Abstract
We prove that the average size of the 3-Selmer group of a genus-2 curve with a marked Weierstrass point is 4.
Highlights
We prove that the average size of the 3-Selmer group of a genus-2 curve with a marked Weierstrass point is 4
In the paper [2], Bhargava and Gross calculated the average size of the 2-Selmer group of the Jacobian of an odd hyperelliptic curve of fixed genus g 2 using a connection with the arithmetic invariant theory of a graded Lie algebra; more precisely, the Z/2Z-graded Lie algebra arising from the element −1 of the automorphism group of a type-A2g root lattice
We exploit the stable Z/3Z-grading of a Lie algebra of type E8 in order to study the 3-Selmer groups of odd genus-2 curves
Summary
In the paper [2], Bhargava and Gross calculated the average size of the 2-Selmer group of the Jacobian of an odd hyperelliptic curve of fixed genus g 2 using a connection with the arithmetic invariant theory of a graded Lie algebra; more precisely, the Z/2Z-graded Lie algebra arising from the element −1 of the automorphism group of a type-A2g root lattice. We exploit the stable Z/3Z-grading of a Lie algebra of type E8 in order to study the 3-Selmer groups of odd genus-2 curves. Using an explicit construction of integral representatives in the square-free discriminant case, we extend the objects in this triple to the complement in Spec Zp[u] of finitely many closed points. The problem of constructing integral representatives can be viewed as the problem of showing that a graded analogue of an affine Springer fibre is non-empty From this point of view, attempting to deform the problem to a case where it can be solved directly is a natural strategy. We develop this technique here just in the case of the stable Z/3Z-grading of E8 and its relation to odd genus-2 curves, it is completely general.
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