@article{20423,
  abstract     = {For any d  2, we prove that there exists an integer n0(d) such that there exists an n × n
magic square of dth powers for all n  n0(d). In particular, we establish the existence of
an n × n magic square of squares for all n  4, which settles a conjecture of
Várilly-Alvarado. All previous approaches had been based on constructive methods and
the existence of n × n magic squares of dth powers had only been known for sparse
values of n. We prove our result by the Hardy-Littlewood circle method, which in this
setting essentially reduces the problem to finding a sufficient number of disjoint linearly
independent subsets of the columns of the coefficient matrix of the equations defining
magic squares. We prove an optimal (up to a constant) lower bound for this quantity.},
  author       = {Rome, Nick and Yamagishi, Shuntaro},
  issn         = {2363-9555},
  journal      = {Research in Number Theory},
  number       = {4},
  publisher    = {Springer Nature},
  title        = {{On the existence of magic squares of powers}},
  doi          = {10.1007/s40993-025-00671-5},
  volume       = {11},
  year         = {2025},
}

@article{19489,
  abstract     = {Let K be a cyclic number field of odd degree over 
𝑄 with odd narrow class number, such that 2 is inert in 𝐾/𝑄. We define a family of number fields {𝐾(𝑝)}𝑝, depending on K and indexed by the rational primes p that split completely in 𝐾/𝑄, in which p is always ramified of degree 2. Conditional on a standard conjecture on short character sums, the density of such rational primes p that exhibit one of two possible ramified factorizations in 𝐾(𝑝)/𝑄 is strictly between 0 and 1 and is given explicitly as a formula in terms of the degree of the extension 𝐾/𝑄. Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals.},
  author       = {Chan, Yik Tung and McMeekin, Christine and Milovic, Djordjo},
  issn         = {2363-9555},
  journal      = {Research in Number Theory},
  publisher    = {Springer Nature},
  title        = {{A density of ramified primes}},
  doi          = {10.1007/s40993-021-00295-5},
  volume       = {8},
  year         = {2021},
}

@article{10874,
  abstract     = {In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler, and Zykin, which allows us to connect invariants of binary octics to Siegel modular forms of genus 3. We use this connection to show that certain modular functions, when restricted to the hyperelliptic locus, assume values whose denominators are products of powers of primes of bad reduction for the associated hyperelliptic curves. We illustrate our theorem with explicit computations. This work is motivated by the study of the values of these modular functions at CM points of the Siegel upper half-space, which, if their denominators are known, can be used to effectively compute models of (hyperelliptic, in our case) curves with CM.},
  author       = {Ionica, Sorina and Kılıçer, Pınar and Lauter, Kristin and Lorenzo García, Elisa and Manzateanu, Maria-Adelina and Massierer, Maike and Vincent, Christelle},
  issn         = {2363-9555},
  journal      = {Research in Number Theory},
  keywords     = {Algebra and Number Theory},
  publisher    = {Springer Nature},
  title        = {{Modular invariants for genus 3 hyperelliptic curves}},
  doi          = {10.1007/s40993-018-0146-6},
  volume       = {5},
  year         = {2019},
}