Papers

The Skolem Problem asks whether a given linear recurrence sequence (LRS) has a zero term, and is equivalent to proving an effective version of the Skolem-Mahler-Lech theorem, which states that a non-degenerate LRS has finitely many zeros. Decidability of the Skolem Problem however, has remained open for many decades. Bilu et al. (2022) showed that the Skolem Problem for simple LRS is decidable subject to the weak $p$-adic Schanuel Conjecture and the Skolem Conjecture (also known as the exponential local-global principle), the latter of which states that an LRS has an integer zero if and only if it has a zero modulo every integer $m$. This paper works towards understanding the Skolem Conjecture. We provide an example showing that a natural strengthening of the Skolem Conjecture, where we restrict $m$ to be a prime power, is false. This failure is accounted for by the relationship between $p$-adic zeros of LRS (which arise naturally in the proof of the Skolem–Mahler–Lech theorem and have been studied algorithmically by Bacik et al. (2026)) and twisted rational zeros (introduced by Bilu et al. (2025)). By studying this relationship further, we are able to prove the main result of this paper: a local-global principle for simultaneous zeros of two coprime LRS, subject to the $p$-adic Schanuel Conjecture. A fundamental step in proving this result is characterising when twisted rational zeros are (or are not) $p$-adic zeros for infinitely many primes $p$, which we do unconditionally. This also answers two open questions of Bilu et al. (2025). Finally, we conjecture that the existence of twisted rational zeros is the only way that the aforementioned strengthened Skolem Conjecture may fail, which is supported by a heuristic argument.

This is the second of two papers dedicated to the explicit computation of the Fargues–Scholze correspondence. We compute the Tate cohomology of cuspidal representations arising from Yu's construction. Combining this with modular functoriality in the Local Langlands Correspondence, independence of $\ell$, and the partial characterization of the Local Langlands Correspondence established in the first paper, we compare the Fargues–Scholze and Kaletha parametrizations. Among the consequences, we deduce that the Fargues–Scholze correspondence has finite fibers and is surjective onto inertial L-parameters.

This is the first of two papers dedicated to the explicit computation of the Fargues–Scholze correspondence. In this first part, we construct (inspired by, and extending, Kaletha's explicit Local Langlands Correspondence for non-singular supercuspidal representations) semisimple inertial L-parameters for all cuspidal representations arising from Yu's constructions, with arbitrary coefficients. We then establish a characterization of this correpondence in terms of certain functoriality properties. This characterization will be used in the sequel paper in order to compute the Fargues–Scholze correspondence after restriction to inertia.

In their previous work devoted to the distribution of rational points near Brownian motion, the authors conjectured the existence of an oscillation principle governing the asymptotic behavior of the number of rational points with bounded denomi\-nators near the graph of a monofractal curve. In this note, a weaker form of this conjecture is shown to hold for a broad class of deterministic fractal curves. These include the classical Takagi and Weierstrass nowhere differentiable functions, and indeed a prevalent (i.e.~"large") class of functions among those which are Hölder continuous with a given exponent of regularity. This constitutes the first instance of deterministic fractal curves for which a precise count of the rational points under consideration is established.

In 2015, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function $\overline{t}(n)$ to be the number of overpartitions of weight $n$ where (i) the difference between two successive parts may be odd only if the larger part is overlined and (ii) if the smallest part is odd then it is overlined. In their work, they proved that $\overline{t}(n)$ satisfies an elegant congruence modulo 3. Since then, a number of authors have studied arithmetic properties satisfied by $\overline{t}(n)$. In particular, in 2023, Hanson and Smith utilized the theory of modular forms to prove the following two congruences modulo 5: For all $n\geq 0$, \begin{align*} \overline{t}(80n+40)\equiv \overline{t}(80n+60)\equiv 0 \pmod{5}. \end{align*} Our goal in this work is to provide a truly elementary proof of this pair of congruences.

We construct a computable irrational number that simultaneously has small continued fraction entropy and base $b$ entropy for every integer base $b\geq 2.$ This addresses a question raised by Bugeaud.

