Papers

For a smooth projective curve $C/\mathbb{Q}$ of genus $\geq 2$ and $L/\mathbb{Q}$ an extension, we write $C(L)_{\text{new}}=\{P\in C(L):\mathbb{Q}(P)=L\}$. Recent work of Khawaja and Siksek conjectures that this set is empty for $100\%$ of degree $n$ number fields $L$, when ordered by absolute discriminant. Moreover, they bring evidence towards this conjecture when $C$ is a degree $n$ cover of $\mathbb{P}^1$. We complement their work by proving analogous results for degree $n$ covers $\psi:C\to E$ of elliptic curves $E$. Our main result shows that, under suitable hypotheses, the number of distinct absolute discriminants at most $X$ of primitive degree $n$ fields $L$ with $C(L)_{\text{new}}\neq\varnothing$ is $O(X^{1/2})$ or $O(X/(\log X)^{\alpha})$, for some $\alpha>0$. In degrees $2,3,4$ and $5$ we show that these fields have density $0$ among all fields of the same degree (in degree $4$, also among the primitive ones). The novelty is for degrees $4$ and $5$, where we use work of Bhargava–Shankar–Wang and McGown–Thorne–Tucker to count fields with specified local constraints. Moreover, we give concrete examples of $8$ bielliptic modular curves $X_0(N)$, for which $X_0(N)(L)_{\text{new}}=\emptyset$ for $100 \%$ of quadratic fields $L$. Lastly, we point out modular covers of degrees $3$ and $5$ in the LMFDB for which similar conclusions hold.

The alternating composition of $N = 2p$ weighted differential operators $w_j(x)\cdot\partial_x^{\,p}$ of strict order $p$ on the line $\mathbb{R} \ni x$ is again an operator of order $p$; its coefficient is the universal constant $c(p)$ times the Wronskian of the weights $w_1,\ldots,w_N$. Lie brackets of vector fields fix $c(p=1)=1$; we want to find $c(p \geqslant 2)$: e.g., $c(2) = 2$ or $c(3) = 90$. Direct symbolic expansion (over $|S_{2p}| =(2p)!$ permutations) fails for $p \geqslant 4$. Taking the monomials $w_j = x^{j-1}$ reduces the summation to the much smaller set $\Phi_p \subseteq S_{2p-1} \subsetneq S_{2p}$ of late-growing permutations. Expressing $c(p)$ as a signed sum of products of falling factorials, we implement and speed up the algorithm that gains all the integer values up to $c(18) = 4.881\ldots \cdot 10^{462}$. The resulting sequence is new, now registered as OEIS A392714; its (sub)leading-order growth rate is $\log c(p) \simeq 2p^2\log p -b p^2 + \overline{o}(p^2)$ for $p\gg 1$, with $b\geqslant 2.6744$.

Recently, Girsch-Kurinczuk have obtained new formulas for local newforms in depth-zero and minimax integral-depth cuspidal representations of $p$-adic $\mathrm{GL}(n)$. One of their main results is an averaging formula which provides an integral representation of the Whittaker newform in terms of the Paškūnas-Stevens Whittaker function. In the present paper, we first prove a reverse averaging formula by establishing an integral representation of the Paškūnas-Stevens Whittaker function in terms of the Whittaker newform. Together, these formulas give an explicit two-way passage between these two distinguished vectors. Finally, we use our averaging formula and earlier work of Kurinczuk-Matringe to give new explicit test vectors in terms of Whittaker newforms for such cuspidal $\mathrm{GL}(n)\times\mathrm{GL}(n)$ pairs, whenever their Rankin-Selberg $L$-factor is not identically equal to $1$.

We denote by $f_k(\Gamma)$ the largest integer with the property that every subset of a finite abelian group $\Gamma$ of size at least $|\Gamma| - f_k(\Gamma)$ is a $k$-fold sumset. Extending a recent result of Alon and Pham, we prove that $$ f_k(\Gamma) \leq \widetilde{O} \left(n^{(2k-1)/(4k-3)}\right) $$ holds for all finite abelian groups $\Gamma$ and integers $k \geq 2$, where $n = |\Gamma|$. Additionally, we also show that the lower bound $f_k(\Gamma) \geq \widetilde{\Omega} \left(n^{1/k}\right)$ holds if $\Gamma$ has no nontrivial element of order dividing $k$. Our upper bound improves a previous result of Balogh, Liu, and Sharifzadeh, and recovers the bound of Alon and Pham in the case $k = 2$. The proof relies on a new upper bound for the independence number of random Cayley sum hypergraphs, which may be of independent interest.

