Papers

In their 1988 monograph, Conner and Hurrelbrink classified all real quadratic and biquadratic fields with odd class number. They also investigated real quadratic fields whose $2$-class numbers are equal to $2$. Their results, together with some well-known results of Kučera, provide a complete classification of real quadratic number fields with $2$-class number $2$. The main aim of this paper is to provide a complete classification of all real biquadratic fields whose $2$-class numbers equal $2$; and moreover, we draw the complete list all real multiquadratic fields with odd class numbers. Finally, as an appendix we give explicit formulas for $\sqrt{\varepsilon_d}$, where $\varepsilon_d$ denotes the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{d})$.

The quadratic sieve (QS) is an irregular, cache-hostile, branch-heavy integer factorization algorithm; prior GPU work accelerates individual stages of it. We present \swname{}, a self-initializing quadratic sieve in which every stage — polynomial initialization, sieving, relation post-processing, large-prime matching, \GFtwo{} matrix construction, Block Wiedemann linear algebra and the square root — executes on the GPU, the CPU restricted to orchestration, setup and I/O, the sieve window 99.9% GPU-busy with no device-wide or event synchronization and no synchronous copy. With it we factored the 512-bit RSA-155 challenge modulus end to end on GPUs: a 64$\times$H100 sieve feeding a single-H100 \GFtwo{} solve, 700.6~GPU-h and 242~kWh in total, 98.4% of it sieving. To our knowledge this is the largest integer factored by the quadratic sieve, exceeding the prior QS record RSA-150 (Buhrow, YAFU, 2025); all comparable general factorizations since 1996, RSA-155's own in 1999 included, used the number field sieve. On a single device we factor RSA-100 in 29.2~s on an H100 and 51.0~s on a consumer RTX~5070~Ti; in a controlled same-modulus measurement, with energy measured on both sides, the H100 is faster than the fastest CPU quadratic sieve, YAFU, on 96 cores of an EPYC~9655 (106–122~s), and than CADO-NFS's SIQS (292~s) and GNFS (263~s). Across four discrete GPUs (a 3.7$\times$ range in CUDA cores) RSA-100 throughput is close to proportional to core count, memory bandwidth being a weaker predictor, and measured cost over seven benchmark factorizations from RSA-100 to RSA-155 grows as $\LN{1/2}{1}^{1.04}$ ($R^2=0.993$). On RSA-150, the identical modulus on which the prior QS record was set, we measure 302.9~GPU-h against that run's 11{,}664 core-hours.

In a recent breakthrough, Fauzan proved the irrationality of $\zeta(5)$. In this paper, we combine Fauzan's determinant method with classical simultaneous approximations of zeta values studied by Zudilin and Dauguet–Zudilin to prove the linear independence of $1$, $\zeta(3)$, and $\zeta(4)$ over $\mathbb{Q}$. As a corollary, $\zeta(3)/\pi^4$ is irrational.

We spread out geometrically irreducible perverse sheaves. Over an arithmetic base scheme, such sheaves extend from the generic fiber to a relatively perverse, universally locally acyclic sheaf over a dense open of the base scheme. As an application, we prove the arithmetic Kashiwara conjecture by Esnault and Kerz for geometric traits in equicharacteristic and generalize the decomposition theorem for arithmetic complexes to fields finitely generated over a separably closed field. We also recover the Hard Lefschetz theorem for arithmetic perverse sheaves.

We show that the two modular curves of level $49$ and genus $9$ left open by the work of Rouse, Sutherland and Zureick-Brown, and of Furio and Lombardo, have no non-CM rational points. Together with the theorem of Furio and Lombardo on $X_{ns}^{+}(49)$, this completes the classification of the $7$-adic images of Galois of non-CM elliptic curves over $Q$. The proof is a Kummer descent on the superelliptic equations $F(x,y) = k\,w^{7}$ of Furio and Lombardo, in the cases $7 \mid k$ that they left open. The covering curves are twists of the Fermat septic, they map to twists of the Klein quartic, and a $7$-adic computation shows that the twist attached to a solution is trivial. The rational points of the Klein quartic were determined by Hurwitz, who reduced the question to Fermat's Last Theorem for exponent $7$, proved by Lamé. Thus, the last open case of the $7$-adic part of Mazur's Program B rests on Fermat's Last Theorem for exponent $7$.

