Equidistribution Questions for Universal Extensions

Abstract

We discuss in detail some equidistribution questions arising from the study of the universal extension of an elliptic curve by a vector group. We will also indicate analogous questions in the case of the universal extension of a Jacobian by a vector group.

2000 AMS Subject Classification::

• 14G15
• 14H25
• 11G05
• 11K36

Keywords:

• elliptic curve
• universal extension
• equidistribution

Notes

¹It is not a lifting of the Verschiebung V_q of E. Indeed, from the relation , we see that V_q acts on as multiplication by q, so only the points in of order dividing q − 1 are fixed by V_q. Our problematic points P in are those of p-power order, so are certainly not fixed by V_q. So although V_qdoes lift to an endomorphism of , this lifting will in general not even act on our .