We study function fields in one variable over the field with one element $\mathbf{F}_1$. It is proved that the Galois extensions of $\mathbf{Q}$ are isomorphic to the curves over $\mathbf{F}_1$ being understood as the Deitmar schemes. Specifically, one gets explicit formulas linking the genus and the number of cusps of an algebraic curve over the extension $\mathbf{F}_{1^m}$ of order $m\ge 1$ of the field $\mathbf{F}_1$ and the $m$-th roots of unity of the corresponding number field. It follows that elliptic curves with one cusp over $\mathbf{F}_1$ are either the cyclotomic fields or the maximal abelian unramified extensions of the quadratic number fields. Our proof depends on representaion of the Drinfeld modules by the bounded linear operators on a Hilbert space and the crossed product structure of the Cuntz-Krieger algebras.

In this paper, we give an explicit formula for the rank and generators (up to finite index) over $\mathbb{Q}(t)$ of all non-trivial elliptic curves of the form $y^2=x^3+At^6+Bt^3+C$, which is a larger class of elliptic surfaces than the one in Kloosterman's paper (2026) and Desjardins and Naskrecki's paper (2024), namely $y^2=x^3+At^6+C$. Our proof provides a new method to find this formula using generators of the geometric Mordell-Weil group while this geometric Mordell-Weil group (which is a $\mathbb{Z}[\omega]$-module) no longer decomposes into rank-one submodules by the methods in Kloosterman's paper (2026) and Desjardins and Naskrecki's paper (2024). Moreover, our proof uses only a few elements in the Galois group of the coordinates of the generators and only light computations by hand.

We establish bijections among spherical orbits on loop symmetric spaces, principal bundles on the twistor $\mathbb P^1$, and tempiric Langlands parameters. Using Vogan's theory of minimal $K$-types, we further establish bijections among irreducible equivariant local systems on loop symmetric spaces, irreducible local systems on principal bundles on the twistor $\mathbb P^1$, and irreducible representations of all strong inner forms of the dual symmetric subgroup. The latter bijections may be viewed as a combinatorial shadow of geometric Langlands duality for real reductive groups.

We establish the dynamical Mordell–Lang conjecture over the complex numbers for a family of radially ramified polynomial skew products in arbitrary base dimension and fiber rank. We assume that one fixed iterate sends the vertical critical locus into the invariant zero section. Our result covers affine-linear bases and, under a degree-gap condition, nonlinear étale polynomial bases. We also establish a uniform Skolem–Mahler–Lech theorem for a fixed nonlinear orbit and a fixed polynomial-coefficient linear recurrence whose trailing coefficient is a nonzero constant. The theorem applies to joint polynomial and rational relations involving the orbit and finitely many successive terms of any solution. A single eventual period works for every such solution and relation, although the finite exceptional set may depend on the relation. Our proof combines controlled nonarchimedean realizations with an orbitwise transfer theorem for superattracting line-bundle dynamics.

We show that partial sums of the Sato–Tate random multiplicative functions introduced by Cogdell and Michel exhibit better-than-square-root cancellation. The proof proceeds via a connection to multiplicative chaos, following Harper's seminal work. By a non-trivial adaptation of Harper's derandomization argument for character sums, we also obtain upper bounds for low moments of Hecke eigenvalue sums and of Hecke eigenforms near the cusp $0$; to our knowledge, this is the first appearance of multiplicative chaos in the context of automorphic forms on $\mathrm{GL}(2)$. A novel ingredient is the introduction of Hecke $s$-norms.

The Cassels–Swinnerton-Dyer conjecture asserts that a cubic hypersurface contains a rational point if and only if it contains a point of degree coprime to $3$, or, equivalently, a zero-cycle of degree $1$. The case of smooth cubic surfaces in characteristic zero has been reduced by Coray (1976) and Voisin (2026) to the case of points of degree $4$. We give two independent proofs of this missing case and use a lifting argument to extend the result to smooth cubic surfaces over arbitrary fields. We further give a separate argument for the case of singular cubic surfaces, extending previous work of Coray over perfect fields. Altogether, this proves the Cassels–Swinnerton-Dyer conjecture for cubic surfaces.

Elements of the Brauer group $\operatorname{Br}(S)$ of a variety $S$ have geometric incarnations as étale-projective $S$-bundles, yet producing minimalist constructions of such bundles, which often power arithmetic applications, remains a difficult problem. When $S$ is a K3 surface with Picard rank $1$, we use the birational geometry of moduli spaces of twisted sheaves on $S$ to construct geometric realizations of nontrivial elements of $\operatorname{Br}(S)$. We recover many known geometric constructions of Brauer classes on K3 surfaces while providing a common moduli-theoretic framework for them. As a by-product, we give a new proof of the period-index theorem for very general K3 surfaces.

