Papers

We consider Hecke modules of orthogonal modular forms attached to definite even ternary lattices of half-discriminant $N^2$, where $N > 1$ is odd and square-free. We prove explicit Jacquet-Langlands style correspondences with spaces of classical modular forms (in the spirit of Birch and Hein-Tornaria-Voight). This is done by applying sequences of Watson-style transformations that reduce the discriminant while preserving the adelic orthogonal group.

Assuming a smoothness property for mixed-characteristic shtuka spaces and hyperbolic localization for twisted sheaves, we determine the inertial restriction of the Fargues-Scholze $L$-parameter for irreducible depth-zero representations. Let $E$ be a non-archimedean local field of residue characteristic $p$, let $G$ be an unramified connected reductive group over $E$, and let $\pi$ be an irreducible smooth depth-zero representation of $G(E)$ over $\overline{\mathbb{F}}_{\ell}$ where $\ell$ is a prime number which does not divide $p|\pi_0Z(G)|$. Let $\varphi_{\pi}^{\operatorname{FS}}$ be the Fargues-Scholze $L$-parameter associated to $\pi$. We describe the restriction of $\varphi_{\pi}^{\operatorname{FS}}$ to the inertia subgroup in terms of Deligne-Lusztig theory. We prove this by computing the inertia action on the cohomology of shtuka spaces with depth-zero level structure. As an application, we prove that the Fargues–Scholze $L$-parameters agree with DeBacker–Reeder $L$-parameters over $\overline{\mathbb{Q}}_{\ell}$.

We establish lower bounds on the orbit cardinalities of $\text{SL}(2,\mathbb{Z})$ on the $\mathbb{F}_p$-points of the generalized Markoff surfaces given by $x^2 + y^2 + z^2 - xyz = \tau+2$. We show that outside of the reductions of a certain exceptional finite set of $\overline{\mathbb{Q}}$-points found by Dubrovin and Mazzocco, every orbit has cardinality $\ge p$. Together with a recent result of Nathaniel Kingsbury-Neuschotz, this establishes, for all but finitely many primes $p$, the classification conjectures of McCullough-Wanderley (2013), and gives a complete classification of the connected components of the Hurwitz stack of $\text{SL}(2,\mathbb{F}_p)$-covers of elliptic curves, only branched above the origin. This extends earlier work of Bourgain-Gamburd-Sarnak and the author in the Markoff case $\tau = -2$. Our method studies certain noncongruence modular curves attached to a $\text{SL}(2,\mathbb{Z})$-orbit. These modular curves parametrize elliptic curves with a $\text{PSL}(2,\mathbb{F}_p)$-cover. We use this moduli interpretation to show that the noncongruence modular curves involved have bad reduction, as a cover of the $j$-line, at some place above $p$. This implies that the degree of the cover is $\ge p$, which translates into the desired bound on orbits. Key inputs include a result of Bouw-Wewers on bad reduction of three-point $\text{PSL}(2,\mathbb{F}_p)$-covers of the projective line, and the general stable reduction theory of Henrio, Obus, Raynaud, and Wewers.

In 1965, Chowla conjectured that the values of the Liouville function become asymptotically uncorrelated at consecutive natural numbers. Recently, the two-point case is resolved by OpenAI. In 2026, Charamaras and Richter conjectured a generalization of the Chowla conjecture to the asymptotic independence of the consecutive prime Omega functions. In this article, we prove the two-point case of their conjecture. As a consequence, we obtain a dynamical generalization of the two-point Chowla conjecture.

We describe Redhill, a Lean formalisation of the major results in Hölzl, Kleine and Stephan's "Improved Lower Bounds for Strong $n$-Conjectures" (arXiv:2409.13439). Design decisions regarding types and data structures are justified as enabling canonical versions of the paper's claims, and from there the most direct proofs of said claims. The formalisation process also uncovered many errors, omissions and optimisation opportunities in the informal proof, all of which are addressed in the final library. Although Redhill is meant to stand alone, a few lemmas were upstreamed to the primary Lean mathematics library mathlib. We discuss the tension between upstreaming and leaving lemmas in separate libraries. Unlike many other contemporary formalisation projects, Redhill was completed without any assistance from large language models (LLMs). We compare our work with LLM-assisted projects, in particular an extension of arXiv:2409.13439 to $n=4$, aiming to establish design and style guidelines that define good Lean formalisations.

