Papers

In this paper we obtain new classes of the second generation of new $\zeta$-functionals and corresponding $\zeta$-equivalents of the Fermat-Wiles theorem. These are based on sets of elementary rectangular $\zeta$-signals generated by the Gramm's sequence.

We prove a Fourier-type toric spectral reciprocity formula over number fields, relating Tate integrals over a quadratic extension to a dual family of Hecke periods over the base field. The formula keeps the spectral parameter free and allows test functions adapted to arithmetic applications. We apply this reciprocity formula to canonical Hecke $L$-functions over CM fields and obtain explicit first moment formulas, including both central values and central derivatives. These formulas are uniform in the weight and the twisting conductor, and require no Heegner-type splitting hypothesis. Together with subconvexity bounds, they yield quantitative nonvanishing results for canonical Hecke $L$-functions and their derivatives. In weight one, this gives a rank-one analogue of the arithmetic applications of Masri–Yang, yielding quantitative Mordell–Weil rank-one results for quadratic twists of the associated CM abelian varieties.

We construct a bilinear pairing for $p$-adic families of cohomology classes associated to locally symmetric spaces of a certain level at $p$ and obtain a comparison between the specialisations of constructed pairing and the natural duality pairings on cohomology of automorphic sheaves under specialisation maps. We show that the considered locally symmetric spaces with the chosen level admit an automorphism which naturally generalises the Atkin–Lehner involution for modular curves and interchanges the ordinary and anti-ordinary parts of the cohomology thereby making it possible to pair two anti-ordinary classes. Our construction is motivated by the pairing constructed by Ohta for modular curves and recovers it.

We give a one-parameter functional equation for the real Rogers dilogarithm, with arguments in $\mathbb{Q}\bigl(u,\sqrt{4-3u^{2}}\bigr)$, and prove it by an explicit array of ten instances of Rogers' five-term relation: the fifty arguments so contributed cancel down to the five of the identity, which admit no shorter relation among themselves. The equation comes from a pair of integrals whose equality is elementary, and we show that the underlying integrand is essentially forced. Specialising the parameter gives identities over $\mathbb{Q}(\sqrt{33})$ and $\mathbb{Q}(\sqrt{17})$, a relation between $\mathbb{Q}(\sqrt{13})$, $\mathbb{Q}(\sqrt{3})$, and $\operatorname{Cl}_2(\pi/3)$ with a new analogue for Catalan's constant, and a pair of dilogarithm ladders of quartic base over $\mathbb{Q}(\sqrt{33})$. The base equations of these ladders, and of four conjectural ones located by integer-relation search, are irreducible quartics with four real roots. All previously recorded ladders of degree four known to us have base equations with two real roots, so these appear to be the first totally real ladders of degree exceeding three.

We prove a new effective bound for cubic Thue–Mahler equations with power-saving dependence on the regulator. This has some applications. First, we improve Stark's bound $\log \max\{|x|,|y|\} \ll_\epsilon |k|^{1+\epsilon}$ for the integer solutions of Mordell's equation $y^2=x^3+k$ ($k$ a non-zero integer) by reducing the exponent $1+\epsilon$ to $1/2+\epsilon$; this is the first power-saving improvement without restrictions on $k$ in more than 50 years. Secondly, for integer squares and cubes of size $\asymp T$ we improve the known unconditional separation lower bound $(\log T)^{1-o(1)}$ obtained by Stark in 1973 to $(\log T)^{2-o(1)}$. Finally, we obtain a power-saving improvement in the conductor aspect of the strongest currently available bounds for Frey's height conjecture (a strengthening of Szpiro's conjecture) in the case of elliptic curves over $\mathbb{Q}$ with integral $j$-invariant.

