@article{21385,
  abstract     = {We prove that the average size of a mixed character sum (math. formular) (for a suitable smooth function w) is on the order of √x for all irrational real θ satisfying a weak Diophantine condition, where χ is drawn from the family of Dirichlet characters modulo a large prime r and where x 6 r. In contrast, it was proved by Harper that the average size is o(√x) for rational θ. Certain quadratic Diophantine equations play a key role in the present paper. },
  author       = {Wang, Victor and Xu, Max},
  issn         = {1473-7124},
  journal      = {Proceedings of the Royal Society of Edinburgh: Section A Mathematics},
  pages        = {1--15},
  publisher    = {Cambridge University Press},
  title        = {{Average sizes of mixed character sums}},
  doi          = {10.1017/prm.2026.10123},
  year         = {2026},
}

@article{12311,
  abstract     = {In this note, we prove a formula for the cancellation exponent  kv,n between division polynomials  ψn  and  ϕn  associated with a sequence  {nP}n∈N of points on an elliptic curve  E  defined over a discrete valuation field  K. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.},
  author       = {Naskręcki, Bartosz and Verzobio, Matteo},
  issn         = {1473-7124},
  journal      = {Proceedings of the Royal Society of Edinburgh Section A: Mathematics},
  keywords     = {Elliptic curves, Néron models, division polynomials, height functions, discrete valuation rings},
  number       = {5},
  pages        = {1646--1660},
  publisher    = {Cambridge University Press},
  title        = {{Common valuations of division polynomials}},
  doi          = {10.1017/prm.2024.7},
  volume       = {155},
  year         = {2025},
}

@article{19407,
  abstract     = {We discuss, in a non-Archimedean setting, the distribution of the coefficients of L-polynomials of curves of genus g over  Fq . Among other results, this allows us to prove that the  Q-vector space spanned by such characteristic polynomials has dimension g + 1. We also state a conjecture about the Archimedean distribution of the number of rational points of curves over finite fields.},
  author       = {Ballini, Francesco and Lombardo, Davide and Verzobio, Matteo},
  issn         = {1473-7124},
  journal      = {Proceedings of the Royal Society of Edinburgh Section A: Mathematics},
  publisher    = {Cambridge University Press},
  title        = {{On the L-polynomials of curves over finite fields}},
  doi          = {10.1017/prm.2025.7},
  year         = {2025},
}