Papers

By applying recent estimates on the number of integer solutions of ternary quadratic forms in boxes due to Browning and Heath-Brown, we obtain new estimates for the number of lattice points close to a curve. This leads to a new result on the distribution of squarefull numbers in short intervals.

Let $P\in \mathbb Z[X]\setminus\{0\}$ be of degree $\delta\ge 1$ and usual height $H\ge 1$, and let $\alpha\in \overline{\mathbb Q}^*$ be of degree $d\ge 2$. As a consequence of general result due to Lang and Galochkin, we have the following transcendence measure: for any $\varepsilon>0$, there exists $c>0$ such that $\vert P(J_0'(\alpha)/J_0(\alpha))\vert>c/H^{4d^2\delta+\varepsilon}$ where $J_0$ is the Bessel function. In this paper, we prove that the exponent $4d^2\delta$ can be replaced by a smaller (explicit) quantity $\mu(d,\delta)\le 4d^2\delta-2d\delta-1$. A similar improvement holds more generally for the logarithmic derivative of any $E$-function $f$ of differential order 2 and such that $f$ and $f'$ are homogeneously algebraically independent. Our method rests upon the optimization of the size of a determinant that appears naturally in the classical Siegel-Shidlovskii method, following the steps of our previous improvement of the transcendence measure of the value $e^\alpha$ for any $\alpha\in \overline{\mathbb Q}^*$.

We consider a possible approach to the Lindelöf hypothesis for Dirichlet $L$-functions. It is based on a special form of the functional equation for the corresponding theta functions. To estimate $L_\chi(0.5+it)$ we need to solve certain systems of linear equations. The entries to the corresponding matrices are formed by the summands to the series for theta functions and their derivatives. Numerical data suggest that the inverse matrices have a deep structure and allow us to state a number of conjectures. In particular, it seems that for a character modulo $q$ certain entries to the inverse matrices tend to finite limits when the sizes of the matrices run over arithmetical progressions with step $2q$. Moreover, these limits belong to the real cyclotomic field $\mathbb{Q}(\cos(\pi/q))$ (up to a scaling factor of $\sqrt{q}$).

Let $E/F$ be a CM extension and let $m\geq2$ be an even integer. Starting from the odd-degree Hermitian Ikeda lift of degree $m+1$ constructed by Yamana, we construct a Hermitian Ikeda lift of degree $m$ from the theta components of its first Fourier-Jacobi coefficient. We determine its global Arthur parameter and its local constituents at every finite place. Using the global multiplicity formula for unitary groups, we obtain an explicit multiplicity-free decomposition of the resulting representation of $\mathrm{U}_{m.m}(\mathbb{A}_{F,\mathbf{h}})$ and show that its isomorphism class is independent of the Fourier-Jacobi index. When $F=\mathbb{Q}$, we compare Fourier coefficients and identify the construction with Ikeda's even-degree Hermitian lift.

In the direction of Szpiro's conjecture for all elliptic curves $E$ over $\mathbb{Q}$, the current best bound is $\log \Delta \ll N \log N$ where $\Delta$ is the absolute value of the minimal discriminant of $E$ and $N$ is the conductor of $E$ (the implicit constant is absolute). This bound dates back to 2013. We obtain the stronger bound $\log \Delta \ll N\log \log N$ which was previously known only under GRH. In addition, for semistable elliptic curves we prove $\log \Delta \ll N$. Our methods build on the theory developed by the author for approaching the $abc$ conjecture via Shimura curves.

We prove two conjectures regarding partition perimeter inequalities. It was conjectured by Gray, Payne, and Watson that Franklin's partition identity becomes an eventual inequality if one replaces the size of the partition with its perimeter. We prove this conjecture by asymptotic analysis of the corresponding generating functions. Gray, Payne, Swisher, and Watson also conjectured that there is a bias for partitions with fixed perimeter to have more odd parts than even parts. We prove this conjecture by deriving the corresponding generating functions and give a positive recurrence for the coefficients. In the case that there are an equal number of odd parts and even parts we provide a connection to peakless Motzkin paths.