A subset of the $d$-dimensional lattice cube $[n]^d$ is sum-free if it contains no solution to the equation $x+y=z$. We study the total number of such subsets. For $d=1$, Cameron and Erdős conjectured that the number of sum-free subsets of $[n]$ is $O(2^{n/2})$, and this was proved independently by Green and Sapozhenko. A recent work by Ghosal solved the case $d = 2$. In this paper, we consider all remaining dimensions and prove that for every fixed integer $d \geqslant 3$, the number of sum-free subsets of $[n]^d$ is $2^{M([n]^d) + O_d(n^{d-1})}$, where $M([n]^d)$ is the maximum possible size of a sum-free subset of $[n]^d$. This verifies a conjecture of Elsholtz and Rackham. Our proof combines the dual weights constructed by Keevash and Lim in their work for $M([n]^d)$, a one-dimensional counting estimate due to Ghosal, a bipartite swapping lemma of Zhao, and a strong fractional entropy inequality of Madiman and Tetali, and it avoids the use of the container lemma or deriving a stability theorem first.

Let $\lambda(n)=(-1)^{\Omega(n)}$ be the Liouville function. We prove a fixed power-logarithmic bound for its logarithmically weighted two-point correlations across the full shift range. There is an absolute $c>0$ such that every sufficiently large $x$ admits a single set $\mathcal E_x\subseteq[1,x]$ with $|\mathcal E_x\cap[1,H]|\ll_A H(\log x)^{-A}$ $(1\le H\le x)$ for every fixed $A>0$, while $\max_{\substack{1\le h\le x\ h\notin\mathcal E_x}}\sup_{1\le y\le x}\left|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right|\ll(\log x)^{1-c}$. The same exceptional-set formulation extends, without an upper cutoff, to all positive integer shifts. Earlier full-range theorems average over the shift; here a fixed saving holds pointwise outside one set whose density in every initial segment is smaller than every fixed negative power of $\log x$. The new middle-scale argument combines a general-good-modulus Liouville deletion lemma with a linear bad-modulus score, a progression Fourier estimate, and a Mellin-localized dilation that separates divisor-dependent endpoints. Maximal fixed-moment bounds evacuate the low prefix and control the long-shift range. Assuming GRH for primitive Dirichlet $L$-functions, we also prove, uniformly for $h\in\mathbb N$ and $1\le y\le x$, $\left|\sum_{n\le y}\frac{\lambda(n)\lambda(n+h)}{n}\right|\le\log(2\min{h,y})+O((\log x)^{1-c_{\mathrm G}})$ for an absolute $c_{\mathrm G}>0$, with no exceptional shifts.

Let $D(G)$ denote the derived category of smooth $G$-representations on $k$-vector spaces, where $G$ is a locally pro-$p$ group and $k$ is a field of characteristic $p$. In this paper we are primarily interested in the graded center of the subcategory of compact objects $Z^*(D(G)^c)$ and variants thereof. When $G$ is a $p$-adic Lie group, without proper open centralizers, we completely determine this center modulo locally nilpotent elements and give various applications.

In the recent work by T. H. Chan, it was shown that there are no three consecutive powerful numbers with the middle term a perfect cube and each of the other two having only a single prime factor raised to an odd power. They get it with Pell equations, elliptic curves, and second-order recurrences. In this paper, with results given by Nagell-Ljunggren, Lebesgue, Ko and some elementary methods, we extend the result to similar cases, whose middle term can be perfect powers for infinitely many powers. With this conclusion, we discuss the solutions to some Diophantine equations.

We study extreme values of normalised $r$-th nearest neighbour distance functions in inhomogeneous Diophantine approximation, and prove Weibull and Fréchet limit laws for these functions when the shift satisfies a natural Diophantine condition. Using these extreme value laws we prove corresponding logarithm laws for these functions. Our approach is based on homogeneous dynamics. We adapt the Poisson approximation strategy of Bj$\ddot{\text{o}}$rklund and Gorodnik (arXiv:2201.05116) combined with an effective multiple equidistribution result, which we deduce using results of Str$\ddot{\text{o}}$mbergsson (arXiv:1309.6103) and of Björklund and Gorodnik (arXiv:2105.05468).