Let $\mathcal{A}\to S$ be an abelian scheme over a normal quasi-projective variety over a number field, and let $\mathcal{X}\subseteq\mathcal{A}$ be a positive-dimensional subvariety. For every integer $t>0$, we prove that, outside the $t$-degeneracy locus of $\mathcal{X}$, at most finitely many points lie in flat group subschemes of relative dimension less than $t$. In particular, this intersection is not Zariski dense whenever $\mathcal{X}$ is $t$-nondegenerate. The proof combines a bounded height theorem with uniform large Galois orbit estimates, o-minimal point counting, and mixed Ax–Schanuel, following the Pila–Zannier strategy. The arithmetic input is an explicit bound for the complexity of endomorphism relations, obtained from geometry of numbers and height estimates for abelian varieties. As applications, we formulate a relative Mordell–Lang conjecture, prove it for section images in powers of certain abelian schemes, and establish injectivity of specialization of the group of sections outside a proper closed subset under variation and dimension hypotheses.

In this paper we give an analytic proof of the Two-point Chowla conjecture for the truncated Möbius function $\mu_M$ for the function field case in the fixed $q$ setting. We reformulate the problem into studying the distribution of $(\omega_M,\omega_M\circ T_a)$. Thus we develop a uniform bivariate Erdős-Kac theorem in terms of the characteristic function for small pair of real numbers $(u,v)$ via the Kubilius model. For this purpose we also develop and employ the pure Brun sieve for polynomials in this paper.

We construct Golod–Shafarevich towers of arithmetic surfaces over the integers. In particular, we obtain an arithmetic surface with a section and infinite geometric etale fundamental group, answering a question raised by Bost and Charles about the existence of such surfaces. Taking generic fibres gives curves over the rationals with infinite geometric etale towers in which a rational point splits completely. This answers questions posed by Ihara and by Frey, Kani and Völklein. Taking the generic fibres of the tower, we also obtain sequence of curves over the rationals whose genera goes to infinity and whose logarithmic conductors grows no more than linearly in the genera, answering a question asked by Venkatesh at PCMI in 2022. The authors worked together on this problem since 2022 and developed a general strategy. They were able to bring it to fruit only after a collaboration with GPT5.6 Sol, Fable and Aletheia a Gemini-powered internal agent developed at Google DeepMind.

Let $q=2^n$ with $n\ge6$, and let $\mathbb{F}_q$ be the finite field with $q$ elements. For a subspace $E$ of $\mathbb{F}_q$, define \[ S(E)=\sum_{x\in E\setminus\{0\}}x^{-1}.\] Let $N_{n,k}$ denote the number of $k$-dimensional $\mathbb{F}_2$-subspaces $E$ for which $S(E)=0$. We prove that, if $k\ge3$ and $n\ge2k+1$, then \[ \left|\frac{2^nN_{n,k}}{\genfrac{[}{]}{0pt}{}{n}{k}_{2}}-1\right|<2^{r^2+r}2^{n(1-r/2)},\] where $r=\lfloor\frac{k-1}{2}\rfloor$, and $\genfrac{[}{]}{0pt}{}{n}{k}_{2}$ is the Gaussian binomial coefficient. In particular, \[ N_{n,k}\sim2^{-n}\genfrac{[}{]}{0pt}{}{n}{k}_{2}\quad (n\to\infty)\] uniformly for $n/3<k\le(n-1)/2$. Combined with the known low-dimensional cases and the symmetry between dimensions $k$ and $n-k$, and the elementary middle-dimensional subfield construction, this proves a conjecture of Carlet: for every $3\le k\le n-3$, there exists a $k$-dimensional $\mathbb{F}_2$-subspace $E$ such that $S(E)=0$.

We prove the heuristically predicted lower bound for the number of rational points of height at most $Q$ lying within $\epsilon/Q$ of a fixed analytic nondegenerate manifold in $\mathbb{R}^n$, provided that $\epsilon \asymp Q^{-\tau}$ for some $\tau \leq 3/(2m+1)$, where $m$ is the codimension of the manifold. Our result establishes the lower bound well beyond the previously conjectured range $\tau \leq 1/m$, and improves upon a recent result of Schindler, Srivastava, and Technau, who established the same lower bound for $\tau \leq 3/(2n-1)$.

Let $d\geq2$ be an integer, and let $\operatorname{MPoly}^d$ be the moduli space of degree-$d$ polynomials. For every number field $K$, we prove that there exists $\epsilon_{K,d}>0$ such that \[ \left\lbrace\alpha\in\operatorname{MPoly}^d(K^{\mathrm{ab}})\colon h_{\mathrm{crit}}(\alpha)<\epsilon_{K,d}\right\rbrace=\left\lbrace\alpha\in\operatorname{MPoly}^d(K^{\mathrm{ab}})\colon h_{\mathrm{crit}}(\alpha)=0\right\rbrace \] is finite, where $h_{\mathrm{crit}}$ is the critical height. In particular, only finitely many $K^{\mathrm{ab}}$-rational points of $\operatorname{MPoly}^d$ are postcritically finite. The proof combines the correspondence method of Ji–Song–Xie with local ramification estimates and the uniform finiteness of intertwined classes in $\operatorname{MPoly}^d$. We also prove an analogous finiteness theorem for non-preperiodic marked points in non-isotrivial one-parameter polynomial families.