Let $\Th(\alpha,L)=\sum_{v\in L}e^{-\pi\alpha|v|^2}$ for $\alpha>0$ and $E(L,s)=\sum_{v\in L\setminus\{0\}}|v|^{-2s}$ for $s>3/2$ be the theta and Epstein zeta functions associated to the lattice $L$, respectively. We are particularly interested in physically relevant dimension three. Fix the covolume of the lattice $L$ to $1$. Up to an orthogonal transformation, we prove that \begin{equation}\nonumber \argmin_{|L|=1}\Th(\alpha,L)= \begin{cases} \boldsymbol{\mathrm{FCC}}\;\;\mathrm{lattice},\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;&\text{if}\;\; \alpha>1, \boldsymbol{\mathrm{BCC}}\;\;\mathrm{lattice}, \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&\text{if}\;\; \alpha<1, \boldsymbol{\mathrm{FCC}}\;\;\mathrm{or}\;\; \boldsymbol{\mathrm{BCC}}\;\;\mathrm{lattice},\;\;&\text{if}\;\;\alpha=1, \end{cases} \end{equation} and \begin{equation}\nonumber \argmin_{|L|=1}E(L,s)= \boldsymbol{\mathrm{FCC}}\;\;\mathrm{lattice},\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\text{if}\;\; s>3/2. \end{equation} Therefore, we prove the Sarnak–Strömbergsson conjecture \cite[Inequalities (43)–(44), Section 5]{SS}.

Let $p>3$ be a prime number. We prove that every principally polarized superspecial abelian surface over $\mathbb{F}_{p^{2}}$ with $p^{2}$-Frobenius $[-p]$ admits a separable polarized isogeny to its Frobenius conjugate with multiplier at most $(p^{3}/2)^{1/5}$. This generalizes a recent result of Aubry, Oyono, and Vincent (2026, arXiV:2607.14624) who proved that every supersingular elliptic curve defined over $\bar{\mathbb{F}}_{p}$ admits an isogeny to its Frobenius conjugate with degree at most $(p/2)^{1/3}$.

We establish a class formula with variable for arbitrary effective Anderson motives. This pursues the pioneering work of Taelman and extends the scope of the work of Anglès-Ngo Dac-Tavares Ribeiro, removing the admissibility assumption and allowing for more general Anderson modules. In the same setting, we further conjecture a ”nondeterminantal” version of the class formula which holds before taking determinants and prove a weak version of it. Finally, we prove an overconvergent version of the class formula and derive from it strong convergence properties for the associated L-series.

We prove the uniform Mordell–Lang conjecture for semiabelian varieties, that is, for a subvariety $X$ of a semiabelian variety, its intersection with a finite-rank subgroup of rational points, can be covered by at most $c$ cosets contained in $X$, where $c$ depends only on the dimension of the semiabelian variety, the degree of $X$, and the rank of the subgroup. Our proof is new even in the abelian case previously proven by Gao–Ge–Kühne.

For a nontrivial finite abelian group $G$, let $\nu(G)$ be the smallest integer $\ell$ such that every zero-sum free sequence $T$ over $G$ of length at least $\ell$ has the following property: all nonzero elements of $G$ that do not occur as a subsequence sum of $T$ lie in a proper coset of some subgroup of $G$. It is easy to check that $\mathsf d (G)-1 \le \nu (G) \le \mathsf d (G)$, where $\mathsf d (G)$ is the small Davenport constant of $G$. A conjecture by Gao from the year 2000 stated that equality should always hold at the lower bound. This conjecture has since been confirmed for many families of groups (including all p-groups and groups of rank at most two). In the current note, we disprove the conjecture.

We develop an extension of Deligne–Lusztig theory to certain (possibly infinite type) disconnected reductive groups arising from the special fibers of point stabilizers in the Bruhat–Tits building, which we call paraductive. We then compute explicit lower bounds for the Tate cohomology of representations of paraductive groups, relating these to Shintani descent, Lusztig restriction, and the Glauberman correspondence. As an application, using Feng's modular functoriality and Scholze's independence of $\ell$, we compute the Fargues–Scholze L-parameters of non-singular depth $0$ cuspidal representations of a (possibly wildly ramified) reductive group over a nonarchimedean local field.

