Number Theory Papers

ALT We extend Dill's generalized Vojta-Rémond inequality to proper adelic curves of characteristic zero, under an explicit Siegel hypothesis. The geometric case is closely related to recent work on Mordell-Lang plus geometric Bogomolov for semi-abelian varieties. Our formulation retains Dill's generality for products of distinct projective varieties. The appendix establishes the Siegel input for arithmetic function fields with nef and big polarizations on the model.

ALT Let H=k(x,y) be the Hermitian function field yq+y=xq⁺¹ over k=Fq², and let G be a subgroup of its k-automorphism group PGU(3,q). We give a uniform construction of a plane model for Hᴳ. For each of x and y, we multiply its distinct translates under the stabilizer S in G of their common pole, then sum the translates of that product over G/S. The two resulting functions U,V generate Hᴳ. In Hᴳ, their only pole is the restriction of the common pole, with coprime orders a=q/|Gₓ|and b=(q+1)/|Gy|, where Gₓ,Gy are the subgroups of G fixing x,y, respectively. Their plane equation has degree max(a,b)≤ q+1 and can be recovered from finitely many Laurent coefficients of U,V at the common pole of x,y. All steps are effective given matrix generators of G. The total delta invariant of this affine model is g(Hᴳₓᴳʸ)-g(Hᴳ)=¹⁄₂(a-1)(b-1)-g(Hᴳ). Consequently g(Hᴳ)≤¹⁄₂(a-1)(b-1); when g(Hᴳ)≥2, the model is smooth if and only if G=GₓGy. The generator construction also applies to other function fields admitti

ALT We provide an (in our opinion) simpler proof of the main results of [Stoll1992,Li2020,Li2021] on the question for which integers a the Galois groups of all iterates of x² + a are maximal (equivalently, the associated arboreal Galois representation is surjective). At the same time, we extend the results to rational parameters a. The main result states that for a subset of rational numbers of relative density 1/2 in the complement of a certain open interval, the maximality of the Galois group of the second iterate is sufficient for the arboreal Galois representation to be surjective. We also discuss experimental evidence suggesting that a similar statement involving the fourth iterate may hold in general. The results were obtained in collaboration with Claude (Anthropic's AI model); they are also formalized in Lean (except for the experimental evidence).

ALT Let F=\F₂^n and let T₁={t∈ F:\Tr_F/\F₂(t)=1}. For an integer u=\sum_i=0^n-1u_i2^i of even binary weight, put L_u(X)=\sum_i=0^n-1u_iX²^i, L₁,u(t)=t+L_u(t^-1). Hollmann and Xiang conjectured that L₁,u permutes T₁ if and only if u∈{0,3,6,10}. We prove the necessity by a moment method; the sufficiency is their Theorem 4.1. The quadratic moment yields a cyclic autocorrelation identity, while the cubic moment gives a two-variable group-algebra identity. An elimination argument reduces the coefficient word to a monomial or a complementary monomial, and explicit cubic and quartic moments complete the classification. We also prove Hollmann and Xiang's Conjecture 4.5. Namely, if n>3 and L_c,d(x)=Lc(x)+Ld(x^-1) is a Kloosterman polynomial on \F₂^n, then its positive linearized part Lc is injective on both trace fibers. This second result follows from relative quadratic and cubic moment identities and a spectral-zero rigidity argument. Consequently every Kloosterman polynomial is obtained by compo

ALT For any real number x, let ‖x‖ denote the distance from x to the nearest integer, and let α be any irrational number. We prove that, for Y=Xᵞ with 7/12<γ≤ 1 and 0<θ<min {(12γ-7)/24, 1/9}, there exist infinitely many X for which there are primes p satisfying ‖αp‖<p⁻ᶿ,  X-Y<p<X. Moreover, we obtain an asymptotic formula for the number of such primes. Our result extends the admissible range of short intervals in the corresponding result of Baier and Roy from 2/3<γ≤ 1 to 7/12<γ≤ 1, reaching Huxley's classical range for primes in short intervals and also improves the exponent θ in 2/3≤ γ≤ 3/4.

ALT We associate algebraic Hecke characters to certain invertible motives for absolute Hodge cycles with coefficients in a number field K, reversing a viewpoint of Schappacher. These motives N include abelian motives and a new notion of algebraic motive. This Hecke character determines the complex and the p-adic periods of N, and is determined by the latter; this gives a proof of the Mumford-Tate conjecture for N. When K and N are abelian, we prove a refinement of the Gross-Deligne conjecture, as well as l-adic and p-adic variants. We also formulate a version of the algebraic relations between rational values of the gamma function given by Koblitz and Ogus in terms of a distribution with values in the Picard group of the category of (abelian) Chow motives.

