Papers

Let $q$ be an odd prime power and $m\geq 1$. Let $Q$ be a nondegenerate quadratic form on $\mathbb F_q^{2m}$. For every sphere $S_j=\{Q=j\}$ with $j\in \mathbb F_q^\times$, we prove the sharp extension estimate $R_{S_j}^*(2\to r)\lesssim_m1$ for $r\ge 2(m+1)/m$, uniformly in $q$, $Q$, and $j$. As an application, we show that if $E\subseteq\mathbb F_q^{2m}$ satisfies $|E|/q^{m+1/3}\to\infty$, then almost every pin $y\in E$ determines $(1-o(1))q$ values of $Q(x-y)$. The proof combines two arithmetic Hecke operator estimates with an induction in the dimension, a centered sphere–cone estimate, and an orthogonal decomposition.

Let $N$ be a square-free, positive integer and let $p\ge 5$ be a prime dividing $N$. In this work, we construct semistable models of Atkin–Lehner quotients of $X_0(N)$ over the Witt ring $W(\overline{\mathbb{F}}_p)$ by taking the corresponding quotients of the Deligne–Rapoport model of $X_0(N)$. We give an explicit description of the special fiber of the models and obtain genus formulae for the Atkin–Lehner quotients of $X_0(N)$.

Let $f$ be a primitive holomorphic cusp form of arbitrary weight, level and nebentypus. We prove $\int_0^T|L(1/2+it,f)|^6\,dt\ll_{f,\epsilon}T^{2+\epsilon}$, extending Jutila's level-one theorem. The new ingredient is a large sieve comparing the stationary phases in the Booker-Milinovich-Ng transformation at different heights. Applications include moments of order $T^{1+\epsilon}$ on every line to the right of the critical line, $\Omega(T^{4/35-\epsilon})$ simple zeros, zero density estimates, and bounds for cubic-field power sums, shifted convolution sums and the general divisor problem.

Colored partitions constitute a well-established and extensively studied area of research with numerous generalizations arising from imposing restrictions on the allowed colors of the parts. In particular, restricted colored partitions have received considerable attention owing to their rich combinatorial and arithmetic properties. Motivated by these developments, we consider a colored overpartition function $\overline{\mathfrak{C}}_{m,r,s}(n)$, which enumerates the overpartitions of $n$ such that each part divisible by $m$ can be assigned any of $r$ colors, while each part not divisible by $m$ may be assigned any of $s$ colors. We investigate an asymptotic formula and several arithmetic properties of this function, including its density properties.

We establish a multiplicity-one refinement of the non-archimedean type-II Howe correspondence: whenever the small theta lift is non-zero, it occurs in the big theta lift with Jordan-Hölder multiplicity one. Furthermore, we apply this result alongside the Godement-Jacquet zeta integral to compute big theta lift for Arthur-type representations. Based on these computations, we also propose a conjecture regarding the irreducibility of the big theta lift.

Motivated by questions arising in noncommutative Iwasawa theory, let $R$ be a (not necessarily commutative) ring, $T \in R$ a regular central element, and $M$ a pseudo-null $R$-module. We investigate necessary and sufficient conditions under which the quotient $M/TM$ is pseudo-null as an $R/TR$-module. We first give necessary and sufficient conditions in terms of associated prime ideals when $R$ is commutative. Then we apply this to the case when $R$ is a Krull domain and obtain a precise relationship between the characteristic ideals of $M/TM$ and the $T$-torsion submodule $M[T]$. We give necessary and sufficient conditions in terms of $\operatorname{Ext}$ groups when $R$ is a noncommutative ring and then compare them with the criterion when $R$ is commutative. Lastly, we give an application to noncommutative Iwasawa theory. Roughly speaking, if `big' dual fine Selmer group is pseudo-null, then `most' specialized dual fine Selmer group is pseudo-null.

Let $\Pi$ be a regular algebraic cuspidal automorphic representation of $\mathrm{GL}_{3}(\mathbb{A}_{\mathbb{Q}})$ of weight $(a, 0, -a)$, and let $p$ be a prime. Generalising work of Loeffler–Williams in the nearly-ordinary case, we construct $p$-adic $L$-functions for each small slope regular $P_{1}$-refinement of $\Pi$ at $p$ consistent with the conjectures of Coates–Perrin-Riou and Panchishkin.

