Papers

We show that the set $\mathbb{P}-1$ of shifted primes contains infinitely many sum-product patterns of the form $\{x,x+y,xy\}$ with $x,y$ arbitrarily large distinct integers. More strongly, we can also show that, for any $k\geq 1$, the set $\mathbb{P}-1$ contains longer patterns of the form $\{x,x+y,\ldots, x+ky,xy\}$ with $x,y$ arbitrarily large distinct integers, a statement that contains the Green–Tao theorem as a special case.

Ribet's 1976 proof of the converse to Herbrand's theorem created a new paradigm in algebraic number theory by illustrating that unramified abelian Galois extensions of number fields can be constructed using Galois representations associated to Eisenstein congruences, i.e., cuspidal eigenforms that are congruent to Eisenstein series. In recent years, further progress has been made in investigating the quantity of such Eisenstein congruences, showing that when there are many such congruences, additional and finer information can be gleaned about the arithmetic structure of the related extensions. In this vein, we use new results on the existence of Eisenstein congruences in tame families to explain a computational observation about splitting behavior of primes in a certain metabelian number field.

We establish a criterion for finiteness of the depth 3 Chabauty–Kim set $X(\mathbb{Q}_2)_3$ for hyperelliptic curves $X$ over $\mathbb{Q}$ with a rational Weierstrass point. To illustrate its applicability, we include an example of a genus $2$ curve satisfying the criterion whose Jacobian has Néron–Severi rank $1$ and Mordell–Weil rank $4$.

Gosper introduced many remarkable formulas involving infinite products. One such formula provides an evaluation of the product $$\prod_{n \geq 1} \frac{1}{e}\left(\frac{1}{3n} + 1\right)^{3n + \frac{1}{2}}. $$ It appears that a complete proof of this evaluation has not appeared in the literature. In this paper, we introduce some techniques to extend Gosper's evaluation. We prove a generalization of Gosper's formula involving three free parameters, which can also be applied to evaluate arbitrary multisections of Gosper's product. Then we investigate higher-order extensions involving higher-degree polynomials in $n$, relative to the polynomials $3n$ and $3n+\frac{1}{2}$ involved in Gosper's product, yielding a generalization of an infinite product evaluation due to Ramanujan and recently studied by Bradley–Thrush [\it Ramanujan J. (2025)]. We also consider connections with infinite products due to Kurokawa and Rovinskiĭ, building on the recent work of the first and fourth authors [\it Ramanujan J. (2025)].

We establish sharp bounds for rational points near compact smooth nondegenerate curves in $\mathbb R^n$, for every $n\geq4$. For a nondegenerate curve $\mathcal C$ in $\mathbb R^n$, let \[ \mathcal R_{\mathcal C}(\delta,Q) :=\#\left\{(\mathbf p,q)\in\mathbb Z^n\times\mathbb N: 1\leq q\leq Q,\ \operatorname{dist}(\mathbf p/q,\mathcal C)\leq\delta/q\right\}.\] We prove that \[\mathcal R_{\mathcal C}(\delta,Q) \lesssim_{\varepsilon,\mathcal C} \delta^{n-1}Q^2+Q^\varepsilon \left(Q+\sum_{k=1}^{n-1} \delta^{k^2/(k+1)}Q^{(2k+1)/(k+1)}\right).\] A key new ingredient is a wave packet method developed by Gan–Maldague–Oh.

To a coprime triple $a+b=c$ we attach two coordinates, the quality $X$ and the logarithmic mass $Y$, whose ratio $\lambda$ measures the balance of the triple and lies asymptotically between $2$ and $3$. In these coordinates the $abc$ conjecture and Szpiro's conjecture for Frey curves become asymptotic bounds with thresholds $X=1$ and $Y=3$, and the direct implications between them follow from the range of $\lambda$. We then place the approximation and power gains of Müller, Taktikos and de Weger in the same plane. Relative to a chosen presentation of the triple, one exponent factor $f$ relates the gains to the coordinates, and Szpiro's conjecture for this family is equivalent to the asymptotic bound $f\le 3$ uniformly over all presentations. Under the two separate asymptotic gain bounds for a fixed choice of presentations, we obtain $\limsup X\le 9/2$ and $\limsup Y\le 27/2$ as the radical tends to infinity. Stronger conclusions depend on the presentation or on a joint bound for the gains. We close with numerical data and two research objectives.

