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Add ell-adic index to Galois rep tables on ECQ pages #6190

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AndrewVSutherland opened this issue Sep 26, 2024 · 0 comments
Open

Add ell-adic index to Galois rep tables on ECQ pages #6190

AndrewVSutherland opened this issue Sep 26, 2024 · 0 comments
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ECQ Elliptic curves over Q feature request Feature request

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@AndrewVSutherland
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AndrewVSutherland commented Sep 26, 2024

Currently the "Galois representations" section of home pages for elliptic curves over Q includes a table that gives the ell-adic image for each prime ell where the image is not maximal; note that this applies to CM curves as well as non-CM curves.

In the non-CM case the index of the ell-adic representation in GL_2(Z_ell) is part of the label that identifies the image, but in the CM case it is not obvious what the index is (relative to the maximal image for an elliptic curve with CM by the same order, as defined in section 12 of RSZB), and even in the non-CM case it is only obvious if one knows the label format.

I think it would be helpful to list the index explicitly as a separate column in the table of non-maximal ell-adic representations. For non-CM curves this can be extracted directly from the second component of the image label, and for CM curves with j(E) != 0,1728 it will always be 2 (it is either 1 or 2, and if it is 1 the image is maximal and will not be listed). To handle j(E) = 0,1728 it would suffice to add an additional table which gives the index of the maximal image at prime power levels for all the relevant prime powers (there won't be more than a few hundred of these).

@AndrewVSutherland AndrewVSutherland added ECQ Elliptic curves over Q feature request Feature request labels Sep 26, 2024
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