We determine exactly how harmonic sums, taken over integers with a given number of occurrences of a fixed block of digits, depend on that number of occurrences. Language factorizations, combined with stochastic radix expansions, relate these harmonic sums to the iteration of an operator whose eigenvectors are the Lebesgue measure and distributional derivatives of singular measures. The dual picture is given by a Riesz basis of eigenpolynomials in a suitable Hardy space. The resulting modal expansion admits a natural interpretation as a block-directed Euler-Maclaurin formula interpolating between Euler-Maclaurin and Taylor expansions.

We give new bounds for list sizes of random linear codes at capacity, max loads of linear hash functions, and Furstenberg sets, over every finite field $\mathbb{F}_q$. 1. Random linear codes over $\mathbb{F}_q$ with rate $1 - H_q(p) - \epsilon$ are $(p, O(q H_q(p)/\epsilon))$-list decodable with high probability for all values of $p, q, \epsilon$, including the high error regime. This nearly matches the list size lower bound of $H_q(p)/\epsilon$ due to Guruswami, Li, Mosheiff, Resch, Silas, and Wootters [IEEE Trans. Inf. Theory 2022]. Our bound is the first uniform improvement for $q > 2$ since Guruswami, Håstad, and Kopparty [STOC 2010]. 2. Linear hash functions over $\mathbb{F}_q$ hashing $n$ balls to $n$ bins achieve maximum load $O(q \ln \ln q / {\ln q}) \cdot \ln n / {\ln \ln n}$, both in expectation and with probability $1-o(1)$. This nearly matches the lower bound of $\ln n / {\ln \ln n}$. Previously, only a polylogarithmic upper bound was known for $q > 2$, due to Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos [J. ACM 1999]. We reduce list decodability and linear hashing to strong Furstenberg set lower bounds, which we prove using a new polynomial method of multiplicity gaps. While previous polynomial methods analyze a set $S$ by studying polynomials that vanish on it, we consider polynomials that vanish everywhere, but with higher multiplicity inside $S$ than outside.

Under mild assumptions, we prove the strong main conjecture for BDP Selmer groups of elliptic curves with good reduction in the sense of Kurihara's refined Iwasawa theory. In particular, the structure of the initial Fitting ideal of BDP Selmer groups over finite subextensions in the anticyclotomic $\mathbb{Z}_p$-extension is determined by the values of the corresponding $p$-adic modular form at CM points.

Let $T(x)=\prod_{j\ge 0}(1-x^{2^j})=\sum_{n\ge 0}t(n)x^n$ be the Thue–Morse generating function, and write $T(x)^m=\sum_{n\ge 0}t_m(n)x^n$ for a positive integer $m$. For a nonzero integer $a$, let $\nu_2(a)$ be the exponent of $2$ in $a$ and put $\operatorname{odd}(a)=a/2^{\nu_2(a)}$. For every $s\ge 1$, we prove that the sequence $(\operatorname{odd}(t_m(n))\bmod 2^s)_{n\ge 0}$ is $2$-automatic when $m=2^r$, $r\ge 1$, or $m=3\cdot 2^r$, $r\ge 2$. In both families, $\nu_2(t_m(n))=\nu_2\binom{n+m-1}{m-1}$. The valuation identity for $m=2^r$ is known; our proof recovers it and also establishes the automaticity assertion. For $m=6$ we prove $\nu_2(t_6(n))=\nu_2\binom{n+5}{5}+\mathbf{1}_{n\equiv 3\pmod 4}$ and show that its odd parts modulo every $2^s$ are $2$-automatic.

Our aim is to find all the prime numbers $p$ such that $p-x^2$ has at most two different prime factors, for all the odd integers $x$ such that $x^2<p$. We solve entirely the cases $p\equiv 1,5,7 \pmod 8$. The case $p\equiv 3 \pmod 8$ is solved with one possible exception.

In this paper, we establish an asymptotic formula of mixed moments of $L(1/2,f)$ and $L(1/2,\mathrm{sym}^2 f)$ for cusp forms $f$ of weight $k$, prime level $q$ and real primitive nebentypus $\chi$ modulo $q$ in the level aspect with a power-saving error term. As an application, we deduce the simultaneous non-vanishing of $L(1/2, f)$ and $L(1/2,\mathrm{sym}^2 f)$ for sufficiently large prime levels.