For a quasi-projective arithmetic surface $\mathcal U/\mathbb Z$ and a compact subset $E\subset \mathcal U(\mathbb C)$ under mild regularity assumptions, we study algebraic integral points on $\mathcal U$ whose Galois orbits lie in $E$. We characterize the collections of local probability measures that arise simultaneously as the local limit distributions of Galois orbits of such points. Our result also works for measures prescribed at a subset of places. This generalizes the results of Smith and Orloski–Sardari on $\mathcal{U}=\mathbb{A}^1$ concerning the archimedean place. As a consequence, we prove an integral version of Szachniewicz's theorem on curves, which states that every integral GVF functional can be approximated by a sequence of algebraic integral points in the GVF topology. We also give several applications: we prove that the essential minimum of height functions on curves can be attained by algebraic integral points; we connect integer Chebyshev constants with the essential minima of certain height functions on $\mathbb{A}^1$ and prove a conjecture of Montgomery; we also answer affirmatively a question of Levenberg–Londhe by showing that the smallest limit of averaged trace of totally positive algebraic integers can be attained by a sequence of totally positive algebraic units.

Let $G$ be the graph on the positive integers in which distinct vertices are adjacent if they differ by one or their quotient in one order is prime. We prove that $6\leq\operatorname{diam}(G)\leq7$ and $4\leq\operatorname{rad}(G)\leq5$. Under Dickson's conjecture for two linear forms, we prove that $\operatorname{diam}(G)=6$ and $\operatorname{rad}(G)=4$.

Choi, Erdős and Szemerédi showed that there exists an absolute constant $C$ such that for all subsets $A \subseteq \{1, 2, \ldots, 2n\}$ with at least $n+C$ elements, there exist four distinct positive integers whose pairwise sums are all contained in $A$. A proof that one can take $C = 3166$ was recently sketched by the first author, and here we show that we actually have $C = 4$ for all $n \ge 6$, which is optimal. The proof we present was originally conceived of by AI, with the final result proved and formalized by Aristotle, the formal reasoning agent developed by Harmonic.

Ekedahl's theory relates diagonal complexes to \(F\)-gauges, but turning this equivalence into explicit calculations requires careful control of its constructions. These notes explain the equivalence and clarify the generators-and-relations definition of \(\rmt\). At each finite level, we construct bounded projective resolutions of bounded coherent gauge complexes by finite sums of the standard generators. Morita equivalence expresses \(\rmt\) as a tensor product functor. Comparing the constructions across levels yields an exact fully faithful embedding of the diagonal heart into \(\Ch(R)\). Finally, we compute three distinct domino types arising from abelian threefolds that are all supergeneral.

Radchenko and Wheeler (RW) recently proved a finite pentagon relation for real quadratic special values of the modular quantum dilogarithm and used it to establish the rank-$1$ twisted convolution identity conjectured by the current authors. RW gave an explicit argument in the principal case and remarked that their proof holds for all rank-$1$ admissible tuples. We extend their proof to all rank-$r$ admissible tuples and provide an explicit dictionary between the modular quantum dilogarithm and the Shintani-Faddeev modular cocycle conventions in the respective papers. Thus, we establish that, if $d,r$ are positive integers such that $r<\frac{d-1}{2}$ and $\frac{d^2-1}{r(d-r)} \in \mathbb{Z}$, then there exist ghost $r$-SICs: i.e., configurations of $d^2$ rank-$r$ subspaces in $\mathbb{C}^d$ that satisfy a non-Hermitian equichordal condition. Under the Stark conjecture, these configurations are Galois conjugate to Hermitian equichordal configurations called $r$-SICs (or rank-$r$ SIC-POVMs).

In this article, we study the asymptotic expansion of the analytic torsion associated with positive Hermitian line bundles on projective spaces. We show that the expansion contains no terms of the form $k^{-i}\log k$ for $i\geq 1$. The proof combines the arithmetic Riemann-Roch theorem with an explicit computation of the arithmetic degree of spaces of global sections.

Let $h\geq2$ be an integer and let $1/h<\theta\leq1/(h-1)$. In this paper, we prove that there exists a strongly minimal asymptotic basis $A$ of order $h$ such that $A(x)\asymp x^\theta$. This solves a problem posed by Nathanson in 1988.

