Selmer groups of symmetric powers of ordinary modular Galois representations
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arXiv
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| Formato: | Preprint |
| Publicado: |
2018
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| _version_ | 1866911870770741248 |
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| author | Zhang, Xiaoyu |
| author_facet | Zhang, Xiaoyu |
| contents | Let $p$ be a fixed odd prime number, $μ$ be a Hida family over the Iwasawa algebra of one variable, $ρ_μ$ its Galois representation, $\mathbb{Q}_\infty/\mathbb{Q}$ the $p$-cyclotomic tower and $S$ the variable of the cyclotomic Iwasawa algebra. We compare, for $n\leq 4$ and under certain assumptions, the characteristic power series $L(S)$ of the dual of Selmer groups $\mathrm{Sel}(\mathbb{Q}_{\infty},\mathrm{Sym}^{2n}\otimes\mathrm{det}^{-n}ρ_μ)$ to certain congruence ideals. The case $n=1$ has been treated by H.Hida. In particular, we express the first term of the Taylor expansion at the trivial zero $S=0$ of $L(S)$ in terms of an $\mathcal{L}$-invariant and a congruence number. We conjecture the non-vanishing of this $\mathcal{L}$-invariant; this implies therefore that these Selmer groups are cotorsion. We also show that our $\mathcal{L}$-invariants coincide with Greenberg's $\mathcal{L}$-invariants calculated by R.Harron and A.Jorza. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1802_08329 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Selmer groups of symmetric powers of ordinary modular Galois representations Zhang, Xiaoyu Number Theory 11F33 11F67 11F80 11R18 11R23 11R34 Let $p$ be a fixed odd prime number, $μ$ be a Hida family over the Iwasawa algebra of one variable, $ρ_μ$ its Galois representation, $\mathbb{Q}_\infty/\mathbb{Q}$ the $p$-cyclotomic tower and $S$ the variable of the cyclotomic Iwasawa algebra. We compare, for $n\leq 4$ and under certain assumptions, the characteristic power series $L(S)$ of the dual of Selmer groups $\mathrm{Sel}(\mathbb{Q}_{\infty},\mathrm{Sym}^{2n}\otimes\mathrm{det}^{-n}ρ_μ)$ to certain congruence ideals. The case $n=1$ has been treated by H.Hida. In particular, we express the first term of the Taylor expansion at the trivial zero $S=0$ of $L(S)$ in terms of an $\mathcal{L}$-invariant and a congruence number. We conjecture the non-vanishing of this $\mathcal{L}$-invariant; this implies therefore that these Selmer groups are cotorsion. We also show that our $\mathcal{L}$-invariants coincide with Greenberg's $\mathcal{L}$-invariants calculated by R.Harron and A.Jorza. |
| title | Selmer groups of symmetric powers of ordinary modular Galois representations |
| topic | Number Theory 11F33 11F67 11F80 11R18 11R23 11R34 |
| url | https://arxiv.org/abs/1802.08329 |