@article{21343, abstract = {The large sieve is used to estimate the density of quadratic polynomials Q ∈ Z[x], such that there exists an odd degree polynomial defined over Z which has resultant ±1 with Q. Given a monic polynomial R ∈ Z[x] of odd degree, this is used to show that for almost all quadratic polynomials Q ∈ Z[x], there exists a prime p such that Q and R share a common root in Fp. Using recent work of Landesman, an application to the average size of the odd part of the class group of quadratic number fields is also given}, author = {Browning, Timothy D and Chan, Yik Tung}, issn = {2270-518X}, journal = {Journal de l'ecole polytechnique mathematiques}, pages = {1677--1691}, publisher = {Ecole polytechnique}, title = {{Solubility of a resultant equation and applications}}, doi = {10.5802/jep.320}, volume = {12}, year = {2025}, } @article{19363, abstract = {For a general family of non-negative functions matching upper and lower bounds are established for their average over the values of any equidistributed sequence.}, author = {Chan, Yik Tung and Koymans, Peter and Pagano, Carlo and Sofos, Efthymios}, issn = {0022-314X}, journal = {Journal of Number Theory}, pages = {1--36}, publisher = {Elsevier}, title = {{Averages of multiplicative functions along equidistributed sequences}}, doi = {10.1016/j.jnt.2025.01.005}, volume = {273}, year = {2025}, } @article{19483, abstract = {We prove matching upper and lower bounds for the average of the6-torsionof class groups of quadratic fields. Furthermore, we count the number of integer solutions on an affine quartic threefold.}, author = {Chan, Yik Tung and Koymans, Peter and Pagano, Carlo and Sofos, Efthymios}, issn = {2036-2145}, journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze}, publisher = {Scuola Normale Superiore - Edizioni della Normale}, title = {{6-torision and integral points on quartic threefolds}}, doi = {10.2422/2036-2145.202412_006}, year = {2025}, } @article{21266, abstract = {For a given elliptic curve E in short Weierstrass form, we show that almost all quadratic twists E D have no integral points, as D ranges over square-free integers ordered by size. Our result is conditional on a weak form of the Hall–Lang conjecture in the case that E has partial 2-torsion. The proof uses a correspondence of Mordell and the reduction theory of binary quartic forms in order to transfer the problem to counting rational points of bounded height on a certain singular cubic surface, together with extensive use of cancellation in character sum estimates, drawn from Heath-Brown’s analysis of Selmer group statistics for the congruent number curve.}, author = {Browning, Timothy D and Chan, Yik Tung}, issn = {1435-9863}, journal = {Journal of the European Mathematical Society}, publisher = {EMS Press}, title = {{Almost all quadratic twists of an elliptic curve have no integral points}}, doi = {10.4171/jems/1704}, year = {2025}, } @article{19486, abstract = {Consider the family of elliptic curves En:y2=x3+n2, where n varies over positive cubefree integers. There is a rational 3-isogeny ϕ from En to E^n:y2=x3−27n2 and a dual isogeny ϕ^:E^n→En. We show that for almost all n, the rank of Selϕ(En) is 0, and the rank of Selϕ^(E^n) is determined by the number of prime factors of n that are congruent to 2mod3 and the congruence class of nmod9.}, author = {Chan, Yik Tung}, issn = {1687-0247}, journal = {International Mathematics Research Notices}, number = {9}, pages = {7571--7593}, publisher = {Oxford University Press}, title = {{The 3-isogeny selmer groups of the elliptic curves y2=x3+n2}}, doi = {10.1093/imrn/rnad266}, volume = {2024}, year = {2024}, } @article{18064, abstract = { We show that the total number of non-torsion integral points on the elliptic curves ED : y 2 = x3 − D2x, where D ranges over positive squarefree integers less than N, is O(N(log N) −1/4+ǫ). The proof involves a discriminant-lowering procedure on integral binary quartic forms and an application of Heath-Brown’s method on estimating the average size of the 2-Selmer group of the curves in this family.}, author = {Chan, Yik Tung}, issn = {1090-2082}, journal = {Advances in Mathematics}, number = {11}, publisher = {Elsevier}, title = {{The average number of integral points on the congruent number curves}}, doi = {10.1016/j.aim.2024.109946}, volume = {457}, year = {2024}, } @article{19487, abstract = {Fix a non-square integer 𝑘≠0. We show that the number of curves 𝐸𝐵:𝑦^2=𝑥^3+𝑘𝐵^2 containing an integral point, where B ranges over positive integers less than N, is bounded by ≪𝑘𝑁(log𝑁)−1/2+𝜖. In particular, this implies that the number of positive integers 𝐵≤𝑁 such that −3𝑘𝐵^2 is the discriminant of an elliptic curve over 𝑄 is o(N). The proof involves a discriminant-lowering procedure on integral binary cubic forms.