A nilpotent group and its elliptic curve: Non-uniformity of local zeta functions of groups |
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Authors: | Marcus du Sautoy |
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Institution: | (1) DPMMS, Centre for Mathematical Sciences, Wilberforce Road, CB3 0WB Cambridge, England |
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Abstract: | A nilpotent group is defined whose local zeta functions counting subgroups and normal subgroups depend on counting points
modp on the elliptic curvey
2=x
3−x. This example answers negatively a question raised in the paper of F. J. Grunewald, D. Segal and G. C. Smith where these
local zeta functions were first defined. They speculated that local zeta functions of nilpotent groups might be finitely uniform
asp varies. A proof is given that counting points on the elliptic curvey
2=x
3−x are not finitely uniform, and hence the same is true for the zeta function of the associated nilpotent group. This example
demonstrates that nilpotent groups have a rich arithmetic beyond the connection with quadratic forms. |
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Keywords: | |
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