Papers

We continue our studies on parametric norm forms $F_t({\bf x})$, with ${\bf x}=(x_0,x_1,\ldots,x_{d-1})$ lying in some parametric linear subvariety $W_t$ and integers $t$ sufficiently large. In a previous paper [Am-Ma-Za2] we proved some effective specialization results for integer solutions $\bf x$ of $F_t({\bf x})=1$. Here we modify our techniques to treat $F_t({\bf x})=q$ for an arbitrary integer $q$. Under mild conditions (not however including the crucial index assumption in [Am-Ma-Za2]) we show that all $\bf x$ are polynomially bounded in terms of $|q|$ and $t$. As in [Am-Ma-Za2] we use the methods of our paper[Am-Ma-Za] based on diophantine approximation techniques to bound certain heights. In particular we do not use linear forms in logarithms and indeed it seems unlikely that those can lead to such polynomial bounds, even for Thue equations in two variables with $x_2=\cdots=x_{d-1}=0$. We present an example with eight variables.

We prove a new asymptotic formula for the second moment of symmetric square L-functions in the weight aspect on average. This result implies that the associated L-function is non-vanishing at the central point for at least 76.69% of holomorphic Hecke cusp forms of bounded weight, improving the previous bound of 70.37%.

Many variants of the multiple zeta values have been studied in recent years. There are a few among them that remain to be real numbers, such as the multiple mixed values defined by the authors as level-two generalizations, which include both Hoffman's multiple $t$-values and Kaneko–Tsumura's multiple $T$-values. A central question is when such a value descends to level one, that is, when it can be expressed as a $\Q$-linear combination of multiple zeta values. We call these values unramified. In this paper, we further consider the alternating version of the above variants and identify five families of unramified alternating multiple mixed values using the descent theory of Brown and Glanois. We conjecture that all unramified truly alternating multiple mixed values are given in this paper.

The purpose of this note is to show that the computation of the universal norms in Iwasawa cohomology for the $p$-adic cyclotomic extension implies the same result for any $p$-adic extension that contains the cyclotomic extension. In addition, we prove a twisted cyclotomic variant, which applies to Lubin–Tate extensions.

Ramanujan's modular equations of degrees $3$, $5$, $7$, $11$ and $23$ are closely related to certain theta function identities. Warnaar generalized the identities arising from the modular equations of degrees $3$ and $7$ to a general theta function identity. Kim subsequently obtained a further general theta function identity associated with the modular equations of degrees $5$, $11$ and $23$, and established several general theta function identities containing known partition theorems as special cases. In this paper, we present an elementary approach to Kim's general theta function identities, yielding $q$-series proofs of these identities within a common framework.

A classical Pythagorean run is an identity in which a block of consecutive squares is equal to the immediately following block of consecutive squares. Boardman's construction gives such a run for every prescribed length. We consider a cubic analogue in which the second block is allowed to have common difference 2. We prove that there are infinitely many positive integer triples \((m,A,B)\), with \(B>A+2m-1\), such that $$ \sum_{j=0}^{2m-1}(A+j)^3 = \sum_{j=0}^{m-1}(B+2j)^3. $$ An explicit infinite family is obtained from the Pell-type equation $$ 48329z^2-156m^2=161, $$ which reduces to a generalized Pell equation in normalized form. The ordering condition ensures that the step-2 progression begins strictly after the consecutive block ends. The resulting sequence of half-lengths satisfies an explicit second-order linear recurrence.

We prove the three symmetric Kanade-Russell identities modulo nine.

Let $G=C_7\times C_7$, and let $S(G)$ denote the set of integer values of its group determinant. We determine $S(G)$ by classifying the values divisible by $7$; the coprime values are already known. Every nonzero divisible value has valuation at least $10$, and every multiple of $7^{12}$ occurs. At valuations $10$ and $11$, we give necessary and sufficient conditions on the cofactor in terms of two additive invariants of ideals in $\mathbb{Z}[\zeta_7]$. The first condition is a bounded signed sum of prime-ideal invariants. The second requires a prime ideal with nonzero invariant pair whose norm divides the cofactor. Neither cofactor set is a union of congruence classes modulo any positive integer. The least positive divisible value is $43\cdot7^{10}$, and the least positive value of valuation $11$ is $8\cdot7^{11}$. The proof combines integral reconstruction from character values with a calculation of the global-unit image modulo $7$. We conclude by identifying the additional local conditions and realization problems that arise at primes at least $11$.

Let $E$ be a non-CM elliptic curve defined over $\mathbb{Q}$. We present models of curves that have 2-adic image as large as possible, given the constraint that they have a rational 2-isogeny. These curves are known to be generic, and one of the aims of this article is to make that fact explicit. Our techniques make use of modular curves and their $j$-maps, as well as classification results on isogeny-graphs that have appeared in the literature recently.