ALT The so-called 𝐶𝑎𝑡 𝑀𝑎𝑝, which is a linear map generated by symplectic matrices A ∈ Sp(2d,ℤ) modulo an integer N≥ 1 is a classical discrete model of quantum ergodicity. For d=1 it has been investigated by P. Kurlberg and Z. Rudnick (2001), J. Bourgain (2007) and A. Ostafe, I. E. Shparlinski and J. F. Voloch (2023). The significantly more difficult case of d>1 has been studied by P. Kurlberg, A. Ostafe, Z. Rudnick and I. E. Shparlinski (2025). All these works apply only to almost all N and are unable to provide an explicit sequence of such “good” values of N. The authors (2026) have given such a construction with a polynomial decay on the corresponding matrix coefficients. Here we use some new ideas and improve this bound.

ALT We study the constraints that a hypothetical least Colossally Abundant (CA) counterexample to Robin's Inequality would place on the non-trivial zeros of ζ(s). In the single-prime case we express the least CA counterexample N as M⋅ p, where M is its CA predecessor and p is prime. Using the von Mangoldt explicit formula for ψ(x), we express ln p in terms of a sum over the non-trivial zeros of ζ(s) and split that expression into its baseline contribution ℛ₀(p) and its off-line distortion. When both converge, we show that the off-line distortion is negative whenever 1+1/(e^ℛ₀(p)+e²ℛ₀(p)+⋯+eᵛ⁺¹ℛ₀(p))≤ln(ln M+ℛ₀(p))/lnln M, where v is the p-adic valuation of M. In the case where the least CA counterexample is the product of its CA predecessor and two primes (the semiprime case), the logarithm of the product of the two primes is the sum of two von Mangoldt expansions. An analogous inequality implies that at least one of the off-line distortions of the two expansions is negative.

ALT We disprove a 1939 conjecture of Erdős and Mahler by constructing, for every real τ≥4, a Cantor set of numbers of irrationality exponent τ, each having infinitely many regular continued-fraction convergents of the form 2ᵃ/3ᵇ.

ALT The Bromberg-Shpilrain-Vdovina hash functions are a family of Cayley hash functions indexed by an integer parameter k≥ 2. Hashes are matrices in SL₂(p) where p is a large prime. We present an attack which finds collisions of length O(log p) whose average-case time complexity is O(p⁰.²⁸) when k=2 and O(p⁰.³⁶) when k=3 under an assumption in the spirit of Zaremba's Conjecture on continued fractions. This is an improvement on the more general attacks of Mullan-Tsaban and Slaughter which have average-case time complexity O(√p). For any k, the average-case time complexity of our attack is O(p¹⁻ᵟₖ⁺ᵒ⁽¹)) where δₖ is the Hausdorff dimension of a certain Cantor set. This tends to O(√p) as k→∞, so our attack has time complexity comparable to Mullan-Tsaban and Slaughter's attacks as k grows. We give examples of collisions for large primes found by our attack when k=2, 3.

ALT We prove that the F₅-gonality and the ℚ-gonality of the modular curve X₁(42) are both 12, improving the previous lower bound of 10. Since J₁(42)(ℚ) is finite, it follows that 12 is the least degree in which X₁(42) has infinitely many points. The lower bound comes from an exhaustive search over Riemann–Roch spaces, following Derickx and van Hoeij, using a new reduction by the orbits of a cyclic group of automorphisms of order 12 on places of degree at most 5 (743 231 Riemann–Roch spaces). An earlier search with the reductions from van Hoeij's code (about 3 million Riemann–Roch spaces, about 6 times more CPU time) gives the same result and serves as supporting evidence. We prove the reductions and record the data needed to reproduce the computations.

ALT Applying the Dirichlet principle to the Weil quadratic form QWλ on L²([λ^-1,λ],d*u)–using only the primes p≤λ² of the Euler product–yields canonical finite-dimensional Hermitian matrices whose ground state has a Mellin transform with zeros proved real in full generality when the lowest eigenvalue is simple and even. While λ²=13 gave 50 zeros of ζ, λ²=250 (λ\sim16) gives the first 1000 zeros with >500 correct decimals (the first with 1341). We explain this via a correspondence between QWλ and Slepian's time-frequency uncertainty operator Uλ=2-Pλ-P̂λ, diagonalized by prolate wave functions with eigenvalues δ_n(λ)\sim c_nλ²n+1e^-4πλ² (Fuchs). Via ℰ(f)(u)=u¹/2\sum_n≥1f(nu) and Gevrey smoothing, ελ≤ Cδ₄(λ)(logλ)³ is a theorem (Connes-Kastoris). Computations up to λ²=300 (ελ(1)\sim10^-1618) show a striking logarithmic coincidence with the Fuchs ladder, confirming this simplicity and evenness. Prolate vectors reproduce all minuscule eigenvectors, and conditioning by them and the pole term rem

ALT We classify modular Nahm sums with a rational symmetric positive-definite two-by-two matrix and rational shifts. Up to coordinate exchange, every modular datum belongs to three explicit rational-point families, is a product of two modular rank-one sums, or is one of thirty-four exceptional triples on twelve matrices. The list consists of Zagier's rank-two examples, the Vlasenko–Zwegers additions and split products; no further data occur. The proof first bounds the joint degree of the distinguished Nahm point by four, using an initial degree bound of twenty-four and exact local exclusions at the prime 7. Explicit height and exponent bounds then give exhaustive arithmetic classifications in degrees one through four. Exact polynomial elimination determines all rational shifts, and published identities of q-series establish modularity of the survivors. The degree bound is sharp. The finite steps are accompanied by exact certificates, verification programs and independently implemented chec