For an integer $g$ not equal to $-1$ or a square, we study the least prime $p_g$ such that $g$ is a primitive root modulo $p_g$. We show that, assuming the Generalised Riemann Hypothesis, all such $g$ with $|g|\in[N,2N]$ have $p_g\leq(\log |g|)^{3.44}$ with at most $O((\log N)^{2.44})$ exceptions. This can be directly compared to the uniform bound $p_g\leq(\log |g|)^{19}$ proven recently by Fan and Pollack. To obtain our result, we prove a new variant of Gallagher's larger sieve, which may be of independent interest. In addition to our conditional result, we discuss other “almost all" bounds for $p_g$ that hold unconditionally.

Let $f: \mathbb{N} \rightarrow (0,\infty)$ be a multiplicative function, and let $\omega > 0$. We consider the frequency with which patterns of the form $$ f(n) > \omega \max\{f(n-1),f(n+1)\}, \quad f(n) < \frac{1}{\omega}\min\{f(n-1),f(n+1)\} $$ occur in the sequence $(f(n))_n$, and give a sharp classification of all positive-valued multiplicative functions for which at least one of these patterns only occurs with logarithmic density $0$. As a particular application, we resolve, in a strong form, two conjectures of A. Sárközy from 2001, classifying all multiplicative functions $f: \mathbb{N} \rightarrow \mathbb{N}$ that have either finitely many local maxima, or finitely many local minima. The proof is based on a natural “pretentiousness” dichotomy for additive functions. Key tools in the analysis include an analogue of the Matomäki-Radziwiłłtheorem for additive functions, due to the author, as well as a new bound for the concentration of the gaps $g(n+1)-g(n)$ of an additive function $g$, which may be of independent interest.

We study the structure of finitely ramified henselian valued fields of mixed characteristic with arbitrary residue fields via their residue rings of higher length, where a residue ring of length $n$ is the quotient ring of the valuation ring by the $n$th power of its maximal ideal. We prove an approximate lifting theorem for homomorphisms between residue rings of higher length and explicitly determine the optimal bound of error of lifting. As applications, we obtain several Ax-Kochen-Ershov principles for relative completeness, relative existential completeness, and existential closedness using the pure ring structures on the residue rings. And we show that any formula on a residue ring of length $k$ is equivalent to a normal form of sentences in pure ordered group structure and special formulas at level $n$, uniformly for all finitely ramified henselian valued fields of mixed characteristic $(0,p)$ and the same initial ramification index $e$. Here, for $n\ge k$, a special formula at level $n$ represents a definable set on a residue ring of length $k$ given by a projection image of a definable set in pure ring structure on a residue ring of length $n$. Also, such $n$ is optimally computed from the precise estimation of error of lifting and depends only on $k$, residue characteristic $p$, and initial ramification index $e$.

The classical More Sums Than Differences (MSTD) problem studies finite sets $A\subset\{0,1,\ldots,n\}$ for which $|A+A|>|A-A|$, where $A+A=\{a_1+a_2:a_1,a_2\in A\}$ and $A-A=\{a_1-a_2:a_1,a_2\in A\}$. As addition is commutative and subtraction is not, it was conjectured that as $n\to\infty$ almost all subsets $A$ chosen uniformly from the power set of $\{0,1,\ldots,n\}$ are difference-dominated, and it was thus a surprise when Martin and O'Bryant proved a positive percentage of sets are sum-dominant. We greatly generalize this model by introducing a Markov-chain framework, where the classical MSTD model is now just a special case. Let $(X_i)_{i=0}^n$ be a stationary two-state Markov chain on $\{0,1\}$ with transition probabilities $P(0,0)=p$ and $P(1,1)=q$, where $p,q\in(0,1)$. We include $i$ in $A$ exactly when $X_i=1$, and define $A=\{i\in\{0,\ldots,n\}:X_i=1\}$. The usual independent Bernoulli model is recovered when consecutive inclusion decisions are independent, equivalently when $p=1-q$. In particular, the uniformly random subset model corresponds to $p=q=1/2$. Using the fringe-middle method from the MSTD literature, we show that the middle sums and differences are filled with high probability, so the comparison between $|A+A|$ and $|A-A|$ is again governed by endpoint fringes. By fringe manipulation, we prove that the probabilities of sum-dominant, difference-dominant, and balanced sets tend to strictly positive limits as $n\to\infty$. We also give numerical estimates of these three probabilities for finite $n$ over a range of values of $p$ and $q$. Through combinatorial methods, we find a closed-form expression for $\mathbb{E}[|A-A|-|A+A|]$ as $n\to\infty$.

