Publication

Elliptic curves over a finite field and the trace formula

Ian Nicholas Petrow
2017
Journal Articles

Abstract

We prove formulas for power moments for point counts of elliptic curves over a finite field k such that the groups of k-points of the curves contain a chosen subgroup. These formulas express the moments in terms of traces of Hecke operators for certain congruence subgroups of SL 2(Z). As our main technical input we prove an Eichler-Selberg trace formula for a family of congruence subgroups of SL 2(Z) which include as special cases the groups 1(N) and (N). Our formulas generalize results of Birch and Ihara (the case of the trivial subgroup and the full modular group), and previous work of the authors (the subgroups Z/2Z and (Z/2Z)2 and congruence subgroups 0(2),0(4)). We use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field, generalizing results of Vladu, Gekeler, Howe and others.

Official source

https://infoscience.epfl.ch/entities/publication/d2275074-a2b1-4c03-801a-c7f47dc1cf5a

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Ian Nicholas Petrow
2017
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/d2275074-a2b1-4c03-801a-c7f47dc1cf5a

About this result

Related concepts (32)

Congruence subgroup

In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.

Normal subgroup

In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group is normal in if and only if for all and The usual notation for this relation is Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group.

Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is a group operation on H. This is often denoted H ≤ G, read as "H is a subgroup of G". The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. A proper subgroup of a group G is a subgroup H which is a proper subset of G (that is, H ≠ G).

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Elliptic curves over a finite field and the trace formula

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