Local densities and explicit bounds for representability by a quadratic form

Hanke, Jonathan

doi:10.1215/s0012-7094-04-12424-8

Cited by 42 publications

(78 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We make use of the reduction maps in the approach that Jonathan Hanke takes to computing local densities in [10], as well as the explicit formulas of Yang [28]. If p ∤ n and p ∤ N(Q), then…”

Section: Eisenstein Contributionmentioning

confidence: 99%

“…First, we consider the case that p ∤ n, and in this case all solutions will be good type solutions. If p ∤ 2D(Q), the table on page 363 of [10] shows that β p (n) ≥ 1 − 1 p 2 . Otherwise, diagonalize Q and let Q 1 be the orthogonal summand consisting of terms whose coefficients are coprime to p. Then, Yang's formula gives…”

Section: Eisenstein Contributionmentioning

confidence: 99%

See 1 more Smart Citation

Integers represented by positive-definite quadratic forms and Petersson inner products

Rouse

2019

Acta Arith.

View full text Add to dashboard Cite

Let Q be a positive-definite quaternary quadratic form with integer coefficients. We study the problem of giving bounds on the largest positive integer n that is locally represented by Q but not represented. Assuming that n is relatively prime to D(Q), the determinant of the Gram matrix of Q, we show that n is represented provided thatHere N (Q) is the level of Q. We give three other bounds that hold under successively weaker local conditions on n.These results are proven by bounding the Petersson norm of the cuspidal part of the theta series, which is accomplished using an explicit formula for the Weil representation due to Scheithauer.

show abstract

Section: Eisenstein Contributionmentioning

confidence: 99%

Section: Eisenstein Contributionmentioning

confidence: 99%

Integers represented by positive-definite quadratic forms and Petersson inner products

Rouse

2019

Acta Arith.

View full text Add to dashboard Cite

show abstract

“…This is implied by our strong local solubility conditions (1.14), whence k * 4 (Q) k ′ 4 (Q). Hanke [11,Theorem 6.3] has also used a modular forms interpretation to examine a quantity similar to k * 4 (Q), but the estimate he arrives at is too complicated to state here. Again, an alternative local solubility condition is employed, which differs from both Schulze-Pillot's and ours.…”

Section: {K}mentioning

confidence: 99%

On the representation of integers by quadratic forms

Browning

Dietmann²

2007

Proceedings of the London Mathematical Society

View full text Add to dashboard Cite

Abstract. Let n 4, and let Q ∈ Z[X1, . . . , Xn] be a non-singular quadratic form. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q = 0, and when Q is positive definite we provide improved upper bounds for the least positive integer k for which the equation Q = k is insoluble in integers, despite being soluble modulo every prime power.

show abstract

“…By applying this to any prime q such that pq, 2dLq " 1, we have rpn 2 a 2 , Lq " 0, for any integer n such that pn, 2dLq " 1. However, by Theorem 6.3 of [5], there is a sufficiently large integer m such that pm, 2dLq " 1 and…”

Section: Strongly S-regular Quaternary Latticesmentioning

confidence: 99%

The number of representations of squares by integral ternary quadratic forms

Kim

2017

Journal of Number Theory

View full text Add to dashboard Cite

Let f be a positive definite (non-classic) integral quaternary quadratic form. We say f is strongly s-regular if it satisfies a regularity property on the number of representations of squares of integers. In this article, we prove that there are only finitely many strongly s-regular quaternary quadratic forms up to isometry if the minimum of the nonzero squares that are represented by the quadratic form is fixed. Furthermore, we show that there are exactly 34 strongly s-regular diagonal quaternary quadratic forms representing one (see Table 1). In particular, we use eta-quotients to prove the strongly s-regularity of the quaternary quadratic form x 2`2 y 2`3 z 2`1 0w 2 , which is, in fact, of class number 2 (see Lemma 5.5 and Proposition 5.6).2010 Mathematics Subject Classification. Primary 11E12, 11E20, 11E45.

show abstract

Local densities and explicit bounds for representability by a quadratic form

Cited by 42 publications

References 24 publications

Integers represented by positive-definite quadratic forms and Petersson inner products

Integers represented by positive-definite quadratic forms and Petersson inner products

On the representation of integers by quadratic forms

The number of representations of squares by integral ternary quadratic forms

Contact Info

Product

Resources

About