We prove an unconditional Cauchy limit for a symmetrically truncated Stieltjes transform of the projected ordinates of the zeros of the Riemann zeta function. A suitable normalization of the logarithmic derivative of $\zeta$ on the critical line has the same limit provided that the sum of the distances to the critical line of the zeros lying to its right, counted with multiplicity up to height $2T$, is $o(T)$; we give an explicit bound on the comparison error. The proof rests on a convergence theorem for the Stieltjes transform of positive stationary point measures whose counting discrepancy satisfies an integrability criterion. This theorem is obtained by comparison with the transforms of periodic point measures, combined with truncation bounds and a transfer theorem. Over finite fields, a classical cotangent identity yields Cauchy limits for the logarithmic derivatives of zeta functions of varieties: the law is exactly Cauchy for nonconstant pure cohomological factors, and Poincaré duality gives explicit errors for full zeta functions, including Cauchy limits for smooth hypersurfaces of increasing degree, uniformly in the base field.

We establish the worst-case hardness of several ideal lattice problems (including SVP and CVP) in the $\ell_2$ norm by providing a dimension-preserving, deterministic polynomial time reduction from their generic lattice versions. The reduction constructs an ideal lattice in the canonical embedding of a number field that approximates some input lattice up to scaling and orthogonal transformation. The integers defining the ideal and the ambient number ring, in particular its discriminant, are all polynomial in bit length relative to the generic input lattice. Furthermore, the ideal is invertible, the ring is monogenic, and the number field is totally real. If the number ring is also required to be a full ring of integers, the reduction conjecturally succeeds in bounded-error quantum polynomial time.

We give a new and shorter proof of a quantitative inverse theorem for the Gowers $U^4$-norm in the setting of high-dimensional vector spaces over finite fields.

We prove the degree-seven case of the Hermite–Joubert problem in characteristic zero: if $F$ is a field of characteristic zero and $E/F$ is a field extension of degree seven, then $E$ is generated by an element $a$ with $\text{tr}_{E/F}(a)=\text{tr}_{E/F}(a^{3})=0$, that is, with minimal polynomial of the form $\lambda^{7}+c_{2}\lambda^{5}+c_{4}\lambda^{3}+c_{5}\lambda^{2}+c_{6}\lambda+c_{7}$, where the $c_i$'s belong to $F$. This is given by an explicit formula: a covariant of the binary septic of coefficient degree seven and order five, evaluated at a generator $\theta$ and divided by the derivative of its minimal polynomial evaluated at $\theta$. We also announce the general theorem, which will be proved in a companion paper in preparation: over every infinite field, in every characteristic, every étale algebra of degree seven contains a primitive element $a$ with $c_{1}(a)=c_{3}(a)=0$. Moreover, every field extension of degree seven of an arbitrary field has a generator $a$ with $c_{1}(a)=c_{3}(a)=0$.

Motivated by Robert Schneider's trailblazing work toward developing a unifying algebraic theory of integer partitions, we explore various binary operations on partitions to identify algebraic structures on sets of partitions. In particular, we discover that several sets of restricted partitions form abelian groups under reduced versions of concatenation, component-wise addition, and component-wise multiplication. One type of restricted partition from a group structure also enjoys a bijection with ordinary partitions of any given size. We extend two partition groups to vector spaces over the finite field $\mathbb{Z}_p$, where $p$ is a prime. We further discover that partitions are equipped with a commutative ring structure. Finally, we consider subgroups, subspaces, and ideals of our algebraic partition structures to investigate properties of related types of restricted partitions. The new examples of algebraic structures described in this paper open the door to partition analysis via algebraic tools, decompositions, extensions, and geometry.