We study theta series with half-integral characteristics attached to odd positive-definite unimodular lattices. In even rank, we get non-vanishing theta series with odd characteristics. We emphasize ranks divisible by four, where the involved rational lattices are integral unimodular neighbors.

We prove that a smooth irreducible proper variety over a number field admits polylogarithmic-size coverings of rational points of bounded height if it admits a tower of finite étale coverings satisfying several conditions. One of the conditions is a height comparison on twists of the finite étale covers. We prove that these conditions hold for abelian varieties.

In this paper, we introduce and study a special function, referred to as a variant of the fractional gamma-regularized zeta kernel, in which the lower incomplete gamma function appears in the numerator, contrasting with the fractional hypergeometric zeta function, where it appears in the denominator. We investigate its analytic properties, convergence behavior, and its connection to the Riemann zeta function. A representation is derived in terms of a multiplicative modification of the Riemann zeta function, and comparisons are made with known zero-free regions and deformation phenomena. We demonstrate that this function defines a deformation of the Riemann zeta function, which interpolates to the alternating zeta function in a specific limit. Additionally, a Dirichlet-type series representation is provided. Potential extensions, including a functional equation and asymptotic analysis, are proposed for future work.

We study the extent to which the invariants appearing in the Birch and Swinnerton-Dyer conjecture remember the elliptic curve, and we give several affirmative answers to this question. For example, the real period $\Omega_E =\frac{\Gamma(1/4)^2}{2\sqrt{\pi}\,17^{1/4}}$ and the Tate–Shafarevich group $ \Sha(E/\mathbb{Q})\cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ remember the elliptic curve $E: y^2=x^3+17x$. Let $K$ be a number field admitting a real embedding, and fix one such embedding $\sigma\colon K\hookrightarrow\mathbb{R}$. We first prove that, for an elliptic curve $E$ over $K$, the real period of $E$ attached to $\sigma$ determines the $K$-isogeny class of $E$. We then prove that an elliptic curve $E$ over $\mathbb{Q}$ is determined up to isomorphism over $\mathbb{Q}$ by its real period, the number of connected components of $E(\mathbb{R})$, and the real period of the quadratic twist $E^{D}$ by a negative square-free integer $D\equiv 1 \bmod 4$ coprime to the minimal discriminant of $E$. Moreover, over $\mathbb{Q}$, both statements already hold when the real periods are known only to sufficiently many decimal places, where the required precision depends on $E$ (and $D$).

We give an example of a proper, regular, integral arithmetic surface over $\mathrm{Spec}\ \Z$ endowed with a section, and whose étale fundamental group admits an infinite pro-$2$ quotient. The construction is a geometric analog of the Golod–Shafarevich towers for rings of integers of number fields.

An important role in the theory of conflict-avoiding codes is played by a quantity involving the multiplicative order of $4$ modulo $d$, with $d$ running over all positive divisors of an odd integer $n$. We establish results on its distribution as $n$ varies over the odd prime numbers, respectively odd integers, using techniques from the study of Artin's primitive root conjecture. We also point out some other interpretations of this quantity, with one of them making a connection with the factorization of $X^n-1$ into irreducibles over the finite field $\mathbb F_2$.

We prove that the Hodge polygon of the generic fiber of a proper smooth $p$-adic formal scheme over $\mathcal O_C$ lies on or above the generalized Hodge–Witt polygon of its special fiber.

Given a finite Cartesian set $S=X \times Y$ and a decreasing set of monomials $\mathcal M$, we call extremal polynomials for $\mathcal M$ over $S$ those that have the maximum number of zeros in $S$ and whose support belongs to $\mathcal M$. Coordinate factorizations give a family of extremal polynomials; we call them canonical. If $\max(\mathcal M)$, taken with respect to divisibility, is a single monomial, all extremal polynomials are canonical. If $|\max(\mathcal M)|=2$, either all extremal polynomials are canonical, or the problem reduces to the case where $\max(\mathcal M)=\{x^{d_x},y^{d_y}\}$. In the latter case, we prove that the existence of noncanonical extremal polynomials depends on finding families of subsets whose elementary symmetric functions agree. This condition is more restrictive than the classical Prouhet–Tarry–Escott problem, which asks for two sets whose elementary symmetric functions agree. We determine the number of triples $(X, Y, h)$, where $h$ is a quadratic noncanonical extremal polynomial. We apply extremal polynomials to coding theory via minimum-weight codewords.