This is the first of two volumes on regular arithmetic functions, and an introduction to their theory. A regular arithmetic function (RAF) is a kernel whose properties come from a single defining equation. Let $G(n,k)$ be a function of two integer variables with $G(n,n)\neq 0$, and for each $\beta$ define a sequence $(a_k)$ by $\sum_{k\le n} a_k G(n,k)=n^{-\beta}$, solved rank by rank with nothing to assume and no convergence to establish. Then $G$ is regular when the partial sums $\sum_{k\le n} a_k$ change behaviour at one exponent. Below it they reproduce the forced rate, above it they absorb it. That tipping point is the regularity index $\alpha(G)$, a quantity belonging to the kernel itself. The index came out of analogies, experiment and observation, and it is arithmetic by nature. Its most visible application is the Riemann hypothesis, which holds if and only if Ingham's kernel $G(n,k)=(k/n)\lfloor n/k\rfloor$ is a RAF of index $1/2$. It is not the only one. On a problem of a quite different kind the same theory improves the known decay bound for the orthorecursive expansion of unity. Seventeen kernels are worked out in a gallery, so that the notion can be handled by the reader. What is conjectural, conditional or open is marked as such and collected in a register. Volume II is given over to that kernel alone, with its connection to the world of Hasse-Weil zeta functions through a gauged system.

Given a collection of algebraic numbers $\mathcal{S}\subset \overline{\mathbb{Q}}$ we study the varieties $V$ in $\mathbb{A}^m_\mathbb{C}$ such that $V(\mathbb{C})\cap\mathcal{S}^m$ is Zariski-dense in $V$. We show that for many classical families of algebraic numbers $\mathcal{S}$—such as the family of roots of generalized Laguerre polynomials $L_n^{(\alpha)}(x)$, for a finite collection of $\alpha\in \mathbb{Q}$—an unlikely intersections theorem holds. For example, in the case $m=2$, we prove that an irreducible curve in $\mathbb{A}^2_\mathbb{C}$ has infinitely many points from $\mathcal{S}^2$ if and only if it is of the form $x_1=x_2,$ or $x_1=s$, or $x_2=s$ for a fixed $s \in \mathcal{S}$. This is an analogue of the classical theorems of Ihara, Serre, and Tate, treating the case of $\mathcal{S}$ consisting of the roots of unity, and of the Manin–Mumford conjecture. We also show that a similar result holds almost surely for roots of a collection of random polynomials of growing degree and bounded height. The proofs rely on a uniform Galois-theoretic criterion ensuring the unlikely intersection property.

The question of whether a 3-by-3 magic square of squares with distinct integer entries exists has been open since the 18th century. We study the geometry of the algebraic surface parameterizing 3-by-3 magic squares of squares, which is a singular complete intersection of six quadrics in projective 8-space. We compute its geometric automorphism group via an argument involving Gale duality. We compute the basic topological invariants and Hodge diamond of its resolution. We provide an explicit rank 518 sublattice of its geometric Picard group, close to the Hodge-theoretic upper bound of 544. Finally, we study the geometry and arithmetic of del Pezzo, K3, and Enriques surfaces that arise as coordinate projections.

Let $K$ be a totally real number field. We prove that the totally positive algebraic integers $\mathcal O_K^+$ of $K$, viewed as an abstract additive semigroup, uniquely determine $K$. We also describe all the additive relations by giving a presentation of $\mathcal O_K^+$ in terms of indecomposables as generators and a certain concrete set of generating relations. In fact, we obtain this presentation in a more general class of discrete semigroups which we call semilattices.

Euler sums, also called alternating multiple zeta values (MZVs), have been shown to play important roles in many research areas in mathematics and theoretical physics. Motivated by a similar conjecture for MZVs by Kaneko and Zagier, the author proposed a conjectural isomorphism between the space of finite Euler sums and the space of classical ones modulo $\zeta(2)$-products. In this paper, we explicitly construct those finite elements (for which we call effective Euler sums) that correspond to the classical Euler sums under this conjecture in depth one and two. In the appendix, we offer another group of plausible candidates of effective Euler sums when depth is two and weight is even, by extending the heuristic argument of Kaneko and Zagier for the double zeta case. Kina recently obtained independently some the same results in the MZV setting. In particular, using one of his results we prove in the appendix that Kaneko and Zagier's heuristically defined effective double zeta values coincides with ours.