For any nonarchimedean local field $k$, we exhibit the first examples of Zariski-dense, discrete subgroups of special linear groups over $k$, which are not commensurable to any lattice in the points of a semisimple group over a local field, and contain an infinite index subgroup isomorphic to a lattice in the $k$-points of a higher rank $k$-semisimple group. We prove that any Gromov-hyperbolic double of a non-elementary free group along a cyclic subgroup embeds as a quasi-isometrically embedded subgroup of $\mathsf{SL}_5(k')\times \mathsf{SL}_5(k')$, where $k'$ is a discretely valued field. More precisely, $k'$ is the field of rational functions $k(t)$ equipped with the Gauss valuation and $k$ is any nonarchimedean local field of characteristic not equal to $2$ or $3$.

Let $p\geq11$ be a prime, and let $\bar\rho:\Gal(\bar\QQ_p/\QQ_p)\rightarrow\GL_2(\bar\FF_p)$ be a reducible and very generic representation. In this paper, we use ghost series \cite{Lghost2} to study the irreducible components of Kisin's potentially crystalline deformation rings of $\bar\rho$. For wildly ramified principal series inertial types and arbitrary regular Hodge–Tate weights, we distinguish the components by the crystabelline slopes on their generic fibers and determine the total number of irreducible components. Combining this with an explicit formula for the total Hilbert–Samuel multiplicity of the special fiber obtained from the Breuil–Mézard conjecture, we compute the Hilbert–Samuel multiplicity of the special fiber of each component. Every such multiplicity is one when $\bar\rho$ is non-split, and is either one or two when $\bar\rho$ is split, according to whether the generic fiber of the corresponding component parametrizes reducible or irreducible Galois representations.

Let $\mu_q$ be the distribution of the normalized Kloosterman sums $\mathrm{Kl}(a;q)$, $a\in\mathbb{F}_q^\times$, and let $\mu_{\mathrm{ST}}$ be the Sato–Tate law. We prove $W_s(\mu_q,\mu_{\mathrm{ST}})\ll q^{-1/3}$ for $1\le s\le2$, $W_s(\mu_q,\mu_{\mathrm{ST}})\ll q^{-1/6-1/(3s)}$ for $2\le s<\infty$, and $W_\infty(\mu_q,\mu_{\mathrm{ST}})\ll q^{-1/6}$, answering a question of Kowalski and Untrau. More generally, we bound $W_s(\mu,\mu_{\mathrm{ST}})$ for all $s$ whenever the Chebyshev moments of $\mu$ satisfy bounds of Katz type. For $s=2$ the bound also follows from a smoothing inequality of Borda and Cuenin; for $s>2$ and $s=\infty$ our bounds are stronger. The proof uses the quantile coupling, which arises from Stieltjes sums over two different intervals whose cells are paired so as to balance two measures, and which is optimal for all $s$ simultaneously; an explicit $W_2$ smoothing inequality relative to $\mu_{\mathrm{ST}}$; and a sharp comparison between $W_s$ and the Kolmogorov distance for laws with square-root edges. The last result shows that $q^{-1/6}$ is the best rate for $W_\infty$ obtainable from Niederreiter's discrepancy bound. Numerically, for primes $10^3<p<10^6$, all these distances appear to decay like $p^{-1/2}$.

We identify Gaussian curvature as a local mechanism for cubic cancellation of Laplace eigenfunctions on Riemannian surfaces, without arithmetic or dynamical assumptions. For $L^2$-normalized local solutions at frequency $\lambda$, cubic integrals against smooth compactly supported weights tend to zero whenever the curvature is nonzero almost everywhere where the weight is nonzero. Finite-order nondegeneracy yields sharp polynomial decay rates. In particular, on closed surfaces with nowhere-vanishing curvature, we obtain the rate $O(\lambda^{-1/6})$, attained by spherical harmonics on the round sphere. This proves the cubic case of the Gaussian moment conjecture for all normalized real eigenfunctions on closed negatively curved surfaces. The local proof rests on an exact identity relating the reduced phase at a Fermat point to Gaussian curvature. On complete hyperbolic surfaces with positive systole, we also prove global $O((\log\lambda)^{-3/2})$ decay for $L^2$-normalized eigenfunctions and smooth weights with bounded covariant derivatives through order two, without assuming finite area.