We prove the automorphic commutator relation conjectured by Liu and Wang arXiv:2504.00275, which is the geometric input for the automorphic Clifford relations in their framework relating higher period integrals to higher derivatives of $L$-functions over function fields. The relation was previously established only under additional cohomological vanishing assumptions and for curves of genus different from one; we remove these assumptions. The main new ingredient is a "universal-local" stage between the local and the semi-local settings: we introduce the classifying stack of formal multidisks with two ordered marked sections, and we construct universal-local analogues of the Hecke stacks, unit objects, Hecke actions and special cohomological correspondences. Fusion and purity on this stack yield a universal-local commutator identity, which is proved using only the local assumptions on the Plancherel algebra that enter the conjecture. Specialization to a curve and the six-functor formalism then give the conjecture for every smooth projective curve.

Let $\varphi$ be the golden ratio and let $R_n=\{\lfloor k\varphi^n\rfloor : k\ge 1\}$ be the $n$-th row of the array $T(n,k)=\lfloor k\varphi^n\rfloor$ (OEIS A128440). In 2022 Kimberling conjectured that the rows $R_{2n-1}$ and $R_{2n}$ are disjoint, and that after the two rows are merged and each entry is replaced by its rank, they become the lower and upper Wythoff sequences. He also conjectured (OEIS A358359) that if $a(N)$ is the number of rows containing $N$, then every positive integer occurs infinitely often among the values of $a$. We show that the first two conjectures follow quickly from the Skolem-Bang theorem, which also yields the exact rule for when two rows are disjoint: $R_i\cap R_j=\emptyset$ ($i<j$) if and only if $j-i$ is odd and divides $i$. We then prove the third conjecture. The main tools are an explicit determination of the rows containing an odd-indexed Lucas number, which extends a result of Noppakaew, Kanwarunyu and Wanitchatchawan, and a "Lucas shift" lemma: if $N+1$ is not of the form $L_{2e}$ with $e\ge1$, then adding a sufficiently large even-indexed Lucas number to $N$ does not change the set of rows containing it. We also show that each value of $a$ is taken on a set of positive natural density, and we report computations up to $10^8$ suggesting that the least $N$ lying in exactly $v\ge 2$ rows is the Lucas number $L_{4v-5}$.

We construct arbitrarily large finite sets $A$ of real algebraic integers such that both $|A+A|$ and $|AA|$ are at most $|A|^{1.95835}$. The construction combines truncated ideal-smooth $S$-unit fibres with a coprime additive factor. A tensor-product rank argument, using a two-dimensional local feature at each selected prime ideal, controls the loss in the additive factor, while a direct estimate for an outer parallel body improves the sumset packing bound. The arithmetic input is an unramified pro-$2$ tower over a known degree-ten field, with simultaneous Frobenius cuts controlling the small prime ideals. We use unconditional Tsfasman–Vlăduţ inequalities for the joint class-number–regulator cost. All finite numerical comparisons entering the exponent are certified by outward rational interval arithmetic. No unproved hypothesis is used.

In this paper, we provide an algebraic proof of the condition for the existence of modular form solutions to the Kaneko-Zagier differential equation of weight $k$. Unlike previous representation-theoretic approaches relying on $SL_2(\mathbb{Z})$, our method employs the connection matrices of the principal congruence subgroup $\Gamma(N)$. By deriving a condition for simultaneous triangularizability from the commutators of the representation matrices and applying Galois theory, we prove that the associated two-dimensional representation of $\Gamma(N)$ is irreducible when the denominator $m$ of the fraction $(k+1)/6 = n/m$ satisfies $m=1$ or $m \ge 7$. Consequently, we identify the weights that admit no modular form solutions and obtain a complete classification of the dimensions of the spaces of such solutions.

Let $s(n):=\sum_{d\mid n,\,d < n} d$ denote the sum of the proper divisors of $n$. Distinct positive integers $n,m$ form an amicable pair if $s(n)=m$ and $s(m)=n$. Let $A(x)$ count the positive integers not exceeding $x$ that belong to an amicable pair. We report on a proof, found by ChatGPT Astra, that $$ A(x)\le x\exp\{-(1/2+o(1))\log x\,\log_3 x/\log_2 x\}$$ as $x\to\infty$. Here $\log_k$ denotes the $k$th iterate of the natural logarithm.