In this paper we explore the world of $t$-sumfree sequences, primarily for $t=2$ and $t=3$. Starting with $t$ increasing positive integers, build an infinite sequence by choosing as the next entry the least positive integer that is not the sum of $t$ distinct previous values. The main open problem concerns the question of the conjectured ultimate periodicity of these sequences. We formulate a sufficient condition to prove that a conjectured preperiod and period are correct. We also prove periodicity for a considerable subclass of 2-sumfree and of 3-sumfree sequences. Finally, we present the results of extensive computations on 3-sumfree sequences.

We study large values of quadratic Hecke $L$-functions in the conductor aspect. Let $K$ be a fixed number field, and assume GRH for its finite-order Hecke $L$-functions. In a fixed ray class component with conductor norm comparable to $X$, we prove that \[ \max_\chi L\left(\frac12+\frac A{\log_2X},\chi\right) \geq\exp\left\{(e^{-A}+o(1)) \sqrt{\frac{\log X\log_3X}{\log_2X}}\right\} \] for every fixed $A\geq0$. Every fixed smaller constant is attained by at least $X^{1-o(1)}$ characters. The same count holds at a suitably slowly moving threshold approaching the displayed constant. The combinatorial input is a sparse squarefree Gál set of cardinality $N$, retaining the known leading constant $2$ and having square multiplicative energy $N^{2+o(1)}$. The energy bound reflects the low degree of the associated Boolean polynomial. Together with the resonance estimate, it yields the abundance bound. We also prove unconditional analogues for quadratic characters with prime conductor away from one fixed place over any global function field of odd characteristic. Finally, we give bounds in the fixed strip and at $s=1$, including the dependence on the residue of the Dedekind zeta function and the prescribed local factors.

We study prime-power congruences for the prime-to-\(p\) Hecke eigensystems of normalized cuspidal level-one eigenforms as the weight varies, and we address the problem of classifying such systems for small prime powers. We construct Hecke-equivariant transfer maps between level-one modular-symbol modules via multiplication by suitable lifts of Dickson invariants; these maps reduce the problem to finite computation and propagate Hecke operator relations through all weights. For \(p=2,3,5,7\), we obtain all-weight classifications modulo \(2^8\), \(3^4\), \(5^3\), and \(7^2\), respectively.

Given a positive integer $M$ with reduced residue system $\Phi_M$, we partition the set of squarefree integers according to some congruence conditions on the number of prime factors in each class $\phi \in \Phi_M$, and prove two relevant equidsitribution results in this setting. We discover the bias phenomenon in some special cases, and formulate a conjecture with numerical support.

Let $p\geq 11$ and let $\bar\rho$ be a très ramifié representation. For a primitive projective-augmented module of the corresponding type, we prove the très ramifié analogues of the local results of Liu-Truong-Xiao-Zhao and thereby establish the local ghost conjecture in this case. The main difference here is that the Iwahori dimension increases by one between consecutive relevant classical weights and may therefore be odd. Accordingly, the proof of the near-Steinberg results must treat both integral and half-integral centres.

In this article, we prove the Defect Conjecture for Cayley graphs of abelian groups, dihedral groups, and groups of the form $\mathbb{Z}/p\mathbb{Z} \rtimes \mathbb{Z}/(p-1)\mathbb{Z}$ where $p\geq 3$ is a prime. We also compute the Iwasawa invariants of the Bowen–Franks groups associated with these graphs in several cases.

We study the equidistribution in arithmetic progressions modulo powers of a fixed odd prime of Hecke eigenvalues of noncuspidal $\mathrm{GL}_3$ automorphic forms, with a focus on the convolution coefficients $\lambda_{1\boxplus f}(n)=(\lambda_f\star 1)(n)$,attached to a level $1$ holomorphic primitive cusp form $f$. Utilizing techniques introduced by Kowalski–Lin–Michel and Milićević, we prove the exponent $\vartheta=1/2+7/458-\varepsilon$ is admissible.