We prove a Cohen-Lenstra type result commenting on the distribution of class group structures amongst function fields of superelliptic curves over finite fields. We also prove a result commenting on the sparsity of such superelliptic curves whose zeta functions vanish at a fixed complex number. These results are proven via point-counting techniques on certain Hurwitz schemes generalizing techniques of Ellenberg-Venkatesh-Westerland and Ellenberg-Li-Shusterman. To obtain these point counts, We prove a unitary big monodromy theorem generalizing big monodromy theorems of Yu and Achter-Pries by appealing to an arithmeticity theorem of Venkataramana.

We introduce a multiplicity one defect that quantitatively measures the failure of multiplicity one for the Jacobian of the Shimura curve \(X^D(M)\) attached to an indefinite rational quaternion algebra of discriminant \(D\) and level \(M\), localized at a maximal ideal of the Hecke algebra. By constructing a certain lattice inside the local Tate module, we obtain an explicit upper bound for the defect in terms of congruences between eigenforms and the lattice-Tate module discrepancies associated with the individual newforms. We then show that the local ratio between the congruence exponent and the modular exponent of modular abelian varieties attached to newforms is bounded above by the multiplicity one defect. Finally, we examine the relation with the Gorenstein property and give an explicit example at level \(126\) where the local Hecke algebra is Gorenstein even though multiplicity one fails. In this example, we compute the multiplicity one defect to be \(3\), which is also the local congruence–modular exponent ratio.

Let $K$ be a number field of degree $n_K$, and let $h(K)$ denote the smallest absolute value of the imaginary part of a non-trivial zero of its Dedekind zeta function. In 1979, Hoffstein proved the unconditional bound $h(K)\leq 0.87+o(1)$ as $n_K\to\infty$. In a 1990 survey article, Odlyzko stated without proof that a different kernel in Weil's explicit formula could produce $h(K)\leq 0.54+o(1)$. No further improvement on Hoffstein's $0.87$ threshold accompanied by a published proof has since appeared. We choose a linear kernel and first show that Hoffstein's original parameters already yield the improved constant $0.825$. We then generalise his linear kernel method using Stechkin's trick and prove the unconditional bound $h(K)\leq 0.6+o(1)$ as $n_K\to\infty$, giving the best proved asymptotic bound to date.

A result by F.-T. Tu dating from 2011 states that the orders in a local matrix algebra of dimension four that can be written as a finite intersection of maximal orders are those having an Eichler Gorenstein closure. In a more recent paper, we gave a similar characterization for orders with a Bass Gorenstein closure. More precisely, we prove that these orders are the ones that can be written as a ghost intersection, i.e., the set of points defined over the base field of an intersection of maximal orders that is defined over a field extension. Although we proved this result only for the local four-dimensional matrix algebra, the statement makes sense also for a local quaternion division algebra, since the latter turns into a matrix algebra when we extend the base field to an even degree extension. In this work, we prove an analog of our previous result in the latter setting.

We deduce zero-density estimates in short vertical strips from short-interval moments of globally tempered automorphic $L$-functions, by adapting the large-value estimate of Guth and Maynard and the Type II zero argument of Maynard and Pratt to the framework of Ye and Zhang. For the Riemann zeta function, we obtain $N_\zeta(\sigma;T,T+T^a)\ll T^{15a(1-\sigma)/(3+5\sigma)+\varepsilon}$ for $2/3\le a\le1$ and $7/10\le\sigma\le4/5$, improving Ye–Zhang's exponent in the interior of this range. Combining the localized density bound with the recent quasi-Riemann theorems, we estimate smoothed prime-power sums with phase $\exp(2\pi i\alpha n^\beta)$ on intervals of length $K^\delta$. For $0<\beta<13/20$, we obtain a power saving whenever $\max\{17/30,1-\beta\}<\delta<1$; in particular the square-root phase admits every $\delta>17/30$. A revised transform estimate retains both the stationary spread and the Fourier width $K/L$. For example, the bound at $\beta=1/2$, $\delta=3/5$ is $K^{15/26+\varepsilon}$, uniformly for frequencies $m\alpha$ with $1\le m\le K^{1/10}$. We also retain power savings for certain self-dual $GL(2)$ twists and self-dual $GL(3)$ cusp forms, without assuming Ramanujan. All three applications have explicit frequency dependence.