After proving the moonshine conjecture, Borcherds asked in 1995 whether there exist only finitely many holomorphic automorphic products of singular weight. We show that, up to conjugation, there are exactly 27 such products on lattices of the form $2U\oplus L$. The classification proceeds by identifying each product at a standard $1$-dimensional cusp with the superdenominator of a BKM superalgebra that carries a graded module structure over an affine Lie superalgebra $\hat{\mathfrak{g}}$. The auxiliary algebras $\hat{\mathfrak{g}}$ are shown to satisfy strong conditions that reduce the possibilities to an initial list of 1372 candidates. An elimination argument leaves exactly 85 algebras $\hat{\mathfrak{g}}$ that correspond to the 27 products. This classification is closely connected with certain distinguished families of vertex operator superalgebras. Our results provide a systematic framework for extending affine Lie superalgebras to hyperbolic ones with modularity, generalizing the work of Feingold and Frenkel (1983).

In this paper, we construct two extremal positive definite even unimodular lattices of rank $88$ using a Hermitian tensor product over $\mathbb{Q}(\sqrt{-23})$, thereby extending the known existence range for such lattices from rank $80$ to rank $88$.

Let $d$ be the divisor function, and let $K(n)$ be the least positive integer $K$ for which $d((n + K)!) \ge 2d(n!)$. Erdős, Graham, Ivić and Pomerance proved that, for infinitely many $n$, \begin{equation*} K(n) > (1/9)(\log n)(\log_{2} n)(\log_{4} n)/(\log_{3} n)^3. \end{equation*} We improve upon this by a factor of order $\log_{3} n$, which brings the bound to the same order as Rankin's 1938 lower bound for gaps between consecutive primes. The two problems are closely related, but a long prime-free interval does not by itself produce a large value of $K(n)$: what is needed is a weighted variant of the Erdős–Rankin construction. We follow the method of Erdős, Graham, Ivić and Pomerance, replacing a key estimate by an averaging argument that permits some integers to remain uncovered. This improvement was formulated and proved during a private interaction with Claude Fable 5.1, a publicly available generative-AI system; the argument is verified and presented here by the author.

We study the algebraic independence of special values of power series $f(\beta^{-1})$, where $\beta$ is a fixed Pisot or Salem number. In particular, we consider the case where $f(X)=\sum_{n\geq 0} t(n) X^{w(n)}$ is not a lacunary series and is not assumed to satisfy any special functional equation, such as a Mahler-type functional equation. In our main results, we give a new criterion of the algebraic independence of three values. Applying our main results, we prove that the following three values are algebraically independent: \[ % \begin{gathered} \sum_{n=3}^{\infty}\lfloor n^{y}\rfloor\beta^{-\lfloor n^{\log \log n}\rfloor},\quad \sum_{n=3}^{\infty}\beta^{-\lfloor n^{\log \log n}\rfloor},\quad \sum_{n=1}^{\infty}\beta^{-\lfloor n^{\log n}\rfloor}, % \end{gathered} \] where $y$ is an arbitrary positive real number. Since our criterion is flexible, we have considerable freedom in choosing the coefficients $(t(n))_{n\geq 0}$ and the exponents $(w(n))_{n\geq 0}$.

Continued fractions have been introduced and studied in the field of $p$–adic numbers by several authors, with the aim of developing analogues of the classical theory of real continued fractions. In this paper, we focus on Browkin's continued fraction algorithm, for which several fundamental questions remain open, especially concerning periodic expansions of quadratic irrationals. In this paper, we solve a conjecture about periodicity left open in [6]. As a consequence, we prove that, for every positive integer $t$, there exist infinitely many square roots of integers whose Browkin continued fraction expansion is periodic with period length $2t$. This also settles a problem originating from previous classical results on the possible period lengths of square roots of integers. Moreover, we provide new sufficient conditions for the periodicity of quadratic irrationals and derive explicit bounds for the corresponding period lengths. These results also restrict the possible behaviour of a quadratic irrational whose Browkin continued fraction expansion is non-periodic, and therefore provide new tools for investigating whether an analogue of Lagrange's theorem can hold. Finally, we also provide a computational study of the Browkin expansions of quadratic irrationals, with particular attention to the behaviour of square roots of integers and to the search for possible non-periodic examples.

We relate the systems of Hecke eigenvalues arising from the (mod $p$) modular forms on the Shimura curve attached to the indefinite quaternion algebra $B$ of discriminant $\delta$ over $\mathbf{Q}$ to the systems of Hecke eigenvalues arising from the (mod $p$) algebraic modular forms attached to the definite quaternion algebra $D$ of discriminant $p\delta$. Moreover, we discuss details of the Hecke structure on the latter spaces, following ideas of Serre. The entire setup can be seen as a Shimura curve analogue of Serre's letter to Tate on quaternions and modular forms. The bijection between the sets of systems of Hecke eigenvalues is a special case of recent work of Terakado and Yu; the novelty of this paper is the explicit nature of the construction, allowing for finer control of its behaviour with respect to weights, as well as the results on the Hecke module structure.