Let $G$ be a finite nonabelian group. The small Davenport constant $\mathsf d(G)$ of $G$ is the largest integer $\ell$ such that there exists a product-one free sequence over $G$ of length $\ell$, while the Gao constant $E(G)$ of $G$ is the least integer $\ell$ such that every sequence over $G$ of length at least $\ell$ contains a product-one subsequence of length exactly $|G|$. A long-standing conjecture of Gao and Zhuang \cite{ZG2005} asserts that $E(G)=\mathsf d(G)+|G|$ for every finite nonabelian group $G$. Let $p$ be an odd prime and let $H_{p^3}=\operatorname{UT}_3(\mathbb F_p)$ be the finite Heisenberg group over $\mathbb F_p$. Godara and Sarkar proved the Gao-Zhuang equality for $H_{27}=\operatorname{UT}_3(\mathbb F_3)$ and asked whether the same equality holds for $H_{p^3}$ for every odd prime $p$. Recently, Volkmann proved that $\mathsf d(H_{p^3})=3p-3$. In this paper, we determine the Gao constant of $H_{p^3}$ and prove that $E(H_{p^3})=\mathsf d(H_{p^3})+|H_{p^3}|=p^3+3p-3$. Together with the known abelian and cyclic-index cases, this completes the verification of the Gao-Zhuang equality for all groups of order $p^3$, for every prime $p$.

Hermite asked in 1848 for a representation of real numbers whose eventual periodicity characterizes cubic irrationals. The totally real case was solved by Karpenkov's $\sin^2$-algorithm; the complex case, signature (1,1), is his Problem 4. We study a deterministic algorithm implementing his suggested analytic extension: the score expression is strictly negative on (1,1) data (closed form proved), the most negative score is selected, and exact score ties are resolved by a declared ordering. On a sample of 205 complex cubic polynomials, every run closes projectively with an exact unit certificate, each transition certified by exact comparisons in $\mathbb{Q}(\alpha)$. An exhaustive campaign over the full box $[-3,3]^3$ closes 194/194. Across 457 deformed bases, the terminal cycle is an invariant of the marked lattice. Certified finite transition graphs are computed for four fields; the plastic case is machine-checked in Lean 4, kernel-only. All data ship in a public archive with a portable verifier.

This is the text of my diploma thesis on Euclidean Rings from 1989, along with an English translation. Its main result is the generalization of Lenstra's idead of exceptional sequences to k-stage Euclidean rings.

We study representations of integers using $k$-generalized Fibonacci numbers. For $k\geq 2$, we first consider signed representations of zero with coefficients in $\{-1,0,1\}$ and give a recursive description of their number. The resulting counting sequences satisfy linear recurrences whose characteristic polynomials are determined explicitly. In the Fibonacci and Tribonacci cases, these recurrences reveal unexpected connections between the corresponding representation counts and Tribonacci and Fibonacci sequences, respectively. We then consider representations with coefficients in $\{0,1\}$ in the Tribonacci case. Using a random inhomogeneous Tribonacci recurrence, we construct a binary-tree model in which representation multiplicities are encoded by a family of polynomials satisfying the product formula $Q_n(x)=\prod_{k=2}^{n-1}(1+x^{T_k})$. This product admits a probabilistic interpretation in terms of weighted Bernoulli sums. After normalization by $T_n$, these sums converge in distribution to the Bernoulli convolution $\sum_{j=1}^{\infty}\varepsilon_j\rho^{-j}$, where $\rho$ is the Tribonacci constant and the $\varepsilon_j$ are independent Bernoulli random variables. The limiting distribution satisfies a natural self-similarity relation, linking the representation problem to self-similar measures associated with Tribonacci scaling.