Let \(A\) be the coefficient ring of a smooth projective curve over \(\mathbb{F}_q\) with a closed point at infinity of arbitrary degree \(\mathrm{N}=\deg(\infty)\ge 1\), and let \(\varphi\) be a rank-one Drinfeld \(A\)-module. In this paper, we prove three main results concerning the module of special functions and its relation to Pellarin's \(\mathcal L(1)\)-series. First, the module \(\mathrm{sf}(\varphi)\) of special functions contains a Tate-algebra unit exactly when it is free of rank one, equivalently when its period lattice is isomorphic to the module of regular differentials. This settles the Gazda–Maurischat conjecture. The proof evaluates Cauchy kernels; on the period lattice all such evaluations agree, and their common characteristic residue recovers the period. Second, for suitable isogeny data to a principal-period target \(\psi\), a strictly normalized shtuka product, together with one characteristic residue, gives an explicit formula for the fundamental period \(\tilde{\pi}\) (the generator of the free lattice \(\Lambda_\psi\)). The same defect equation also yields the residue formula for the Drinfeld logarithm. Third, the first Frobenius twist of the normalized shtuka differential is identified with a pairing built from finite torsion traces, whose coefficients are finite étale traces over the field of definition. As an application, Pellarin's \(\mathcal L(1)\)-series is expressed exactly as the first Frobenius twist of the generating differential at a fundamental period. A notable feature of these results is that they hold for arbitrary $\mathrm{N} \geq 1$, and the source period lattice need not be free.

We record a finite-field census of the split-cubic elliptic curves $E_{a,b,c}^{(n)}:y^2=(ax+n)(bx+n)(cx+n)$ attached to positive integral $D(n)$-triples. For $n\in\{-3,3,5,8,12,20\}$ there are $164$ such triples with $1\le a<b<c\le100$, allowing zero squares. Their reductions at the ten primes $\operatorname{NextPrime}(10^k)$, $3\le k\le12$, give $1640$ records, of which $45$ have cofactor four relative to the largest prime divisor of the group order. We collect the standard split-cubic and quadratic-twist identities in the explicit normalization $\lambda=b(c-a)/(c(b-a))$ and $\delta=cn(b-a)$, and use them to audit the point counts. The archive records group orders, factorizations, cofactors, embedding degrees and transfer-field bit lengths. A worked example compares the main curve with its nonsquare twist. The census is deterministic and descriptive; it provides neither an asymptotic density estimate nor cryptographic parameter recommendations.

We construct degree-three isogenies of Hesse elliptic curves over an arbitrary field of characteristic different from three, which correspond to a step of the cubic arithmetic-geometric mean. As an application, we describe the Serre-Tate canonical lift of an ordinary elliptic curve in characteristic three by the 3-adic cubic arithmetic-geometric mean. We also discuss the relations of the Hesse curves with complex and finite hypergeometric functions.

