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| Format: | Preprint |
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2018
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/1806.04240 |
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| _version_ | 1866913539503947776 |
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| author | Bellaïche, Joël Pollack, Robert |
| author_facet | Bellaïche, Joël Pollack, Robert |
| contents | We study the variation of mu-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the p-adic zeta function. This lower bound forces these mu-invariants to be unbounded along the family, and moreover, we conjecture that this lower bound is an equality. When U_p-1 generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the p-adic L-function is simply a power of p up to a unit (i.e. lambda=0). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1806_04240 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Congruences with Eisenstein series and mu-invariants Bellaïche, Joël Pollack, Robert Number Theory 11F33, 11R23 We study the variation of mu-invariants in Hida families with residually reducible Galois representations. We prove a lower bound for these invariants which is often expressible in terms of the p-adic zeta function. This lower bound forces these mu-invariants to be unbounded along the family, and moreover, we conjecture that this lower bound is an equality. When U_p-1 generates the cuspidal Eisenstein ideal, we establish this conjecture and further prove that the p-adic L-function is simply a power of p up to a unit (i.e. lambda=0). On the algebraic side, we prove analogous statements for the associated Selmer groups which, in particular, establishes the main conjecture for such forms. |
| title | Congruences with Eisenstein series and mu-invariants |
| topic | Number Theory 11F33, 11R23 |
| url | https://arxiv.org/abs/1806.04240 |