The subset $A = \{a_i:i \in I\}$ of a normed vector space is a $B_h$-set if every element of the sumset $hA$ has a unique representation as a sum of $h$ elements of $A$. Let $\varepsilon = \{\varepsilon_i:i \in I\}$ be a set of positive real numbers. An $\varepsilon$-perturbation of $A$ is a set $A' = \{a'_i:i\in I\}$ such that $|a_i'-a_i|<\varepsilon_i$ for all $i \in I$. Let $\Delta_{hA} = \inf\{|x'-x| : x,x' \in hA \text{ and } x\neq x'\}$. It is proved that if $A$ is finite or countably infinite set with $\Delta_{hA}>0$, then there is a $B_h$-set $A'$ that is an $\varepsilon$-perturbation of $A$.

Classically, the places of $\mathbb{Q}$ are often regarded as a one-point compactification of $\mathrm{Spec}(\mathbb{Z})$, with the real place corresponding to a formal “prime” added at infinity. Re-examining this picture from a topos-theoretic perspective reveals a subtler geometry: while the non-Archimedean places are identified with singletons indexed by the non-zero prime ideals of $\mathbb{Z}$, the Archimedean place is represented by the space of upper reals $\overleftarrow{[0,1]}$, which may be informally thought of as the unit interval equipped with a non-Hausdorff topology. On a technical level, our analysis brings together geometric logic and descent techniques from topos theory, distinguishing standard descent from lax descent toposes both at the level of sheaves and of the geometric theories they classify. More broadly, this paper brings into conversation two parallel distinctions: on the number-theoretic side, between Archimedean and non-Archimedean phenomena, and on the topos-theoretic side, between standard and lax descent. Looked at from a high level, these perspectives begin to converge on a common theme: how should the connected and the disconnected interact?

We ask how small an increasing integer model of a finite real set can be while preserving all equalities between equal-length sums of at most $q$ elements, with repetitions allowed. The increasing correspondence must preserve exactly which sums are equal; the signs of unequal comparisons may change. Let $H_m(q)$ be the least diameter sufficient for every ordered real set of size $m$. For every integer $q\ge2$ we prove $H_4(q)=q(q+1)$ and $H_5(q)=q^2(q+1)$, and we determine $H_6(2)=24$. The proofs use the linear space determined by the additive equalities and the inequalities prescribing the order of the elements. They also give bounds for larger sets, determine the first two asymptotic terms for each fixed $m\ge4$, and yield further families of sharp examples. An application bounds the integer alphabets needed to realize finite additive-square-free spectra. The supplementary data support the finite six-element classification.

Let $k=\mathbb F_q$. We classify, up to pre- and post-composition by $k$-Möbius transformations, all separable $k$-indecomposable rational functions $f\in k(X)$ of degree greater than one whose Galois closure has genus zero. The classification is valid in arbitrary characteristic and includes exact arithmetic conditions and class counts over the prescribed field. Semilinear Frobenius descent determines the finite-field forms and which geometric decompositions descend to $k$. For every separable $f\in k(X)$ of degree greater than one with Galois closure of genus zero and every $m\geqslant1$, we prove that $f$ permutes $\mathbf P^1(\mathbb F_{q^m})$ if and only if it is exceptional over $\mathbb F_{q^m}$, meaning that it permutes $\mathbf P^1(L)$ for infinitely many finite extensions $L/\mathbb F_{q^m}$. No indecomposability assumption or lower bound on $q$ is needed. The argument combines fixed-point averaging with ramification on the Galois-closure curve. If the full constant field is $\mathbb F_{q^d}$, these properties depend only on $\gcd(m,d)$. For each classified family we determine the permutation extension degrees explicitly and characterize polynomial representatives.

Let $(a_n)_{n\geq0}$ be a sequence of complex numbers, and define \[ b_n=\sum_{k=0}^n \binom{n}{k} a_k, \qquad c_n=\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}a_k. \] Let $\beta>1$, put $\alpha=\sqrt{\beta^2-1}$, and suppose that $b_n,c_n=O(\beta^n)$. Mashreghi and Ransford proved that \[ \limsup_{n\to\infty}\frac{|a_n|}{\alpha^n} \leq \kappa \left(\limsup_{n\to\infty}\frac{|b_n|}{\beta^n}\right)^{\!1/2}\! \left(\limsup_{n\to\infty}\frac{|c_n|}{\beta^n}\right)^{\!1/2} \] with a universal constant satisfying $2/\sqrt{3} \leq \kappa\leq2$. We prove that the optimal constant is indeed $\kappa=2/\sqrt{3}$. The proof, which uses exponential generating functions, Phragmén-Lindelöf estimates, and Cauchy's formula, is compared to the classical one of Mashreghi and Ransford.

Let \(p(n)\) denote the ordinary partition function. Motivated by analogous questions concerning Euler's totient function and its complementary counting function, we study the range of the partition-derived sequence \(p(n)-n\). We give combinatorial interpretations of this sequence and investigate both the attained and omitted positive integers. We obtain exact and asymptotic information about the gaps between consecutive attained values and show that the range is remarkably sparse: its counting function has order \((\log x)^2\), and consequently the range has natural density zero. We also extend the discussion to partitions whose Durfee square has side at least a fixed positive integer.