Let $p$ be an odd prime, $\psi$ a nontrivial additive character of $\mathbb F_p$, $\psi_q=\psi\circ\mathrm{Tr}_{\mathbb F_q/\mathbb F_p}$ and $\lambda\in\mathbb F_p^\times$. For $h\in(\mathbb F_q^\times)^n$ let $\Sigma_n(h)$ be the sum of $\psi_q(u_1^{-1}+\cdots+u_n^{-1})$ over $u\in(\mathbb F_q^\times)^n$ with $h_1u_1+\cdots+h_nu_n=-\lambda$, an additively twisted average of products of Kloosterman sums. For every $n\ge2$, the cohomology computing $\Sigma_n(h)$ is concentrated in one degree and has dimension $2^n-1$. It has $r(n)=2^n-\binom n{\lfloor n/2\rfloor}$ eigenvalues of weight $n-1$, and its other eigenvalues are the integers $q^{k-1}$, $1\le k\le\lfloor n/2\rfloor$, with ballot numbers as multiplicities. The part of top weight forms a lisse sheaf $\mathcal W_n$ of rank $r(n)$ on the torus $\mathbb G_m^n$. We determine its local monodromy at the coordinate boundary and its determinant: $\det\mathcal W_n$ is the Artin–Schreier sheaf of the sum $-2^{n-1}\lambda^{-1}(h_1+\cdots+h_n)$ of the critical values of the phase, twisted by $p^{(n+1)r(n)/2}$. The geometric monodromy group of $\mathcal W_n$, and the arithmetic one after normalization to weight $0$, are equal to $\{g\in\mathrm{GL}_{r(n)}:\det(g)^p=1\}$. Consequently the normalized sums become equidistributed like traces of Haar-random elements of $\{g\in\mathrm U(r(n)):\det(g)^p=1\}$. The proofs use Laumon's Fourier transform, middle convolution, the Cayley cubic surface and Larsen's alternative, but no classification of finite groups.

Let $$ S_n=\left\langle \genfrac{[}{]}{0pt}{}{n}{1},\ldots,\genfrac{[}{]}{0pt}{}{n}{n-1}\right\rangle $$ be the numerical semigroup generated by the nontrivial unsigned Stirling numbers of the first kind in the $n$th row. Writing $a=\genfrac{[}{]}{0pt}{}{n}{n-1}=\binom n2$ and $b_j=\genfrac{[}{]}{0pt}{}{n}{n-2j}$, we determine the complete gcd filtration of the reduced generating system. More precisely, if \[ M_j=\frac12\operatorname{lcm}\{m\ge1:\varphi(m)\le2j\}, \] then $$ \gcd(a,b_1,\ldots,b_j)=\frac{a}{\gcd(a,M_j)}. $$ The proof uses prime-power block polynomials and exact $p$-adic support at threshold degrees. We then establish a divisor-support property for canonical mixed-radix reductions and combine it with an elementary-symmetric growth estimate to prove that, after inactive generators are removed, the resulting generating sequence is telescopic. Consequently, the Apéry set of $S_n$ is rectangular, yielding an explicit Frobenius formula. The same structure shows that $S_n$ is free and symmetric and gives explicit formulas for its genus and conductor, while its type is one.

We prove that any counter-example to the two dimensional Jacobian Conjecture with integral coefficients and determinant of its Jacobian matrix equals one, has only paths of asymptotic values that can contain only finitely many points of the planar integral lattice. The second part of our paper gives some nontrivial applications of the theory of certain Theta functions to the two dimensional Jacobian Conjecture over the real field, and over the complex field.

We prove that the irrationality exponent of $\zeta(2)=\pi^2/6$ satisfies $\mu(\zeta(2))<5.0193784$. The best published bound is 5.095412 (Zudilin, 2014); a bound 5.0495243 with a machine-checked proof was made public in September 2026 by J. Kleid. We use Zudilin's two hypergeometric constructions at a new choice of parameters. The two families of linear forms in 1 and $\zeta(2)$ so obtained coincide for every sufficiently large index: this coincidence, an instance of an identity conjectured by Zudilin, is established by comparing an arithmetic bound for the denominators of the difference with an analytic bound for its size, without creative telescoping. All numerical constants are certified in ball arithmetic.

We prove both Erdős-Ulam conjectures on monochromatic union-closed families. Every two-colouring of the subsets of a finite set contains a monochromatic union-closed family whose size grows faster than any fixed power of the size of the set, while suitable colourings admit no monochromatic union-closed family of exponential size. Both results hold for any number of colours and are verified in Lean.