Let $C\subset X(1)\times X(1)$ be a smooth curve defined over $\mathbb{C}_p$ with irreducible reduction. We study the intersection loci of $C$ with the modular subvarieties $Y_0(n)$. If the curve $C$ avoids points where both coordinates have supersingular reduction, or if a sequence $Y_0(a_n)$ is taken where the numbers $a_n$ have increasing divisibility by $p$, then these loci equidistribute to the unique canonical point of the analytification, $C^\text{Berk}$. If neither condition is satisfied, we expect equidistribution to fail and give a family of examples whose behaviour is believed to be typical. We also study the accumulation points of the set $C\cap\bigcup_nY_0(n)$ and show that this set is always non-discrete.

We establish that only finitely many powers of $3$ can have a simple binary representation.

Let $q$ be an odd prime power, and $\mathcal{C}, \mathcal{D}$ be two smooth plane conics defined over $\mathbb{F}_q$ with transversal intersection. We present an exact formula for the number of points in $\mathbb{P}^2(\mathbb{F}_q)$ that are internal/external to $\mathcal{C}$ and internal/external to $\mathcal{D}$. This refines the $\frac{q^2}{4}+O(q^{3/2})$ asymptotic estimate for this quantity due to Asgarli and Yip \cite[Theorem 1.2]{Asgarli}. In particular, we show that the error term is of size at most $q+\sqrt{q} + 1$. By studying the geometry of the incidence variety related to this problem, we link the exact point counts directly to the Frobenius traces of two associated elliptic curves, and the number of $\mathbb{F}_q$-rational intersection points of $\mathcal{C}$ and $\mathcal{D}$ and of the corresponding dual conics $\mathcal{C}^*$ and $\mathcal{D}^*$. Lastly, We provide a remark explaining the challenges in generalizing this method to the study of higher-dimensional quadrics.

Let $p(m,n)$ denote the number of partitions of a rectangle $m\times n$ into integer-sided rectangular blocks, where two partitions are indistinguishable if they consist of the same multiset of blocks, regardless of their geometric arrangement. We present an elementary approach to show that, for every fixed positive integer $m$, $$ \log p(m,n)=\pi\sqrt{\tfrac{2mH_m}{3}}\sqrt{n}+O(\log n), \qquad \text{as }n\to\infty, $$ where $H_m$ denotes the $m$-th harmonic number. This confirms a conjecture recently posed by the authors and generalizes the Hardy–Ramanujan formula for integer partitions.

Let $D_n$ denote the supremum of the orthogonality defects over all $n$-dimensional Hermite–Korkine–Zolotarev (HKZ)-reduced bases. This paper shows that, as $n\to\infty$, \[ \log D_n=2n\log n-n\log\log n+O(n). \] We also determine the exact value $D_4=4375/1024$.

Let d=2g+1>4 and let A^{d-1} be the coefficient space parametrizing hyperelliptic families C_a: y^2 = x^d + a_{d-1}x^{d-1} + ... + a_1x+t over the t-line. We show that for a Zariski-generic coefficient vector a=(a_1,...,a_{d-1}) in \bar{Z}^{d-1}, for every prime p large enough, C_a has generically ordinary reduction at every prime above p. For arbitrary a in \bar{Z}^{d-1}, the set of ordinary primes of the hyperelliptic curve C: y^2=x^d+a_{d-1}x^{d-1}+...+a_1x+t over Q(a)(t) has natural density at least 1/[Q(a,\zeta_d):Q(a)]; after base change to Q(a,\zeta_d)(t), it has natural density 1.

We study congruences between special values of zeta functions and modular $L$-functions in cyclotomic towers. The underlying mechanism is an integral norm congruence for finite-level Iwasawa-theoretic elements, which becomes stronger as one ascends the tower. In the $\mathrm{GL}_1$ setting, this gives congruences for Dedekind zeta values over totally real fields, with applications to higher $K$-groups, generalized Bernoulli numbers, and Euler-Poincaré characteristics. In the $\mathrm{GL}_2$ setting, we use Mazur-Tate elements and their distribution relations to control the primitive twisted special values appearing at each cyclotomic level. Via Artin formalism, these give congruences for the corresponding base-change $L$-values over successive cyclotomic fields.