This work is devoted to a systematic study of slope stability for $\mathbf Z_p$-towers of the projective line over finite fields of characteristic $p$, ramified only at infinity. Our aim is to understand to what extent geometric stability of the genus sequence, or of the ramification break sequence, implies arithmetic stability of the Newton slopes of the zeta functions of the curves in the tower. We prove positive results and construct counterexamples that clarify the boundary between slope stability and its failure, with sharp bounds in several cases. For towers with eventual minimal ramification break ratios, a condition stronger than genus stability, Kosters and Zhu proved slope stability under an additional degree-gap condition. We construct towers with minimal break ratios that are not slope stable, answering negatively both their question of whether the gap can be removed and their broader question of whether genus stability implies slope stability. Under the minimal-break hypothesis alone, we prove exact partial slope stability on growing intervals approaching the first block, with uniform comparisons over explicit boundary annuli and finite flat spectral pieces. Further counterexamples distinguish partial from complete first-block propagation, first-block from two-block propagation, and full classical stability from a uniform annular slope law below one. We obtain improved bounds for the classical degree-gap constant, the partial-stability loss, and the sufficient starting levels, all independent of the constant-field degree. The proof uses a separate precision theorem for the semilinear Frobenius on finite contact blocks.

We study the set of integers $n \le x$ for which $\sum_{i=1}^{n}\frac{1}{\gcd(n,i)}$ takes an integer value. We show that the number of such integers is $\ll x^{2/3+\epsilon}$ for all fixed $\epsilon > 0$.

We prove that the absolute Galois group of every discretely valued field of mixed characteristic is ab-faithful. We also prove ab-faithfulness for higher local fields of positive dimension and give further examples using finite split embedding problems. Finally, we give a formal Hilbert~90 criterion for center-freeness and slimness of finite-step solvable quotients. We apply this criterion to ab-faithful absolute Galois groups and obtain slimness for several classes of fields, including higher local fields of positive dimension.

For an arbitrary prime $p$ and arbitrary positive integers $r$ and $d_{1}, \ldots, d_{r}$ with $d_{r} > \cdots > d_{1}$, we show that $x^{d_{r}} + x^{d_{r-1}} + \cdots + x^{d_{2}} + x^{d_{1}} + p$ removed of all of its monic irreducible reciprocal factors is irreducible. As a consequence, we establish a number of results related to a conjecture of Charles Nicol that, over the rationals, the sum of two cyclotomic polynomials $\Phi_{n}(x) + \Phi_{m}(x)$ is a product of cyclotomic polynomials times either $2$ or an irreducible polynomial, where $n$ and $m$ are integers exceeding $1$. We also establish similar results for $\Phi_{n}(x)\Phi_{m}(x)+1$.

We study triples of distinct nonconstant polynomials $a,b,c$ over $\mathbb{R}$, $\mathbb{Q}$ or $\mathbb{Z}$ for which $$ ab+1,\qquad ac+1,\qquad bc+1,\qquad abc+1 $$ are all polynomial squares. We show that, up to permutation and a common invertible affine change of variable, there is exactly one such triple of quadratic polynomials in $\mathbb{R}[X]$, the one arising from a recent construction of Jurasić, and that none exists in $\mathbb{Q}[X]$ or $\mathbb{Z}[X]$. Moreover, for arbitrary nonconstant entries over $\mathbb{R}$ or $\mathbb{Q}$, we prove that the degree pattern $(2n,2m,2l)$ with $1\le n\le m\le l$ satisfies either $l=m$ or $l>n+m$. For the case $l=m$, we derive factorization systems and reducibility restrictions. In the equal-degree case over $\mathbb{Q}$, we further show that every entry has at least two distinct irreducible factors. As an application, we show that in a natural regular parametrized family of polynomial Diophantine triples in $\mathbb{Q}[X]$ the fourth expression $abc+1$ is never a square when the parameter is constant or the first entry is a square.

We establish sharp mean value estimates for various exponential sums lacking translation-dilation invariance properties. For example, suppose that $k\ge 1$ and $s>0$. Then, for each $\varepsilon>0$, one has \[ \int_{[0,1)^k}\biggl| \sum_{1\le h\le X}\sum_{1\le z\le X}e(hz\alpha_1+\ldots +hz^k\alpha_k)\biggr|^{2s}\,{\rm d}\boldsymbol \alpha \ll X^\varepsilon\bigl( X^s+X^{2s-k(k+3)/2}\bigr), \] confirming the main conjecture for a new family of exponential sums. Our methods are based on the application of very recent work of the author employing the nested efficient congruencing method.