Let $g$ be a fixed primitive Hecke–Maass cusp form, and let $f$ vary with spectral parameter $k_f=K$, at a fixed level and with trivial nebentypus. We study smooth sums of $\lambda_f(n)\lambda_g(n)e(\alpha n^\beta)$ for fixed real $\alpha\ne0$. The crucial input is the smooth spectral second moment of Lau, Liu, and Ye, for which the window length can be taken to be $L=K^{1/3+a}$ for arbitrarily small fixed $a>0$. We show that, for every $\kappa>0$, there is $\beta_0>0$ such that the sum is $O(K^{2/3+\kappa}X^{1/2})$ for $1\le X\le K^2$ and $0<\beta\le\beta_0$. We give a detailed Baire category argument which makes the dependence on the smooth weight uniform through one finite derivative norm. A second argument keeps the phase explicit, applies Voronoi summation and shifted convolution estimates, and gives the numerical bound $O(K^{423/224+(5/7)\beta+\varepsilon})$ at $X=K^2$ for $0<\beta<5/32$.

The involution numbers $1,1,2,4,10,26,\ldots$ repeat modulo $3$, $5$ and $7$. We explain this, and many similar patterns, by one rule. Many sequences count permutations in which every cycle of length $m$ carries a weight $w_m$ (involutions, derangements, "sets of lists", and many others in the OEIS). Writing $A(n)$ for such a count, the rule is $A(n+k)\equiv A(n)A(k)$ modulo $\mathrm{lcm}(n,k)$. It generalizes our earlier proof of the case modulo $k$ for one such sequence, but now it holds for all weights at once, as a coefficientwise congruence between universal polynomials, and we find the largest modulus that works for the universal polynomials, $nk/\mathrm{rad}(\gcd(n,k))$, and show that nothing larger does. We then show that passing from $k$ to $pk$ behaves like raising to the $p$-th power modulo a high power of $p$, with a sharp exponent for the polynomials, which gives $A(p^r)\equiv(w_1+(p-1)w_p)^{p^{r-1}}$ modulo $p^r$ for odd $p$, and explains why the prime $2$ is special. A family of sequences defined by Lagrange inversion fits the same framework, which proves Bala's conjecture for the whole family (the single sequence A278070 had been proved earlier by an automated system) and the divisibility observed in the entries A251568 and A251569. As further applications we prove OEIS conjectures of Richman, Hanna, van Hoeij and Bala (A227937), and the theorems apply to 101 OEIS sequences, whose congruences are recorded in a supplementary catalogue. Higher-order versions of the content formula are treated in a separate paper.

We study the Iwasawa cohomology complex attached to a pseudorigid family of $(\varphi, \Gamma)$-modules and prove coadmissibility of its cohomology groups, perfectness and compatibility with base change. We apply this result to give another proof of perfectness of the Herr complex in pseudorigid family. We also use it to prove perfectness of Selmer complexes and obtain a result on variation of ranks of Selmer groups in analytic family.

Bugeaud, Haynes and Velani (2011) resolved a metric version of the $p$-adic Littlewood conjecture. We establish a fully inhomogeneous generalisation of their result, as well as a partially inhomogeneous generalisation of a result of Harrap and Haynes (2013) in the more general mixed setting.

Linnik's approximation to Goldbach's conjecture concerns the representation of sufficiently large even integers as the sum of two primes and a bounded number of powers of two. We prove that $6$ powers of $2$ suffice unconditionally and that $5$ suffice under the GRH, improving previous works.

Let $(T_n)_{n\ge1}$ be the Tribonacci sequence, defined by $T_1=T_2=1$, $T_3=2$, and $T_n=T_{n-1}+T_{n-2}+T_{n-3}$ for $n\ge4$. In this paper, we study the Tribonacci numbers whose decimal representation is a palindromic concatenation of two distinct repdigits, that is, numbers of the shape $\overline{d_1\cdots d_1\,d_2\cdots d_2\,d_1\cdots d_1}$ with $1\le d_1\le9$, $0\le d_2\le9$, $d_1\ne d_2$. We prove that no Tribonacci number is a palindromic concatenation of two distinct repdigits. The proof combines lower bounds for linear forms in logarithms, the Baker-Davenport reduction method, and a finite computational verification.