Let $n$ be even, let $\zeta$ be a primitive $n$-th root of unity, and let $\Delta_n$ be the set of derangements in the affine group $\overrightarrow{GL}(\mathbb{Z}/n\mathbb{Z})$. We evaluate the Wang–Sun type sum \[ S_n=\sum_{\pi\in\Delta_n}\mathrm{sign}(\pi) \prod_{x\in\mathbb{Z}/n\mathbb{Z}} \frac{1+\zeta^{x-\pi(x)}}{1-\zeta^{x-\pi(x)}}. \] The main step is an arithmetic grouping: for an affine map $\pi(x)=vx+u$, the product depends only on $\sigma=\gcd(1-v,n)$ and on the residue class of $u$ modulo $\sigma$. After summing the translation signs, the full sum becomes a sum over even divisors with explicit local multiplicities. The only remaining analytic blocks are alternating even cotangent power sums; a recent formula of Liu and Xin then gives a closed expression in Bernoulli polynomials and universal coefficients. Moreover, using results by Cvijović and Klinowski we can evaluate the cotangent power sums using exclusively rational arithmetic. This supplies a general formula for the full affine sum whose initial values were computed using the definition only in an earlier work. The formula also lets us prove some general patterns of the sum: it has integral value for powers of $2$, prime denominator for $2p$ for primes $p>3$ and it has alternating signs.

Let $F$ be a rank-two symmetrizable Nahm sum, with any symmetrizer $\operatorname{diag}(d_1,d_2)$, a positive-definite rational matrix and rational shifts, and let $K$ be the field generated by its Nahm point; write $m=d_2/d_1$. We prove that if $q^cF$ is modular, then $[K:\mathbb{Q}]\le 24$. Modularity forces the higher coefficients of the logarithmic radial asymptotic expansion to vanish; we study the resulting polynomial equations through their Newton polygons and a pole-order analysis along the determinant conic, and for $m=1$ show that their common components occur only on one explicit family, whose Nahm point is rational. For equal row sums, Bloch-group torsion gives $[K:\mathbb{Q}]\le 2$. For complementary Nahm coordinates, the bound is $2$ when $m\ne 1$ and $4$ when $m=1$, using the first two asymptotic corrections in the latter case. Several steps of the proofs rely on exact computation.

In this paper, we obtain extreme values of Dirichlet $L$-functions on the critical line with a growing prime modulus. Our result yields a larger lower bound for extreme values than that from Soundararajan's classical resonance method. Consequently, we show that one branch of the dichotomy established by Bondarenko, Darbar, Hagen, Heap, and Seip (2023) can be realized for this family of Dirichlet $L$-functions. The proof relies on the resonance method together with precise estimates for GCD sums.

We prove that the wedge local model for a ramified unitary group of signature $(r,s)$ ($r>s$) at self-dual level is flat when $p\geq 2s-1$, confirming Pappas's conjecture in this case. The proof first reduces the problem to determining the defining equations of a nilpotent orbit closure of self-adjoint matrices, and then to computing the global sections of the corresponding Kempf resolution. The computation is carried out by reducing to the type $A$ case using the Koszul complex.

We present two number-theoretic compression techniques. The first is local, and compresses individual numbers. The second is global, and compresses the entire set of numbers using a prime wavelet tree, which is a data structure of independent interest. Our techniques apply to any set of numbers satisfying a Korselt-like criterion. We demonstrate our compression techniques using the recently completed tabulation of all $308{,}279{,}939$ Carmichael numbers less than $10^{24}$ that occupies $18.4$ gigabytes as text. When combined with standard compression techniques and a heuristic choice of divisors, the resulting file is $588.7$ megabytes ($31.2$ times smaller than the text file). The initial $80$-bit numbers now use about $15.3$ bits.

Let $\delta \in [0,1/2]$ and $Q\geq 1$. Given a compact $C^{\infty}$-curve $\mathcal{C}$ in $\mathbb{R}^n$, denote by $N_{\mathcal{C}}(\delta, Q)$ the number of pairs $(\mathbf{a}, q)\in \mathbb{Z}^n\times [Q/2,Q]$ such that the rational point $\mathbf{a}/q$ is $\delta/q$-close to $\mathcal{C}$. We investigate the range of $\delta$ in terms of $Q$ which is necessary for the heuristic $N_{\mathcal{C}}(\delta, Q)\asymp\delta^{n-1}Q^2$ to be correct for the moment curve. For a nondegenerate curve $\mathcal{C}\subset \mathbb{R}^n$ and for sufficiently large $Q$, Hickman and the second author previously established that $N_{\mathcal{C}}(\delta, Q)\lesssim_{\nu} \delta^{n-1}Q^2$ for all $\delta\in[Q^{\alpha(n+1)+\nu},1/2)$ and any $\nu>0$. Here $\alpha(n)=-4n^{-2}+O(n^{-3})$ was explicitly computed. We show that this range is surprisingly close to being sharp in the asymptotic sense as $n\to\infty$, and can at most be extended to $\delta\in[Q^{\alpha(n)},1/2)$. In particular, the sharp exponent can only be about $O(n^{-3})$ better, even though it was widely believed before that an improvement of $O(n^{-1})$ should be possible. The complementary upper bound in this remaining range has been recently established by Gan–Guo–Oh.