We introduce new techniques to study mixed-characteristic special loci in moduli spaces and give several applications, focusing in particular on the moduli space $\mathsf{M}_g$ of curves. These applications include Zariski density results for split Jacobians, non-density results for supersingular loci, and lifting results for algebraic cycles, all under appropriate numerical conditions. Our methods generalize to more-or-less arbitrary moduli spaces mapping to Hodge-type Shimura varieties (and even some settings beyond the Shimura case), and relate to questions raised by Oort and de Jong.

Let $L$ be a number field containing a CM field $K$. We compute an explicit current representing the $\mathrm{GL}_{\mathcal{O}_K}(\mathcal{O}_L)$-equivariant algebraic Eisenstein-Kronecker classes constructed by Kings-Sprang. In the Bianchi case, where $K$ is imaginary quadratic and $[L:K]=2$, this yields a purely algebraic construction of Eisenstein cohomology classes, even though the associated locally symmetric space is not an algebraic variety. We express their constant terms at the cusp in terms of partial Hecke $L$-values of $K$.

Let $E/F$ be a CM extension and let $1\le r<n$. We determine primitive Fourier coefficients of Hermitian Klingen-Eisenstein series on $\mathrm{U}_{n,n}$ induced from cusp forms on $\mathrm{U}_{r,r}$, at principal congruence level. Using the pullback formula for Hermitian Eisenstein series, we express a normalized primitive Fourier coefficient of a Klingen-Eisenstein series as the corresponding primitive Fourier coefficient of a Siegel Eisenstein series multiplied by a Petersson inner product with an explicitly constructed holomorphic automorphic form. The automorphic form in the Petersson inner product is a finite sum of products of Hermitian theta series and Siegel Eisenstein series of degree $r$. Under some hypotheses, we also prove the $p$-integrality of these normalized primitive Fourier coefficients.

In this paper, we study two questions concerning perfect powers in $S$-unit equations and sparse representations of perfect powers. Corvaja-Zannier \cite{corvaja2013finiteness} proved that there are only finitely many odd perfect powers in $\N$ having precisely four non-zero digits in their binary expansion. At first, we prove the finiteness of the set of solutions to the equation \begin{equation*} y^d=1+c_1g^{m_1}+c_2g^{m_2}+c_3g^{m_3}, \quad 0<m_1<m_2<m_3, \end{equation*} where $c_i$ are $S$-units for a suitable set $S$. The proof combines consequences of the Subspace Theorem concerning integer values of analytic series at $S$-unit points with Roth's theorem for number fields. As a second result, we extend the work of Bennett and Bugeaud \cite{bennettbugead2013perfect} by proving that, for every fixed integer base $g\geq 2$, there exists a constant $q_0(g)$ such that, for every prime $q>q_0(g)$, no perfect $q$-th power admits a representation with five non-zero digits in base $g$. The proof relies on explicit lower bounds for linear forms in both Archimedean and non-Archimedean logarithms.

We extend a result by Ikeda and Suriajaya (2025) to the average number of representations of an integer as a sum of an arbitrary number $j\ge 3$ of primes. We show that the corresponding asymptotic formula holds in the general case. In the proof we use new ingredients like averages for the singular series of the generalized Goldbach problem.