Let $q$ be an odd prime power. We revisit Florea's asymptotic formula for the first moment of quadratic Dirichlet $L$-functions over the odd-degree hyperelliptic ensemble $\mathcal H_{2g+1}$, and show that its secondary term is not the complete one. The square-dual generating function has three double poles of the same modulus on the secondary circle: the positive real one reproduces Florea's polynomial, while the two nonreal conjugate poles contribute at the same order. The complete secondary term is $q^{2g/3}(\alpha_{g\bmod3}g+\beta_{g\bmod3})$, with real coefficients whose dependence on the genus has minimal period exactly three, and on at least two of the three residue classes the nonreal poles contribute a term of order $gq^{2g/3}$. We also prove the formula for every odd prime power $q$, whereas Florea assumes $q\equiv1\bmod4$ throughout; this is achieved by a phase-normalized quadratic Gauss sum that restores the multiplicativity and the local evaluations her argument uses. Exact finite-ensemble moments for $q=3$ exhibit the three residue-class trends.

We study how the noncentral zeros of an elliptic-curve $L$-function enter the Birch–Swinnerton-Dyer (BSD) leading coefficient and the prime-scale crossings of an elliptic Chebyshev race. Under the elliptic-curve Riemann hypothesis (ECRH), the completed $L$-function yields an exact Hadamard decomposition, and the zero statistics $J_E$ and $V_E$ satisfy the positive identity $J_E=V_E/8+\Delta_E$. For seventeen rank-one isogeny classes of conductor $50700$, the first zero contributes on average $82.4%$ of $\Delta_{E,20}$, while independent edge values reveal a discrete $35/36$ zero-count split beyond height $20$. For Haar $SO(2N+1)$ we prove a fixed-dimensional hard-edge law $\Pr{J_N>y}\sim C_Ne^{-3y/2}$. A finite zero truncation also defines a Morse function on a phase torus, whose level sets and coarea density provide a topological model for crossing statistics. Motivated by false Chebyshev primes, we define elliptic crossing primes through the midpoint and half-jump of a logarithmically weighted Frobenius race. A four-stage conductor-$50700$ computation shows agreement at the level of ensemble profiles, while omitted-zero noise prevents resolution of individual shrinking-window events at height $20$. The experiment therefore identifies a quantitative resolution barrier rather than a failure of the prime-sampled transfer.

For a prime number $p$, we denote by $k_p$ the greatest odd divisor of $p-1$. Tonelli–Shanks algorithm is a classical algorithm to compute a square root of a quadratic residue $n \in \mathbb{Z}$ modulo $p$ as the product of $n^{\frac{k_p+1}{2}}$, which we call {\it Tonelli–Shanks power of $n$}, and a correction term given as a power of a quadratic non-residue modulo $p$. We prove the transcendence over $\mathbb{Q}$ of the images of the following in the ring $\mathscr{A}$ of integers modulo infinitely large primes: the greatest odd divisor $k_p$ of $p-1$, the composite $f(\frac{p - 1}{k_p})$ of any injective map $f \colon \mathbb{Z} \to \mathbb{Z}$ and the greatest $2$-power $\frac{p - 1}{k_p}$ dividing $p-1$, Tonelli–Shanks power $n^{\frac{_p+1}{2}}$ of any $n \in \mathbb{Z} \setminus \{-1,0,1\}$, and the correction term of Tonelli–Shanks algorithm for such $n$.

A set $\{a,b,c\}$ of distinct positive integers is called a Diophantine triple if $ab+1$, $ac+1$, and $bc+1$ are perfect squares. Dujella formulated the conjecture that the only integral $x$-coordinates on the attached elliptic curve $$y^2=(ax+1)(bx+1)(cx+1)$$ are $0,d_-,d_+$, together with $-1$ when the triple contains $1$, where $d_{\pm}=a+b+c+2abc \pm2\sqrt{(ab+1)(ac+1)(bc+1)}$. The weaker question was stated as Problem 4.8 in Dujella's list of open problems: must every integral point with $x\ne-1$ make $ax+1$, $bx+1$, and $cx+1$ all perfect squares? We construct infinitely many triples $\{5,115,c\}$ admitting an integral point with $x\ne-1$ for which all three factors are nonsquares.