In this article, we study the monogenity of the number field generated by a root of an irreducible factor of the generalized $(k,t)$-Fibonacci and $(k,t)$-Lucas polynomial sequence. We use Dickson polynomials and their connection with cyclotomic fields to prove our main results. As special cases, we recover the monogenity results for the classical Fibonacci and Lucas polynomials and obtain corresponding monogenity results for Chebyshev polynomials of the first and second kinds.

We give an exposition of some of the ideas leading up to Jacob Tsimerman's groundbreaking work developing the theory of o-minimal definability in arithmetic and algebraic geometry.

We investigate Miller's bias conjecture for general one-parameter families of elliptic curves over $\mathbb{Q}$ with non-constant $j$-invariants. Utilizing methods developed by Michel, we view the second moment as the sum of distinct cohomological components: a main $p^2$ term, a $p^{3/2}$ term arising from the first cohomology over the projective line, and terms of order $p$ or lower corresponding to singular fibers. By assuming the generalized Sato-Tate conjecture, we employ representation and motive theory to demonstrate that the coefficient of the $p^{3/2}$ term averages to zero in the limit. We then independently analyze the lower-order contributions without assuming Sato-Tate. By applying the Chebotarev density theorem and analyzing Galois representations, we show that the contributions to the order $p$ term from additive singular fibers also average to zero. Consequently, the only remaining average contribution to the $p$ term is $-B_pp$, where $B_p$ denotes the number of “bad” points where the discriminant vanishes. Because $B_p$ averages to a strictly positive value, the overall coefficient of the $p$ term averages to a strictly negative value in the limit, thereby establishing the bias conjecture for these families.

c-Polynomials are polynomials whose (non-zero) coefficients are all equal to 1. Taking a theorem by Carlitz and Moser as a motivation, we prove a structure theorem for the factorisations of the c-polynomial $(x^n-1)/(x-1)$ into c-irreducible factors by relating c-torisations into c-polynomials to joint ordered factorisations arising from the integer n. The question of how many different joint ordered factorisations there are leads to the precise chromatic polynomial for path graphs, counting the ways of path graph colourings with $m$ colours under the condition that all colours are used. Moreover, we give similar counting formulae for joint ordered factorisations under the constraints that all factors are primes, or that all factors are square-free.

Let $p\geq 5$ be a prime, and let $f$ be a $p$-ordinary cuspidal newform of even weight $k = 2r \geq 2$ and level $N$, with $p \nmid N$ and trivial nebentype. In this article, we prove Kato's Iwasawa main conjecture for $f$, formulated in terms of the $p$-adic $L$-function of $f$, under the sole assumptions that the associated Galois representation of $f$ satisfies the big-image hypothesis and that its residual representation is absolutely irreducible. This generalises the main result of Burungale–Castella–Skinner, which treats elliptic curves over $\mathbb{Q}$ with good ordinary reduction at $p$. Our proof follows their general strategy. A key new ingredient is the construction, over an imaginary quadratic field in which $p$ splits, of a zeta element attached to $f$, together with a proof of its explicit reciprocity laws, generalising the corresponding work of Burungale–Skinner–Tian–Wan. These results provide the higher-weight analogue of a crucial input in the argument of Burungale–Castella–Skinner and allow their method to be carried out for higher-weight modular forms.

In this paper we study Euclidean triangles associated with Diophantine $D(4)$-triples. For a $D(4)$-triple $\{a,b,c\}$, $a<b<c$, we prove that the lengths $r=\sqrt{ab+4}$, $s=\sqrt{ac+4}$ and $t=\sqrt{bc+4}$ associated with this $D(4)$-triple form a non-degenerate triangle if and only if the triple is regular and $r>b-a$. For a fixed shortest side $r$, we describe all triangles associated with $D(4)$-triples in terms of the factorisations of $r^2-4$, give a divisor criterion for their number, and determine their angle type. We also prove that two similar triangles associated with $D(4)$-triples are necessarily congruent. Finally, for the family of $D(4)$-triples $\{r-2,r+2,4r\}$, we obtain a Pell-equation characterisation of the Heronian triangles associated with these $D(4)$-triples.

