $p$-adic $L$-function $L_p(s,\theta ^{-1}\omega ^2)$, we compute the image of $\varpi _{\theta ,\mathfrak {p}}$ in a local Galois cohomology group and prove that it is an isomorphism. Also, we prove that the residual Galois representations associated to the cohomology of modular curves are decomposable after taking the same localization.## References

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## Additional Information

**Sheng-Chi Shih**- Affiliation: Fakultät für Mathematik, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
- MR Author ID: 1066891
- ORCID: 0000-0003-1607-2482
- Email: sheng-chi.shih@univie.ac.at
**Jun Wang**- Affiliation: Morningside Center of Mathematics, No. 55, Zhongguancun East Road, Beijing, 100190, People’s Republic of China
- Email: jwangmathematics@gmail.com
- Received by editor(s): September 27, 2020
- Received by editor(s) in revised form: February 4, 2021, March 1, 2021, and March 10, 2021
- Published electronically: June 9, 2021
- Additional Notes: The first author was supported by the Labex CEMPI under Grant No. ANR-11-LABX-0007-01, by I-SITE ULNE under Grant No. ANR-16-IDEX-0004, and by Austrian Science Fund (FWF) under Grant No. START-Prize Y966. The second author was supported by Sujatha Ramdorai and Morningside Center of Mathematics in his postdoctoral studies
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 8531-8546 - MSC (2020): Primary 11R23, 11F33, 11F80, 11R34, 11S25
- DOI: https://doi.org/10.1090/tran/8433
- MathSciNet review: 4337920