We establish two quantitative results concerning additive decompositions of the set $\mathbb{P}$ of primes. First, we prove that, if $A,B\subseteq\mathbb N$ have at least two elements and $A+B$ differs from the primes by only finitely many elements, then $$ \frac{\sqrt{x}}{\log x}\ll A(x),B(x) \ll \sqrt{x}. $$ Second, we show that, if $A,B,C\subseteq\mathbb N$ have at least two elements each, then $$ ((A+B+C)\mathbin{\bigtriangleup}\mathbb P)(x) \gg \frac{\sqrt{x}}{(\log x)^2}. $$ An analogous estimate is obtained for sumsets of $k\ge 3$ sets. The proofs use Montgomery's large sieve, Gallagher's larger sieve, a mean-value argument for a certain multiplicative function, and a submultiplicative sumset inequality.

We count the integral solutions of x_1^2+x_2^2-x_3^2=D weighted by Phi(x_3/M), for M smaller than sqrt(D). Such solutions correspond to binary quadratic forms of discriminant D, and the count becomes a sum, over the geodesics of discriminant D, of a function of their distance to the point i, which we evaluate by the spectral theory of automorphic forms. For square D=n^2 the geodesics run from cusp to cusp, and Hecke theory reduces them to a single geodesic. For every n and both parities of x_3 we obtain an asymptotic formula with error O(n^{1+theta+eps} M^{-1/2}), theta=7/64, using the Kim-Sarnak bound and the large sieve of Deshouillers and Iwaniec; it is non-trivial for M >= n^{71/96+eps}. This improves the range n^{5/6+eps} of the elementary method and confirms the corrected main term of the author's earlier work. For non-square D=f^2 D_0 the geodesics are closed, and a Hecke relation for their traces reduces everything to the fundamental discriminant D_0. For odd D_0 we obtain an error O(D^eps f^{1+theta} D_0^{2/3} M^{-1/2}), using the formulas of Duke-Imamoglu-Toth and Baruch-Mao together with Young's cubic moment. For D = 5 mod 8 this counts all solutions, and the range M >= D^{1/2-1/1332} of Friedlander and Iwaniec improves to M >= D^{4/9+eps}. This paper was written in collaboration with Claude (Opus 5.5, High), an AI model developed by Anthropic.

Let \(\alpha, \beta \in \mathbb{T} = \mathbb{R}/\mathbb{Z}\), and let \((\xi_n)_{n \ge 1}\) be independent random variables taking the values \(\alpha\) and \(\beta\) with equal probability. We study the random walk \[X_n = \xi_1 + \cdots + \xi_n \pmod 1\] and present a deterministic Diophantine condition on \((\alpha, \beta)\) that implies almost sure Poisson pair correlation. To the best of our knowledge, this is the first result providing explicit examples in this framework where the increment measure is purely discrete. The classical Fourier contraction method fails in this setting; a different approach is required. Our proof uses the parabolic geometry of returns to \(1\) and an exact covariance formula for overlapping blocks.

Let $q=2^n$ with $n=2m+1$ for an integer $m\ge1$, and let ${\mathbb F}_q$ be the finite field with $q$ elements. Let $W_n(x)=x^{2^m+3}$ be the Welch function on ${\mathbb F}_q$. For $1\le k\le n$ and $b\in{\mathbb F}_q$, denote by $N_{n,k}(b)$ the number of $k$-dimensional ${\mathbb F}_2$-subspaces $E$ of $\mathbb F}_q$ such that \[ \sum_{x\in E}W_n(x)=b. \] In this paper, we determine $N_{n,k}(b)$ for all $k$. The only nontrivial case is $k=3$, for which the answer is given in terms of a binary Kloosterman sum. As a consequence, we prove a recent conjecture of Hou and Zhao on the third-order sum-freedom of the Welch function.