Questions concerning representations of integers as the sum of a prime and a power of two go back to the correspondence between Euler and Goldbach in 1752 and to de Polignac's work of 1849. Romanoff proved that the set of integers admitting such a representation has positive lower density. Following the complementary direction studied by Kalmynin and Konyagin, we consider long intervals containing no Romanoff numbers. We use truncated products of divisor sums to derive the bound $G_{\mathcal{R}}(X)\gg\log\log X$ for the longest block in $[1,X]$ containing no integer of the form $p+2^n$, with $p$ prime and $n\ge1$. This improves the bound $G_{\mathcal{R}}(X)\gg\log\log X/\log\log\log X$ of Kalmynin and Konyagin. The key step extends a translate lemma to sets of $O(\log X)$ integer shifts of absolute value at most $X$: auxiliary primes dividing differences of shifts are removed at negligible cost. After preliminary sieving, this allows all remaining shifts to be covered simultaneously. The argument is unconditional and also applies to $p+a^n$ for every fixed integer $a\ge2$.

Given integers $k,\ell \geq 2$, define the Erdős–Selfridge curve \begin{align*} \mathcal{C}_{k,\ell} : y^\ell = x(x+1)\cdots (x+k-1). \end{align*} Bennett and Siksek proved that if $(x,y)$ is a rational point on $\mathcal{C}_{k,\ell}$ with $\ell$ a prime and $y \neq 0$, then $\ell$ is bounded (doubly-exponentially) in terms of $k$. We improve this bound on $\ell$ when $k$ is sufficiently large. Our method uses combinatorial and sieve-theoretic arguments, as well as results on solutions to generalized Fermat equations.

We prove a geometric version of Poonen's "Mordell–Lang plus Bogomolov" theorem for semi-abelian varieties in characteristic zero. This is a generalization of the Mordell–Lang conjecture and the geometric Bogomolov conjecture.

Let $M$ be an imaginary quadratic field. Let $A$ be a polarised abelian threefold defined over $M$ with geometric endomorphism algebra isomorphic to $M$. We study residual Galois representations attached to $A$. In particular, we describe the endomorphism field and the natural restrictions placed on the residual representations by their endomorphisms. We then devise criteria for the image of residual Galois representations to be large, and provide efficient algorithms suitable for large scale calculations. Subsequently, we apply this algorithm to millions of curves in several families, whose Jacobians are abelian threefolds with imaginary multiplication, and to Sutherland's dataset of $7$-smooth Picard curves. We produce an explicit example of a Picard curve which appears to have an isogeny of degree $13$ defined over $\mathbb{Q}(\zeta_3)$. The Jacobians of all other curves, in the range of our computation with endomorphism algebra $M$, have mod-$\ell$ image as large as possible for any prime $\ell>7$. This allows us to realise the group $\Gamma\mathrm{U}_3(\ell)$ of unitary semisimilitudes as a Galois group over $\mathbb{Q}$ for all $\ell \not\equiv 1, 25, 121 \pmod{168}$. Our algorithms also led us to the discovery of several interesting rational families of Picard curves whose generic members appear to have endomorphism algebra of dimension $6$ which is not a CM field. These represent curves on the Picard modular surface and appear to be Shimura curves. Some of them appear to parametrise non-principally polarised abelian surfaces with quaternionic multiplication.

In 2018, Andrews defined and studied the partition function $\mathcal{EO}(n)$, which counts the number of partitions of $n$ where each even part is less than each odd part. Thereafter, Uncu considered a different subset of such partitions, namely those partitions counted by $\mathcal{EO}(n)$ where there are no repeated even parts (we denote the number of such partitions by $\mathcal{EO}_u(n)$). In this paper, we prove several congruences modulo powers of $2$ for $\mathcal{EO}_u(n)$. We also prove some infinite families of congruences as well as a recurrence for the number of this class of partitions. Our methods involve elementary, algorithmic, and modular form techniques.

We prove a conjecture of Smyth on the Mahler measure of non-reciprocal trinomials of height $1$, determining exactly when their Mahler measures are greater than or less than $M(x+y+1)=\rho=1.381356\ldots$, conjectured to be the smallest limit point of $M(P)$ for non-reciprocal polynomials. Conjecturally, our result gives the full spectrum of $M(P)<\rho$ for all non-reciprocal polynomials, not just for trinomials.