The purpose of this note is to give conditional explicit formulas for the Galois action on real multiplication values of the Shintani–Faddeev modular cocycle. We assume the order 1 abelian Stark conjecture for real quadratic fields. We conclude this note by presenting our own conjecture about the Galois action. Numerical evidence for this conjecture will be provided in forthcoming work.

Let A = F_2[T]. The orthogonal multiplicity m(f) of a monic f in A of degree d >= 1 is the number of g in A with deg g < d, gcd(g,f) = 1, and all partial quotients of g/f of degree one. Blackburn (1998) asked how the proportion p(d) of monic polynomials of degree d with m(f) > 0 behaves as d tends to infinity. We attach to each prime divisor P of f a residue psi_P(f) in F_2 of the differential dT/(T(T+1)f), and prove two unconditional facts about this family: a reciprocity relation, namely that the psi_P(f) sum to zero whenever f is prime to T(T+1), and an exact local distribution, namely that psi_P vanishes for a proportion exactly (|P|-2)/(2(|P|-1)) of the admissible completions when P exactly divides f, and for exactly half of them when P^e exactly divides f with e >= 2. Granting that these local conditions pool independently, we obtain p(d) ~ L d^(-1/2) with L = (9/(4 sqrt(pi))) times the product over the monic irreducibles P of A of degree at least two of (1 + 1/(2|P|)) (1 - 1/|P|)^(1/2), equal to 1.2116120743... The same Euler product evaluated at a second parameter returns the exact identity that the m(f) of a given degree sum to 2^d, which fixes its normalisation. We give the exact value of 2^d p(d) for 1 <= d <= 33; the predicted values agree with these to within 5 10^(-5) for 27 <= d <= 33.

We establish asymptotics for the degree and height growth rate functions of partially commutative semigroups of endomorphisms of varieties, i.e., semigroups where the only relations between generators, if any, are commuting ones. To do this, we prove a general result connecting the growth rates of two abstract functions, one multiplicative and one simply nonnegative, subject to a Tate telescoping hypothesis and a sufficiently nice normal form on the underlying semigroup. Finally, we construct several new families of endomorphisms on which our asymptotics apply. To do this, we generalize the standard ping-pong lemma for elements to a ping-pong lemma for subsemigroups.

Determining whether an algebraic number field admits a power integral basis is a classical problem, but it can be difficult for fields defined by polynomial compositions and dynamical iterates. In this paper, we study the monogeneity of compositions $f(h(x))$, where $f(x)$ and $h(x)$ are monic polynomials over an arbitrary Krull valuation ring and $h(x)$ is a trinomial. We derive explicit formulas for the discriminant of the composition and use them to characterize when $f(h(x))$ generates a monogenic field. In particular, we relate the monogeneity of the composition to that of $f(x)$ and to explicit square-free conditions on the critical values of $h(x)$. We further extend the results to polynomial iteration and obtain a criterion for the monogeneity of binomial iterates, yielding new infinite families of monogenic fields. Finally, we give quantitative results and illustrate our criteria with several examples.

We improve the error term in the prime geodesic theorem for the modular group. More generally, our result applies to any congruence subgroup of any indefinite quaternion algebra over $\mathbb{Q}$.

In this paper we prove a generalized version of Mel'nikov's Criterion, which has also been proved by Haran, and use it in order to connect the realization as absolute Galois groups of free pronite and free pro-p products of a sheaf of pro-p groups.

