Bilinear forms in Weyl sums for modular square roots and applications

  title={Bilinear forms in Weyl sums for modular square roots and applications},
  author={Alexander Dunn and Igor E. Shparlinski and Alexandru Zaharescu},
  journal={arXiv: Number Theory},
On the distribution of modular square roots of primes
We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be
Reconstructing points of superelliptic curves over a prime finite field
  • Jaime Gutierrez
  • Computer Science
    Advances in Mathematics of Communications
  • 2022
An upper bound is provided on the number of roots of such bivariate polynomials where the roots have certain restrictions is motivated by the predictability problem for non-linear pseudorandom number generators and, other potential applications to cryptography.
Bounds on bilinear forms with Kloosterman sums
. We prove new bounds on bilinear forms with Kloosterman sums, complementing and improving a series of results by ´E. Fouvry, E. Kowalski and Ph. Michel (2014), V. Blomer, ´E. Fouvry, E. Kowalski,
A problem in comparative order theory
Write ordp(·) for the multiplicative order in F×p . Recently, Matthew Just and the second author investigated the problem of classifying pairs α, β ∈ Q \ {±1} for which ordp(α) > ordp(β) holds for
On the cubic Weyl sum
We obtain an estimate for the cubic Weyl sum which improves the bound obtained from Weyl differencing for short ranges of summation. In particular, we show that for any ε > 0 there exists some δ > 0
Energy bounds for modular roots and their applications
We generalise and improve some recent bounds for additive energies of modular roots. Our arguments use a variety of techniques, including those from additive combinatorics, algebraic number theory
Energy bounds, bilinear forms and their applications in function fields
Bilinear Forms With Modular Square Roots and Twisted Second Moments of Half Integral Weight Dirichlet Series
We establish new results on equations and bilinear forms with modular square roots. The main motivation and application of these results is our new bound on the fourth moment of the error term in


Points on curves in small boxes en applications
We introduce several new methods to obtain upper bounds on the number of solutions of the congruences $f(x) \equiv y \pmod p$ and $f(x) \equiv y^2 \pmod p,$ with a prime $p$ and a polynomial $f$,
Small prime $k$th power residues for $k=2,3,4$: A reciprocity laws approach
Nagell proved that for each prime $p\equiv 1\pmod{3}$, $p > 7$, there is a prime $q 0$, and each prime $p\equiv 1\pmod{3}$ with $p > p_0(\epsilon)$, the number of prime cubic residues $q <
Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums
The key geometric idea is a comparison statement that shows that even when the "sum-product" sheaves that appear in the analysis fail to be irreducible, their decomposition reflects that of the "input"Sheaves, except for parameters in a high-codimension subset.
Bounds for the first several prime character nonresidues
Let $\varepsilon > 0$. We prove that there are constants $m_0=m_0(\varepsilon)$ and $\kappa=\kappa(\varepsilon) > 0$ for which the following holds: For every integer $m > m_0$ and every nontrivial
Primes represented by positive definite binary quadratic forms
  • Asif Zaman
  • Mathematics
    The Quarterly Journal of Mathematics
  • 2018
Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types
On moments of twisted L-functions
We study the average of the product of the central values of two $L$-functions of modular forms $f$ and $g$ twisted by Dirichlet characters to a large prime modulus $q$. As our principal tools, we
On short products of primes in arithmetic progressions
  • I. Shparlinski
  • Mathematics, Computer Science
    Proceedings of the American Mathematical Society
  • 2018
Several families of reasonably small integers and real positive integers, such that the products of p_1, p_k s, and s, are primes, represent all reduced residue classes modulo $m$.
Elliptic curves and lower bounds for class numbers
Simultaneous non-vanishing for Dirichlet $L$-functions
We extend the work of Fouvry, Kowalski and Michel on correlation between Hecke eigenvalues of modular forms and algebraic trace functions in order to establish an asymptotic formula for a generalized