}, author = {Chan, Yik Tung}, issn = {1432-1807}, journal = {Mathematische Annalen}, number = {3}, pages = {2275--2288}, publisher = {Springer Nature}, title = {{Integral points on cubic twists of Mordell curves}}, doi = {10.1007/s00208-023-02578-x}, volume = {388}, year = {2023}, } @article{19490, abstract = {Abstract. We study integral points on the quadratic twists ED : y2 = x3 − D2x of the congruent number curve. We give upper bounds on the number of integral points in each coset of 2ED(Q) in ED(Q) and show that their total is (3.8)rank ED(Q). We further show that the average number of non-torsion integral points in this family is bounded above by 2. As an application we also deduce from our upper bounds that the system of simultaneous Pell equations aX2 − bY 2 = d, bY 2 − cZ2 = d for pairwise coprime positive integers a, b, c, d, has at most (3.6)ω(abcd) integer solutions.}, author = {Chan, Yik Tung}, issn = {1088-6850}, journal = {Transactions of the American Mathematical Society}, number = {9}, pages = {6675--6700}, publisher = {American Mathematical Society}, title = {{Integral points on the congruent number curve}}, doi = {10.1090/tran/8732}, volume = {375}, year = {2022}, } @article{19491, abstract = {Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let D denote the set of positive squarefree integers having no prime factors congruent to 3 modulo 4 . Stevenhagen [19] conjectured that the density of d in D such that the negative Pell equation x2−dy2=−1 is solvable with x,y∈Z is 58.1% , to the nearest tenth of a percent. By studying the distribution of the 8 -rank of narrow class groups Cl+(d) of Q(√d) , we prove that the infimum of this density is at least 53.8% .}, author = {Chan, Yik Tung and Koymans, Peter and Milovic, Djordjo and Pagano, Carlo}, issn = {2050-5094}, journal = {Forum of Mathematics, Sigma}, publisher = {Cambridge University Press}, title = {{The 8-rank of the narrow class group and the negative Pell equation}}, doi = {10.1017/fms.2022.40}, volume = {10}, year = {2022}, } @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{19492, abstract = {Kuroda’s formula relates the class number of a multiquadratic number field K to the class numbers of its quadratic subfields ki. A key component in this formula is the unit group index (math formular). We study how Q(K) behaves on average in certain natural families of totally real biquadratic fields K parametrized by prime numbers.}, author = {Chan, Yik Tung and Milovic, Djordjo}, issn = {1432-1823}, journal = {Mathematische Zeitschrift}, number = {2}, pages = {1509--1527}, publisher = {Springer Nature}, title = {{Kuroda’s formula and arithmetic statistics}}, doi = {10.1007/s00209-021-02823-6}, volume = {300}, year = {2021}, } @article{19493, abstract = {In 2016, Balakrishnan, Ho, Kaplan, Spicer, Stein and Weigandt produced a database of elliptic curves over Q ordered by height in which they computed the rank, the size of the 2-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over Q whose rational torsion subgroup is isomorphic to Z∕2Z×Z∕8Z. Conditional on GRH and BSD, we compute the rank of 92% of the 202,461 curves with parameter height less than 103. We also compute the size of the 2-Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.}, author = {Chan, Yik Tung and Hanselman, Jeroen and Li, Wanlin}, issn = {2329-907X}, journal = {The Open Book Series}, pages = {173--189}, publisher = {Mathematical Sciences Publishers}, title = {{Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion}}, doi = {10.2140/obs.2019.2.173}, volume = {2}, year = {2019}, } @article{19494, abstract = {Starting from any given rational-sided, right triangle, for example, the (3,4,5)-triangle with area 6, we use Euclidean geometry to show that there are infinitely many other rational-sided, right triangles of the same area. We show further that the set of all such triangles of a given area is finitely generated under our geometric construction. Such areas are known as “congruent numbers” and have a rich history in which all the results in this article have been proved and far more. Yet, as far as we can tell, this seems to be the first exploration using this kind of geometric technique.}, author = {Chan, Yik Tung}, issn = {1930-0972}, journal = {The American Mathematical Monthly}, number = {8}, pages = {689--703}, publisher = {Taylor & Francis}, title = {{Rational right triangles of a given area}}, doi = {10.1080/00029890.2018.1495491}, volume = {125}, year = {2018}, }