Supersingular abelian surfaces are essential in isogeny-based cryptography. Despite this, we have no efficient algorithm to verify if a given abelian surface is supersingular. In this work, we initiate this research topic by giving an efficient Monte Carlo algorithm to verify if an abelian surface over $\mathbb{F}_p$ is supersingular in $O(\log p)$ with negligible failure probability, and an efficient conclusive algorithm if the order is smooth. We derive this algorithm by a careful analysis on the structure of supersingular Jacobians over $\mathbb{F}_p$. Furthermore, we derive efficient algorithms to verify if an abelian variety of any dimension is minimal or maximal, and to verify if a Jacobian of any dimension is superspecial.

For $n\ge2$ let $F(n)$ be the least $k\ge2$ such that $n!$ is the largest factor in a product of $k$ distinct factorials that is a perfect square, and let $D_k(X)$ be the number of $n\le X$ with $F(n)=k$. Erdos and Graham asked for the order of growth of $D_k(X)$ for $3\le k\le6$, and conjectured that $D_6(X)\gg X$. We prove that $D_3(X)=\kappa_3\sqrt X+O_\varepsilon(X^{2/5+\varepsilon})$ with an explicit constant $\kappa_3=2.7097\ldots$, and that $D_5(X)\asymp D_6(X)\asymp X$. Together with classical facts, this determines the order of growth of $D_k(X)$ for every $k$. The exponent $2/5$ comes from balancing a uniform bound for Pell equations against Gallagher's larger sieve, with residue restrictions supplied by the Weil bound. For five and six factors we restrict to integers with a prime factor exceeding $X^{1-\alpha}$, where $\alpha>0$ is small and fixed. We exclude shorter representations by combining an equidistribution estimate for primes of Matomaki, Radziwill, Shao, Tao and Teravainen with the large sieve. In an appendix we use a zero-sum theorem for finite abelian groups to construct, for every $m\ge2$, perfect $m$-th powers that are products of a bounded number of distinct factorials with arguments given by fixed affine functions.

We bound the error term in the counting function of cubic extensions of $\mathbb{Q}$ by $\mathcal{O}_\epsilon\left(X^{\theta+\epsilon}\right)$, with $\theta < 2/3$, improving upon results due to Bhargava, Taniguchi, and Thorne. The proof proceeds by relating a certain discriminant-reducing sieve, due to Bhargava, Shankar, and Tsimerman, to averages of Artin characters associated with the Artin $L$-function $\zeta_K(s)/\zeta(s)$, with $K$ a cubic field. Conditional on the Generalised Riemann Hypothesis (GRH) for Dedekind zeta functions $\zeta_K(s)$, we obtain further savings and show that the error term is $\mathcal{O}\left(X^{2/3-1/51+\epsilon}\right)$. Moreover, for the analogous smooth counting problem, we obtain a bound $\mathcal{O}(X^{1/2+\epsilon})$ for the error term, conditionally improving results of Shankar, Södergren, and Templier. Our arguments can also handle finitely many splitting conditions. The methods we use to study cubic fields are natural extensions of methods which are used to study the one-level density associated with the above family of Artin $L$-functions. In this direction, we improve the admissible support $[-\sigma,\sigma]$, from the previously known admissible value $\sigma < 2/5$, to $\sigma < 1$, conditional on the GRH. This also proves that at least $75\%$ of these $L$-functions are non-vanishing at the central point, conditionally improving upon results of Shankar, Södergren and Templier. Finally, one may study similar questions over a rational function field $\mathbb{F}_q(T)$, with $q$ coprime to $2$ and $3$. Here, we obtain a proportion of non-vanishing of at least $75\%$ for the analogous family of $L$-functions. Furthermore, we obtain a bound $\mathcal{O}(X^{1/2+\epsilon})$ for the error term in the counting function of cubic function fields, improving previous results of the author. Over function fields, our results are unconditional.

We show that all closed geodesics in any arithmetic complex hyperbolic surface of second type are simple. These include the fake projective planes studied by Mumford, Klingler, Prasad-Yeung, and Cartwright-Steger.