We show that any integral cubic form in ten or more variables has a nontrivial integer zero. Like earlier authors on this topic, we use geometric methods to produce a zero in some cases, and the circle method in all other cases. New ingredients are required on both fronts: in the first case, bounds on the dimension of singular loci and incidence varieties, and in the second case, a stratified geometric sieve, together with stratification results for point counts over finite fields and a cube-root Hensel lifting analysis.

The 150 years old Sylvester's conjecture asserts that every prime $p\equiv 4,7,8\mod 9$ is the sum of two rational cubes. Recently, a proof of this old problem is obtained by the work of mine for the case 4,7 \cite{Yin} and the work of Burungale-Tian for the case 8 \cite{BT}. In this paper we give a shorter proof by deep mining my result in \cite{Yin} combining with the ideas of my previous work on case 8 in \cite{Yin8}. In particular, we use the ordinary trace and the Kronecker congruence of modular functions, but do not use the $\sqrt{-3}$-division boundary and Fermat isogeny as in \cite{BT}.

Let $K$ be a finite field of characteristic two with $|K| = 2^{n}$, let $\gcd(k,n) = 1$, let $d_{k} = 4^{k} - 2^{k} + 1$ be the Kasami exponent, and let $\Delta_{k} = \{(b+1)^{d_{k}} + b^{d_{k}} + 1 : b \in K\}$ be the image of the normalised derivative of the Kasami monomial in the direction $1$. We show that, for all distinct nonzero $v_{1},v_{2} \in K$, \[ \bigl|\{(x,y,z) \in \Delta_{k}^{3} : v_{1}x + v_{2}y + (v_{1}+v_{2})z = 0\}\bigr| = 2^{2n-3}. \] This establishes the cyclic-additive difference-set condition introduced by Carlet and later posed for the Kasami functions at NSUCRYPTO~2019. Starting from the known half-size property of the derivative image, we express the Fourier correction as twisted root counts and prove their required nonnegativity by an incidence argument on the Fermat cubic. An exact average over the slopes then forces equality pointwise. The argument covers every admissible pair $(n,k)$ and has been formalised and machine-checked in Lean~4 with Mathlib.

We prove the Mumford-Tate conjecture in every degree for hyper-Kähler varieties over fields finitely generated over $\mathbb{Q}$. The proof establishes semisimplicity of $\ell$-adic cohomology by eliminating the unipotent radical of the algebraic monodromy group of total cohomology. We also prove the weight-monodromy conjecture for hyper-Kähler varieties over $p$-adic fields. For hyper-Kähler varieties over number fields with $b_2\geq 4$, we establish, after suitable finite extensions, strong compatibility of the associated Weil-Deligne representations valued in Mumford-Tate groups and its integral refinement away from finitely many primes.

We prove a sharp quantitative form of the van der Corput property for squares. For all sufficiently large $N$, we construct a real trigonometric polynomial $P(\theta)=a_0+\sum_{1\le n\le\sqrt N}a_n\cos(2\pi n^2\theta)$ satisfying \begin{equation}\tag{1}\label{eq:abstract-vdc} P(\theta)\ge0\quad(\theta\in\mathbb{R}/\mathbb{Z}),\qquad P(0)=1, \end{equation} with \[0\le a_0\le\exp(-c\sqrt{\log N\log\log N}),\] where $c>0$ is an absolute constant. As a consequence, if $A\subseteq\mathbb{N}\cap[1,N]$ has no nonzero square difference, then \[|A|\le N\exp(-c\sqrt{\log N\log\log N}),\] improving upon a recent result of Green and Sawhney. Our construction builds on previous work of Slijepčević, Slijepčević–Ninčević, and Fan–Lott, and the proof of the upper bound is elementary and self-contained. We also show that every real trigonometric polynomial of the above form satisfying \eqref{eq:abstract-vdc} has \[a_0\ge\exp(-C\sqrt{\log N\log\log N})\] for an absolute constant $C>0$. Thus the van der Corput property cannot be used to obtain a power-saving bound for the Furstenberg–Sárközy theorem.

