Counting rational points on smooth cyclic covers

Heath‐Brown, D. R.; Pierce, Lillian B.

doi:10.1016/j.jnt.2012.02.010

Cited by 15 publications

(19 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Later, T. D Browning [3] developed the polynomial sieve which extends both the square sieve of D. R. Heath-Brown and d-power sieve of R. Munshi. In [8], D. R. Heath-Brown and L. Pierce improved results of R. Munshi [11] for n ≥ 8.…”

Section: Introductionmentioning

confidence: 92%

Bounds for del Pezzo surfaces of degree two

Ghosh¹,

Kumar²,

Mallesham³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 92%

Bounds for del Pezzo surfaces of degree two

Ghosh¹,

Kumar²,

Mallesham³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…• Pierce [Pi06] introduced the ideas of q-vdC to the square sieve of Heath-Brown, which enables her to derive the first non-trivial bound for the 3torsion of the class group of Q( √ −D). In their joint work on a conjecture of Serre concerning the number of rational points of bounded height on a finite cover of projective space P n−1 , Heath-Brown and Pierce [HBP12] can succeed in the special case of smooth cyclic covers of large degrees invoking the ideas of q-vdC to the power sieve.…”

Section: Definition 23 (Fourier Sheaf)mentioning

confidence: 99%

Arithmetic exponent pairs for algebraic trace functions and applications

Wu¹,

Xi²

2016

Preprint

View full text Add to dashboard Cite

We study short sums of algebraic trace functions via the q-analogue of van der Corput method, and develop methods of arithmetic exponent pairs that coincide with the classical case while the moduli has sufficiently good factorizations. As an application, we prove a quadratic analogue of Brun-Titchmarsh theorem on average, bounding the number of primes p X with p 2 + 1 ≡ 0 (mod q). The other two applications include a larger level of distribution of divisor functions in arithmetic progressions and a sub-Weyl subconvex bound of Dirichlet L-functions studied previously by Irving. Contents 14 4. Producing new exponent pairs: the first approach 19 5. Proof of Theorems 4.1, 4.2 and 4.3 21 6. Producing new exponent pairs: the second approach 26 7. Arithmetic exponent pairs with constraints 29 8. Explicit estimates for sums of trace functions 30 9. Proof of Theorem 1.1: Quadratic Brun-Titchmarsh theorem 31 10. Proof of Theorems 1.2 and 1.3: Divisor functions in arithmetic progressions and subconvexity of Dirichlet L-functions 40 Appendix A. Several lemmas 41 Appendix B. Estimates for complete exponential sums 43 References 45

show abstract

“…Our proof of Theorem 1.1 relies on a variant of the square sieve worked out by Pierce [15], which allows for an application of Heath-Brown's q-analogue of van der Corput differencing. This approach was already put to use by Heath-Brown and Pierce [8] to study cyclic covers of P n and our proof is inspired by their work. Ultimately, for suitable primes p, the proof of Theorem 1.1 is reduced to estimating a certain 4-variable exponential sum W p = W p (λ, h, µ) defined over F p .…”

Section: Introductionmentioning

confidence: 99%