Let $h(x)$ be the number of pairs $1\le a<b<x$ with $\gcd(a,b)=1$ and $\sigma(a)=\sigma(b)$. Erdős Problem #824 asks whether $h(x)>x^{2-o(1)}$. By an argument of Pollack and Pomerance, this follows if for every fixed $\delta>0$ there are $y^{1-o(1)}$ primes $p\le5y$ with $p+1$ free of prime factors exceeding $y^\delta$. We write this implication out with explicit parameters and formalize it in Lean 4. We then check that the argument of the recent OpenAI preprint "Weighted dilation graphs, smooth shifted primes and totient fibers", which claims the corresponding statement for $p-1$, applies to $p+1$ after a change of sign in two of its sections. Consequently $h(x)=x^{2-o(1)}$, conditionally on the correctness of that preprint, which has not been refereed and whose main new estimates we have not independently verified.

It follows from the compactness of the connectedness locus that the locus of PCF polynomials in the moduli space of polynomials of degree d (at least 2) over the complex number is a countable union of zero-dimensional subvarieties. Over fields of characteristic p the analogous claim is certainly false, so it is natural to ask how large the components of the PCF locus can be. We show that the components of the PCF locus in the moduli space of polynomials of degree d over an algebraically closed field of characteristic p>0 have dimension at most d/p and that this bound is sharp, in the strong sense that for every p and d there are infinitely many distinct PCF components whose dimension is the greatest integer no greater than this bound (except in the case $p=d=2$).

We construct a positive-definite symmetric integral $12\times12$ matrix $A$ such that, for every solution $Q$ of Nahm's equations $1-Q=Q^A$ with $Q_i\ne0,1$ for all $i$, the associated element $\sum_i[Q_i]$ is torsion in the Bloch group of $\mathbb{Q}(Q_1,\ldots,Q_{12})$. However, the Nahm sum $F_{A,B,C}(\tau)$ is nonmodular for every $B\in\mathbb{Q}^{12}$ and $C\in\mathbb{Q}$. Thus the torsion condition for all solutions in Nahm's conjecture is not sufficient for the existence of a modular Nahm sum.

Let \(E(X)\) count the even integers between \(4\) and \(X\) that are not sums of two primes. We combine Pintz's explicit formula for the major arcs with the exponential-sum estimate of Maynard, Pandey, and Radziwiłł to prove \(E(X)\ll_\varepsilon X^{7/12+\varepsilon}\) whenever all Dirichlet \(L\)-functions have a common fixed zero-free half-plane \(\Rea s>\theta_0\), with \(\theta_0<1\). The uniform \(7/8\) theorem reported in the OpenAI mathematics release supplies this input. Under the stronger hypothesis that every nontrivial Dirichlet zero has real part at most \(\theta\), where \(1/2\leq\theta\leq2/3\), we prove \[ E(X)\ll_{\theta,\varepsilon}X^{g(\theta)+\varepsilon}, \qquad g(\theta)=\frac{\theta(5-4\theta)}{2(2-\theta)}. \] Thus every fixed \(\theta<2/3\) gives an exponent strictly below \(7/12\). The latter argument uses a classical fixed-modulus zero-density estimate and is independent of the new exponential-sum theorem.

In this article, a new criterion is established to obtain the linear independence of some values of the $p$-adic Hurwitz zeta function at odd integers $s\geq 3$. We generalize a theorem of F.~Beukers, who gave, among others, an alternative proof—using Padé approximation theory—of a result originally due to F.~Calegari on the irrationality of the $p$-adic Riemann zeta value $\zeta_p(3)$ for $p = 2, 3$. To prove our main theorem, we work in a general setting, where we construct and study new Padé approximants of Laurent series satisfying the functional equation $f(z) = f(1-z)$. This complements our previous work on the linear independence of $p$-adic polygamma values.

These notes introduce crystals and crystalline Dieudonné theory through the example of abelian varieties. After discussing $p$-divisible groups, $F$-isocrystals, slope filtrations, and overconvergence, we explain the parabolicity theorem for $F$-isocrystals and some of its applications to monodromy and Hecke orbits. Two appendices formulate research projects on isocrystals over formal schemes and on monodromy beyond the constant-slope setting. The exposition is intended for readers familiar with basic algebraic geometry and the étale and fppf topologies; prior knowledge of Witt vectors is helpful but not required.