Let $K/\Qp$ be a finite extension, $\bG$ a connected reductive group over $K$, and set $G = \bG(K)$, considered as a $p$-adic Lie group. We show first that the Bernstein center $\cC_G$ of the category of solid locally $\Qp$-analytic representations of $G$ is isomorphic to the center of the locally analytic distribution algebra $D^\la(G)$. We then consider the Emerton-Gee stack $\frX_{n,K}$ of rank-$n$ $(\vphi,\Gamma)$-modules over the Robba ring for $K$ and determine its connected components. The latter are in canonical bijection with the primitive idempotents of the ring $\cC_{\GL_n(K)}$. Moreover, under the assumption that the ring of global functions on $\frX_{n,K}$ has no non-zero locally nilpotent elements, we show that this ring is isomorphic to a Fréchet completion of $\cC_{\GL_n(K)}$.

Consider an arithmetic progression inside the finite field $\mathbb{F}_{q^n}$. We study the existence of progressions whose terms are all primitive and at least one of them is normal. Using a combination of sieve estimates, ramified bounds, and computational arguments, we improve significantly upon previous literature for progressions of length $m=4$ and obtain partial results whenever $5\leq m\leq 10$. Finally, we provide a general bound which outperforms previous results in the range $m\geq 53$.

We study how the two cusp expansions of the Rogers-Ramanujan continued fraction control its ordinary and logarithmic Mellin moments. Ramanujan's factorizations express the logarithmic transform as a combination of products of zeta and quadratic Dirichlet $L$-functions. Its adjacent integer and half-integer values give an invertible change of coordinates for the corresponding odd special values. We prove that every reciprocal-cusp coefficient is a quadratic irrational algebraic integer, with an explicit minimal polynomial, and that every logarithmic coefficient is quadratic irrational. These facts determine the exact poles of three normalized-moment generating functions. The logarithmic one has an absolutely convergent partial fraction expansion with a pole at every positive square. For the ordinary moments, a two-cusp incomplete-gamma expansion gives meromorphic continuation at every positive splitting point. A general Mellin lemma yields the corresponding nonholonomicity statements. We also establish reflection identities, vertical strip decay and an explicit truncation estimate.

We show that an ir{\-}reducible cuspidal eta quotient of level~$N$ is uniquely identified by its orders of vanishing at $O((\log_2 N)^{r-1})$ cusps, where $r$ denotes the number of distinct prime divisors of $N$. In particular, all such eta quotients which are Fricke eigenforms of a prime power level, are uniquely identified only by their orders of vanishing at~$\infty$. We also prove that for all $k\in\N$, there exists a primitive and irreducible cuspidal eta quotient of weight $k/2$. It follows from the exhaustiveness of Zagier's list of the primitive holomorphic eta quotients of weight~$1/2$ that the 1st Mersmann level, $\cl{M}_1=12$. Here we show that for all $k\in\N$, the $k$-th Mersmann level is divisible by $2^{2k}$.

In this paper we further examine the structure of the graded paramodular Hecke algebra $\mathcal H$. We define a commutative subring $\mathcal{H}^\alpha$ of $\mathcal{H}$ as the elements fixed by a certain automorphism $\alpha$, and we show that the elements $T(q^k)$, defined as the sum of the double cosets of degree $k$, are contained in $\mathcal{H}^\alpha$. We prove that the formal power series in the indeterminate $t$ with coefficients $T(q^k)$ is an explicit rational function in $t$ with coefficients from $\mathcal{H}^\alpha$. We also determine the structure of $\mathcal{H}^\alpha$ and its relation to $\mathcal{H}$.

Let $m,n\geq2$ be integers. We study the least exponent $t_{m,n}$ such that the $m$-fold additive energy of any subset $A$ of the discrete cube $\{0,1,\cdots,n-1\}^d$ in any dimension $d$ satisfies $E_m(A)\leq |A|^{t_{m,n}}$. For every fixed $m$, we obtain the asymptotic formula $$ t_{m,n}=2m-1-(1+o(1))\log_n \frac{(2m-1)^{m-1/2}}{(2m)^{m-1}} \qquad(n\to\infty). $$ For $m=2$, this gives $$ t_{2,n} = 3 - (1+o(1)) \log_n\frac{3\sqrt{3}}{4}, $$ proving a conjecture of the first author.

