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Non-abelian congruences between L-values of elliptic curves
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Delbourgo, Daniel and Ward, Thomas (2008) Non-abelian congruences between L-values of elliptic curves. Annales de l’institut Fourier, Volume 58 (Number 3). pp. 1023-1055. doi:10.5802/aif.2377 ISSN 0373-0956.
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Official URL: http://dx.doi.org/10.5802/aif.2377
Abstract
Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove weak forms of Kato’s $K_1$-congruences for the special values $ L\bigl (1, E/\mathbb{Q}( \mu _{p^n},\@root p^n \of {\Delta } )\bigr ) . $ More precisely, we show that they are true modulo $p^{n+1}$, rather than modulo $p^{2n}$. Whilst not quite enough to establish that there is a non-abelian $L$-function living in $K_1\bigl ( \mathbb{Z}_p[[ \rm {Gal} ( \mathbb{Q}( \mu _{p^\infty },\!\!\@root p^\infty \of {\Delta } )/\mathbb{Q}) ]] \bigr )$, they do provide strong evidence towards the existence of such an analytic object. For example, if $n=1$ these verify the numerical congruences found by Tim and Vladimir Dokchitser.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Annales de l’institut Fourier | ||||
Publisher: | Association des Annales de l'Institut Fourier | ||||
ISSN: | 0373-0956 | ||||
Official Date: | 2008 | ||||
Dates: |
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Volume: | Volume 58 | ||||
Number: | Number 3 | ||||
Page Range: | pp. 1023-1055 | ||||
DOI: | 10.5802/aif.2377 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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