We study the Laurent coefficients of the Barnes multiple zeta function with complex parameters in a common open half-plane and a fixed holomorphic determination of the logarithm. At the highest pole, we derive explicit limit formulae for every regular Laurent coefficient in terms of finite multiple sums and logarithmic correction terms. At the lower possible poles, we establish all-order relations with the Taylor coefficients at the origin and their parameter derivatives. We also determine the large-order behavior by subtracting all principal parts. The remaining function is entire, so Cauchy's estimate separates explicit residue contributions from a remainder that decays faster than any fixed geometric rate. The neighboring residues yield the limiting even and odd subsequences, with vanishing residues and cancellations accounted for. The constant-term cases recover known finite-part representations; the main focus is their extension to higher Laurent coefficients.

We construct an infinite family of imaginary biquadratic fields with large class numbers. The construction of this family is done by composing two distinct families of imaginary quadratic fields: the first one is the family Q(sqrt(k^2-p^l)) introduced by Banerjee and Hoque [1], subject to specific constraints on the integer parameters, l, k and p, and the second one is the parametrized family of the form Q(sqrt(-F_(50s+25))) based on Fibonacci numbers, introduced by Kishi [2]. One of the main tools that is used is Kuroda's class number formula of imaginary biquadratic fields. We choose these two families with the sole aim to get high order elements in the class group of the resultant biquadratic fields.

We obtain a Furstenberg-Sárközy-type result for sets $A\subset [N]$ whose difference set $A-A$ does not contain the sum of $s$-many $k$-th powers of positive integers, with $k>1$ odd and $s>0$ even. Namely, we prove that such sets must satisfy a power-saving bound $|A| \, \ll \, N^{1-\frac1k\min\{s \, \sigma_k, \, 1/2\}+\epsilon}$ for any fixed $\epsilon>0$, where $\sigma_k >0 $ is any admissible saving in a classical one-variable Weyl estimate. In particular, we can take $\sigma_k=\max\left\{{2^{1-k}}, \, \frac{1}{k(k-1)}\right\}$ using the classical theory and the best currently available bounds for classical Weyl sums. A greedy construction produces a set $A\subset[N]$ with $|A|\gg N^{1-s/k}$ for which $A-A$ contains no sum of $s$-many positive $k$-th powers, so our power-saving bound is of the correct shape.

Let $p$ be an odd prime and let $K_\infty/K$ be a $\mathbb{Z}_p$-extension. For each finite layer $K_n$, let $G_n$ be the Galois group of its maximal unramified pro-$p$ extension. We study the augmentation and Zassenhaus filtrations of the groups $G_n$ as $n$ tends to infinity. Assume that the classical Iwasawa $\mu$-invariant is positive. If $d_n$ denotes the minimal number of generators of $G_n$, then there is an integer $\delta>0$ such that $d_n=\delta p^n+O(1)$, while the minimal number of relations is $O(p^n)$. For each fixed $m\geq2$, as $n\to\infty$, we show that the $m$-th augmentation quotient has the same leading asymptotic as that of a free pro-$p$ group on $d_n$ generators. We also obtain corresponding asymptotics for the Hilbert series, the exponential growth rate of the augmentation filtration, and the Zassenhaus quotients.

We determine the Aharonov invariants of order at least two of every elliptic double cover by evaluating its projective kernel. In orders at least three, the formula separates the invariant into a constant Eisenstein term and an Eisenstein–Kronecker function evaluated under multiplication by two. The formula determines the ramification principal parts, torsion specializations, and isogeny traces, including the correction from two-torsion in an isogeny kernel. It also realizes the same intrinsic de Rham tensor through every degree-two projection: in higher orders this is the image of an Eisenstein section, while order two gives the classical Weierstrass complement to the Hodge line. A separate differential calculation shows that the Bernoulli-normalized invariant of order $p-1$ reduces to the quotient of the $p$th and first iterates of the defining derivation. For an elliptic invariant derivation this quotient is the Hasse invariant, independently of the rational function on its separable locus. The universal double cover provides a global specialization, integral away from two-torsion, whose reduction extends regularly across that locus. Classical Eisenstein zero theorems and the supersingular divisor congruence then describe the exactness loci and their reduction.