Constants in Titchmarsh divisor problems for elliptic curves

Bell, Renee; Blakestad, Clifford; Cojocaru, Alina Carmen; Cowan, Alexander J.; Jones, Nathan D.; Matei, Vlad; Smith, G.; Vogt, Isabel

doi:10.48550/arxiv.1706.03422

Search citation statements

Order By: Relevance

Paper Sections

Select...

Averages For Invariant Factors Of E(f Q )1

Counting Curves Containing a Prescribed Subgroup1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2017

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The method of proof of Theorem 5 also allows one to compute averages of several types of arithmetic functions of n 2 (E) over C, such as the divisor function, E q (τ (n 2 (E))), or a Dirichlet character, E q (χ(n 2 (E))). Such applications were suggested to us by Cojocaru at the Arizona Winter School 2016, see the related work of her project group [1]. On the other hand, computing averages of arithmetic functions of n 1 (E) over C seems more challenging.…”

Section: Averages For Invariant Factors Of E(f Q )mentioning

confidence: 99%

See 1 more Smart Citation

Elliptic curves over a finite field and the trace formula

Kaplan

Petrow

2017

Proc. London Math. Soc.

View full text Add to dashboard Cite

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 Γ 1 (N ) and Γ(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 Γ 0 (2), Γ 0 (4)). We use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field, generalizing results of Vlǎduţ, Gekeler, Howe, and others.

show abstract

Section: Averages For Invariant Factors Of E(f Q )mentioning

confidence: 99%

“…U k−2 (t, q)P q (C(A, t)), (26) where U k−2 (t, q) are the normalized Chebyshev polynomials defined in (1).…”

Section: Counting Curves Containing a Prescribed Subgroupmentioning

confidence: 99%