ALT For finite sets in a commutative semigroup, Gyarmati, Matolcsi and Ruzsa proved |A₁+…+A_n|^n-1 ≤ prod_i=1^n|A₁+…+A_i-1+A_i+1+…+A_n|. They asked whether in an arbitrary group |A₁⋯ A_n|^n-1 ≤ prod_i=1^n max_a∈A_i |A₁⋯ A_i-1 a A_i+1⋯ A_n|, where each omitted set is replaced by the largest slice obtained by fixing one of its elements. Nair, Namayandeh and Zhu proved that this inequality does not hold when n=4 and G=D₄¹4, of group order 8¹4. We develop its double-coset mechanism and construct counterexamples in many other groups. For n=4, the smallest group found has order 518 400, namely A₆²× C₂²; no absolute minimality is claimed. For n=6 there is a witness in a group of order 1728, for n=10 a class-three 3-group of order 243 suffices, and for n=4 a single class-two 2-group of order 2²3 works. We also obtain torsion-free examples. The violation ratio R_n=|A₁⋯ A_n|^n-1/prod_iN_i is greater than one exactly when the inequality fails; explicit four-set constructions attain R₄≈1.29 in A₈², an

ALT For every fixed integer k≥2 and every fixed nonzero integer h, we prove that, as x→∞, ∑ₓ<ₙ≤₂ₓΛ(n)τₖ(n+h) ∼ Cₖ,ₕx(log x)ᵏ⁻¹, where τₖ is the k-fold divisor function and Cₖ,ₕ>0 is a constant depending only on k and h.

ALT In this paper, we propose a comparison of relative trace formulas to study the Gan–Gross–Prasad conjecture. We unfold the relative trace formulas, match their regular orbits, and formulate the corresponding fundamental lemma and smooth transfer conjecture. Finally, we prove the fundamental lemma for the unit element of the Hecke algebra.

ALT We study linear and algebraic independence of Hasse–Weil L-functions of abelian varieties over ℚ, with Dirichlet series as coefficients. For nonzero varieties that are pairwise non-isogenous over \̅mathbbQ, we prove that their L-functions with arbitrary Dirichlet twists, together with all their derivatives and 1, are linearly independent when the coefficient series use only primes from a set of natural density zero. For untwisted L-functions, this independence is equivalent to the condition that the varieties are pairwise non-isogenous over ℚ. If the individual monodromy groups are connected, the result for varieties that are pairwise non-isogenous over \̅mathbbQ extends to larger coefficient rings: the prime supports of coefficients are unrestricted outside a prescribed set of positive upper natural density and meet it in a set of natural density zero. We also characterize algebraic independence of untwisted L-functions in terms of the multiplicities of the simple isogeny factors of t

ALT We classify the genera of primitive positive definite ternary quadratic forms in the extremal cases of Lehman's level and discriminant conditions. The genera are parametrized by their level, discriminant, and genus conductor, which also distinguishes local forms having the same isotropy. Using the correspondence with quaternion orders of level (N₁,N₂), we express their weighted representation numbers and class numbers in terms of generalized modified Hurwitz class numbers. The formulas allow arbitrary admissible prime power exponents and include the additional genera obtained by Watson's transformation. As an application, we determine all 255 one-class genera in these families and give an explicit representation formula for each. Of these formulas, 110 concern nonsquarefree N₁N₂.

ALT We prove weighted cancellation under small-value conditions on Hecke eigenvalues of finitely many non-CM holomorphic newforms. We also obtain many squarefree integers of each sign with fixed normalized amplitude, simultaneously small Hecke eigenvalues, and a fixed bound on the number of prime factors, uniformly in growing reduced residue classes. Our questions are motivated by Xi's noncorrelation conjectures and the mixed-moment estimates of Jiang and Lau. Their estimates imply that small Hecke eigenvalues can occur together with non-small Kloosterman values. Our result retains this property with a fixed prime-factor bound, both signs, and the residue conditions. Using OpenAI's zero-free half-plane theorem for Dirichlet L-functions, we give an example with at most 83 prime factors, normalized amplitude at least 10⁻²¹, and moduli M≤ C¹¹/²⁰⁰.

ALT We prove Motohashi's conjecture on primes of the form u²+v²+1, with Iwaniec's corrected constant. Let b denote the characteristic function of the positive integers representable as a sum of two squares. Then, as x→∞, \sum_p≤xb(p-1)=κSprx/(log x)³/2 +O(x/(log x)³/2√(loglog x)), where κ is the Landau–Ramanujan constant and \Spr=prod_{p≡3 (mod 4)}(1-1/(p(p-1))). The proof combines auxiliary prime factors with a mean square estimate for bilinear sums. We obtain the minor arc bound through harmonic switching and high moments, replacing arithmetic divisibility conditions by independent projective lines. We remove the auxiliary weights using Selberg's sieve.