For integers $m\geq 2$ and $m\mid N$, let $G(N;m)=\gcd\{\binom{N}{k}:0<k<N,\ m\mid k\}$. We prove a complete $p$-adic valuation formula for $G(N;m)$ at primes $p\equiv -1\pmod m$, under the hypotheses $m\geq 3$, $m\mid N$, and $m<N$. Writing $N=\sum_i d_i p^i$, put $A=\sum_{i\text{ even}}d_i$ and $B=\sum_{i\text{ odd}}d_i$. Then $v_p(G(N;m))$ is $2$ in the exceptional mixed case $(A,B)=(1,1)$ with $p=m-1$, is $1$ in the mixed case $(A,B)=(1,1)$ with $m<p$, is $1$ in the one-parity cases $(A,B)=(m,0)$ and $(A,B)=(0,m)$, and is $0$ otherwise. The proof uses Kummer's theorem to translate the problem into digitwise borrow counts and a minimal signed zero-sum classification. A source-level literature audit through 25 September 2026 located no equivalent prior theorem for the full minus-one classification: McTague's corrected same-residue extension covers the one-parity $p>m$ subcases, but not the mixed-parity branch or the $p=m-1$ regime. The theorem, its plus-one companion, a scaling reduction, and the $m=3,4,6$ specializations have been formalized in Lean 4 / Mathlib.

The 290-Theorem of Bhargava-Hanke completely classifies all universal quadratic forms over the rational integers in terms of a finite criterion set: a positive definite quadratic form is universal if and only if it represents every integer in this finite set. In this article, we study diagonal forms of higher degree over a totally real number field and prove the existence of a unique finite criterion set, minimal with respect to inclusion, for universal diagonal forms of higher degree. As part of the proof, we establish an analog of the asymptotic local-global principle for such forms and develop a local theory for the representation of integers by diagonal forms of higher degree.

Let the theta and Epstein zeta functions be $\Theta(\alpha,L)=\sum_{v\in L}e^{-\pi\alpha|v|^2}$ for $\alpha>0$ and $E(L,s)=\sum_{v\in L\setminus\{0\}}{|v|^{-2s}}$ for $s>2$, respectively. We consider full-rank lattices $L\subset\mathbb R^4$. Let the covolume of $L$ be one, and let $\mathcal D_4=2^{-1/4}D_4$ be the root lattice $D_4$ rescaled to covolume one. We prove that, up to orthogonal transformations, \begin{equation}\nonumber arg\,min_{covol(L)=1}\Theta(\alpha,L)=\mathcal D_4 \qquad\text{if}\quad \alpha>0, \end{equation} and this implies that \begin{equation}\nonumber arg\,min_{covol(L)=1}E(L,s)=\mathcal D_4 \qquad\text{if}\quad s>2. \end{equation} This proves that the root lattice minimizes the theta and Epstein zeta functions among all lattices in dimension four, as conjectured by Sarnak-Strömbergsson (\cite{SS}, 2006), who established the local minimality of $\mathcal D_4$. Thereby, this resolves the Rankin-Sobolev problem in dimension four dating back to Éndibaev (\cite{End1978}, 1978) and Shushbaev (\cite{Shu1978}, 1978).

We prove an analog of the Ax-Schanuel theorem for the Drinfeld $j$-function in odd characteristic. Roughly speaking, if the graph of $\boldsymbol{j}\colon\Omega^n\rightarrow\mathbb{A}^n_{\mathbb{C}_\infty}$ and its derivatives has an atypical intersection $\mathcal{V}$ with an algebraic variety, then $\mathcal{V}$ projects to a weakly-special subvariety in $\mathbb{A}^n_{\mathbb{C}_\infty}$. More generally, we prove a positive characteristic analog of the differential Galois theoretic Ax-Schanuel theorem of Blázquez-Sanz, Casale, Freitag and Nagloo. Our main theorem for $\boldsymbol{j}$ follows by applying this result to a suitable foliation. A theorem of Pink allows us to conclude that the special varieties in the sense of Blázquez-Sanz et al. in this context agree with the classical weakly-special subvarieties in the Drinfeld sense.