Let $K$ be a number field and $S$ a finite set of places including the archimedian ones. Let $(a_1,a_2,a_3)\in K^*$ we consider the $S$-unit equation $a_1x+a_2y=a_3$ with $x,y\in U_S$. In this paper we prove that for fixed $a_1,a_2$ the equation has at most $\frac{|S|-1}2$ solutions provided that the height of $a_3$ is sufficiently large. We want to emphasize that our results are effective.

We show that for any henselian valued field of mixed characteristic $(0,p)$, the (existential) $\mathcal{L}_{\mathrm{val}}$-theory of the valued field is determined by the (existential) theory of the value group in $\mathcal{L}_{\mathrm{oag}}$ with a constant for $v(p)$ and the (existential) theory of the residue ring $\mathcal{O}_v/(p)$ in an expansion $\mathcal{L}_{\mathrm{Witt}}$ of the language of rings, provided $\mathcal{O}_v/(p)$ is semi perfect. We moreover show that the $\mathcal{L}_{\mathrm{Witt}}$-structure on $\mathcal{O}_v/(p)$ is $\mathcal{L}_{\mathrm{ring}}$-definable using constants, and that this is exactly the structure induced on $\mathcal{O}_v/(p)$ by the ambient valued field. As a consequence, we obtain a relative quantifier elimination result (eliminating $K$-quantifiers) in a suitable language for the theory of roughly deeply ramified henselian valued fields of mixed characteristic $(0,p)$.

We revisit Zhuravlev's 1974 quantitative form of Selberg's sign-change method for the zeros of the Riemann zeta function on the critical line. Zhuravlev's work, originally in Russian and little known in the West, established an explicit positive proportion of such zeros. Using modern techniques we optimise the arithmetic mean value at the heart of the method, which has an exact closed form for reciprocal-square-root coefficients and is improved further by a positive-semidefinite family of mollifiers. We thereby obtain the strongest result yet known from Selberg's method, proving that at least $7\%$ of the non-trivial zeros of the Riemann zeta function lie on the critical line.

We investigate how Heegner points detect the finiteness of the $p$-primary part of Tate-Shafarevich groups of semi-stable elliptic curves over the rationals of arbitrary rank with any good reduction prime $p \geq 5$ with irreducible mod $p$ representation and partially recover Çiperiani–Wiles' theorem on solvable points on genus one curves. As a key ingredient of the proof of these results, we also give a concise proof of Kolyvagin's conjecture for semi-stable elliptic curves with supersingular reduction with a relevant choice of an imaginary quadratic field. Our approach is independent of the cyclotomic Iwasawa main conjecture for elliptic curves with supersingular reduction.

Let $\Omega$ be a lattice in $\mathbb{C}$ with algebraic invariants and complex multiplication, let $\mathcal{E}$ be the elliptic curve associated with $\Omega$, and let $\wp$ be the Weierstrass function relative to $\Omega$. Set $ k:=\operatorname{End}(\mathcal{E}) \otimes_{\mathbb{Z}}\mathbb{Q}.$ We prove that if $t_1,\dots,t_s$ are $\mathbb{Q}$-linearly independent algebraic numbers and $p_1,\dots,p_n$ are $k$-linearly independent algebraic numbers, then the $s+n$ numbers \[ \mathrm{e}^{t_1},\dots,\mathrm{e}^{t_s}, \wp(p_1),\dots,\wp(p_n) \] are algebraically independent over $\overline{\mathbb{Q}}$. The proof uses the Tannakian description of the Lie algebra of the unipotent radical of the $1$-motive associated with these points.

We prove that the bound \[ \max \{ |16A|,|16f(A)| \} \gg_m |A|^{\frac{3}{2}+\frac{1}{162}} \] holds for any polynomial $f$ with degree $m \geq 2$ and any finite $A \subset \mathbb R$. This shows that the classical Jarník obstruction to growth beyond exponent $3/2$, which occurs for general strictly convex functions, cannot occur for polynomial functions.