Let $1,\alpha_1,\alpha_2$ be a basis for a cubic number field $K\subset\mathbb{R}$, and for $\tau_1,\tau_2\ge 1$ let $G(\mathbf{\alpha},\mathbf{\tau})$ denote the number of distinct gaps between the fractional parts of the numbers $m_1\alpha_1+m_2\alpha_2$, with $m_1,m_2\in\mathbb{Z}$, $0\le m_1<\tau_1$ and $0\le m_2<\tau_2$. We show that $G(\mathbf{\alpha},\mathbf{\tau})$ is unbounded as $\mathbf{\tau}$ varies. When $K$ is totally real this is deduced from a density theorem of Shapira for orbits of the diagonal group on the space of lattices. When $K$ has a complex embedding we give an explicit construction, which also shows that there are constants $c,\kappa>0$ for which $G(\mathbf{\alpha},\mathbf{\tau})\ge c(\log\max\{\tau_1,\tau_2\})^{\kappa}$ for infinitely many $\mathbf{\tau}\in\mathbb{N}^2$.

Mod $p$ Hecke eigenforms for congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$ which do not lift to characteristic $0$ have received much attention, and were named ethereal modular forms by G. Schaeffer. In this paper, we study liftability for mod $p$ Bianchi modular forms: computations suggest that a lift to characteristic zero may always be possible by increasing the level, as in the case of classical modular forms. We develop efficient algorithms for computing degeneracy maps in cohomology, which allow us to rigorously verify eigenvalue congruences. For small primes and certain fields, we also prove that lifts always exist as a consequence of the recent modularity theorem due to Caraiani-Newton. A Magma implementation, complemented by LMFDB data due to Cremona, shows that in favourable parameter ranges, most mod $p$ Bianchi modular forms lift.

We prove two new results about the number of classes of positive definite eutactic forms in n variables.

We prove that the Boca–Cobeli–Zaharescu (BCZ) map is strongly mixing with respect to normalized Lebesgue measure on the BCZ triangle.

We obtain new and essentially optimal bounds on the number of solutions of diagonal systems of $k$ polynomial equations in $s$ variables from an arbitrary finite subset of a finitely generated group in a field of characteristic zero. Our bounds generalise and sometimes improve previous results of J. Bourgain, M. Z. Garaev, S. V. Konyagin, and I. E. Shparlinski (2014), for $k=1$, and of A. Manning, A. Ostafe and I. E. Shparlinski (2026), for $k=2$. In turn, this improvement leads to sharpening a result of A. Manning, A. Ostafe and I. E. Shparlinski (2026) concerning arithmetic statistics of certain matrices. We also obtain similar results for $s$-tuples of linear recurrence sequences, in the case of linear equations and quadratic equations.

A conjecture of Erdős asserts that every odd integer $n > 1$ can be written as $n = 2^a + m$, where $a \ge 0$ is an integer and $m$ is a squarefree positive integer. We study this conjecture for primes $p \equiv 7 \pmod{36}$. Writing $x = (p-1)/6$, the candidate remainders take the shapes $p - 1 = 6x$, $p-2 = 6x-1$, $p-4 = 3(2x-1)$, and $p-8 = 6x-7$, in which the primes $2$ and $3$ are completely controlled: every obstruction to squarefreeness comes from a prime $q \ge 5$. For such $q$ a separation principle applies — $q^2$ divides at most one of the remainders, since two of them differ by at most $7 < 25$ — and it persists across any window of exponents of width less than $20$. Separation collapses the square-divisor problem for the quartic $x(6x-1)(2x-1)(6x-7)$ into independent linear problems, and we deduce unconditionally that the primes of the class representable with $a \le 3$ have relative density $1 - \rho_4$, where $\rho_4 = 1.8637 \times 10^{-5}$ is an explicit Euler-type constant; in particular, infinitely many primes of the class are not so representable. Computationally, we verify the conjecture for all primes of the class below $10^8$, always with $a \le 5$, and we exhibit the eleven primes below $10^8$ requiring $a = 5$. An appendix corrects the error bound in a lemma of Hercher, whose conclusions are unaffected.