This sequel to our paper `On the “main conjecture” of equivariant Iwasawa theory' studies the possibility of the vanishing of $SK_1(QG)$, when $QG$ is the total ring of fractions of the Iwasawa algebra $\Lambda G=Z_p[[G]]$, with $G$ the Galois group of the Galois extension $K/k$ of loc. cit. The vanishing is equivalent to $SK_1(D)=0$ for all division algebras $D$ in the Wedderburn components of $QG$, so can be studied via the classification $D_{r,s,F}$ of these $D$'s, which provides a crossed product order $\Delta$ in $D$ and the valuation $v_\bullet$ with valuation ring $\Delta_\bullet$. $\Delta$ contains a special element $\Pi$, which generates a maximal subfield of $D$ over its centre with $V_\bullet(\Pi)=1$ and $\Pi\Delta\Pi^{-1}=\Delta$. The induced $\Pi$-filtration on $\Delta$ enables a study of the reduced norm built on a congruence for $nr(d)$ mod $\Pi\Delta$ for $d\in\Delta$. Given $d\in\Delta^\times$ with $nr(d)=1$, and $n\ge 1$ maximal with $d\in 1+\Pi^n\Delta$ (called the level of $d$), the main problem is to find a suitable commutator product $c\equiv d$ mod $\Pi^{n+1}\Delta$. Then $c^{-1}d\equiv 1$ mod $\Pi^{n+1}\Delta$ hence, setting $d'=c^{-1}d$, has $nr(d')=1$ and level $n'>n$. Repetition ends in $[\Delta^\times,\Delta^\times]$ when the level gets sufficiently large.

In 2025, Boaz Klartag proved that there exists a lattice sphere packing in $\mathbb{R}^n$ whose density is at least \[ c n^2 2^{-n}, \] where $c>0$ is an absolute constant. Our aim is to refine the analysis of Klartag's argument and to show that the constant can be taken as \[ c = \frac{1}{2e} - o(1), \qquad n\to\infty. \]

A curious property of integers, first observed by D.~R. Kaprekar in 1949, is as follows: write the digits of a 4-digit number in both descending and ascending order, and then find the positive difference between these integers. Iterating this process on any 4-digit number - except for multiples of 1111 - eventually produces the number 6174. The process ends here, since 6174 is a fixed point of the procedure. In general, if we start with any integer and iterate this process, it either ends at a fixed point (as with 4 digits) or enters a cycle of numbers, called a Kaprekar cycle. In 2011, Dolan classified all fixed points of this process. We study Kaprekar cycles of length 2. We find general properties of numbers in a Kaprekar 2-cycle, and we classify all 2-cycles such that the two integers have the same smallest digit.

We consider families of lines that cover every point of the integer lattice $\mathbf{Z}^2$ in the plane, subject to the constraint that no two lines of different direction in the family meet at a lattice point. Restricting to lattice lines (lines containing at least two, hence infinitely many, lattice points, equivalently of rational direction), we show that the set of directions occurring in such a covering can be made dense in the space of line directions. The construction is a recursive splitting of $\mathbf{Z}^2$ into nested rank-2 sublattice cosets, each handed off to a freshly chosen direction; the key technical point is a steering lemma showing that at every stage of the recursion a new direction arbitrarily close to any prescribed target can still be realized, via an elementary sieve bound.

We construct a theory of Fourier transform for rational étale motives on derived vector bundles over derived Artin stacks over a field of positive characteristic, defined as the integral transform with kernel the Artin-Schreier motive. The basic properties are established: involutivity, base change, functoriality, duality, monodromicity, constructibility, and compatibilities with various natural transformations.