Kanade and Russell initiated a family of conjectural Rogers–Ramanujan type identities of moduli $9$ and $12$, which ultimately comprised five modulo $9$ identities and eleven modulo $12$ identities. The eleven modulo $12$ conjectures were subsequently settled through the work of Bringmann, Jennings-Shaffer, and Mahlburg and of Rosengren. In this paper, we prove all five modulo $9$ Kanade–Russell sum-product identities, four individual generalized Nahm-sum dual identities, and a product formula for the natural dual companion of the fifth Kanade–Russell identity, which is expressed as a linear combination of two negative-mixed-term generalized Nahm sums. The first three individual dual identities settle Conjecture~3.6 of Wang and Wang, while the fourth proves the corresponding conjecture of Li and Wang. Our results also connect directly with the recent Dynkin-diagram framework of Sun and Wang for generalized Nahm sums. They identified the rank-two pairs $(T_1,G_2)$ and $(G_2,T_1)$ as unresolved cases whose modularity would follow, respectively, from the first modulo $9$ Kanade–Russell identity and its Wang-Wang dual. The present results prove precisely these two required identities and hence establish the corresponding modularity statements unconditionally.

For coprime integers $a$ and $b$, it is known that exactly one of the two Diophantine equations $$ ax+by\ =\ \frac{(a-1)(b-1)}{2} \qquad\text{and}\qquad 1+ax+by\ =\ \frac{(a-1)(b-1)}{2} $$ admits a nonnegative integer solution, and that this solution is unique. We first generalize this result by replacing the right-hand side with an arbitrary integer $m$ and its complement $ab-a-b-m$. This framework enables us to study the existence and uniqueness of nonnegative integer solutions to $$ ax+by\ =\ \frac{(a-1)(b-1)}{k} \qquad\text{and}\qquad 1+ax+by\ =\ \frac{(a-1)(b-1)}{k}, $$ where $k$ is a fixed positive integer. We then obtain explicit results when $a$ and $b$ are consecutive Fibonacci numbers. Finally, we examine the original pair of equations in several particular settings, including when $b\equiv \pm1\mod a$, when $b$ is replaced by a higher power, and when the parameters are squared.

Let $p$ be a prime, $d\ge 2$, $H\in [1,p)$, and $\ln\ln p = o(\ln H)$. We prove that $$ \min_{1\le n \le H} n \left\|\frac{a_1 n}{p}\right\|\ldots \left\|\frac{a_d n}{p}\right\| \approx \frac{1}{(\ln p)^{d-1} \ln H} $$ for "almost all" $a \in ({\Bbb Z} / p{\Bbb Z})^d$.

Let $h\geq 3$ be an integer and $0<\alpha<1/h$. In this paper, we prove that there exists a minimal asymptotic basis of order $h$ with asymptotic density $\alpha$. This solves an open problem posed by Erdős and Nathanson in 1988.

Polynomial modular reduction is central to binary finite-field arithmetic and repeated Frobenius powering. Top-down shift/XOR folding can use few operations when the nonleading support is sparse, but a tap near the leading term creates a long feedback chain. We formulate this recurrence as inversion of a nilpotent shift operator and factor its inverse by characteristic-two Frobenius powers. The resulting Frobenius-factorized reduction (FFR) applies to every monic binary modulus without materializing a reciprocal or dense reduction matrix, and its shifts can be generated online without a persistent modulus-specific schedule. For degree $m$ and nonempty nonleading support of size $s$, with nearest-tap distance $\Delta_{\min}$, FFR uses exactly $\lceil\log_2(m/\Delta_{\min})\rceil$ sequential feedback stages and has scheduled work $O(ms(1+\log(m/s)))$. A portable-C evaluation on 1,096 supports through degree $131072$ identifies distinct FFR, López–Dahab, and gf2x-backed Barrett regions. On four certified irreducible moduli, FFR makes complete Rabin irreducibility testing $1.35$–$8.04$ times faster than NTL and $1.60$–$6.81$ times faster than the matched Barrett implementation.

For a given number $N$, we consider the problem of computing two integers $1\leq r,f < N$ such that the set $$\mathcal{X}(N,r,f) = \{(a+b)-(f+\frac{Nr+1}{f}): ab=Nr\}$$ consists only of positive integers. Computing a solution to the problem is equivalent to finding a pair $(r,f)$ satisfying $l(Nr) < f \leq l(Nr+1)$, where $l(x)$ is the largest divisor of $x$ bounded by $\sqrt{x}$. This requires factoring both $Nr$ and $Nr+1$. We present a simple randomized algorithm that - avoiding factoring - computes pairs $(r,f)$. We give an exact formula for the total number of possible pairs $(r,f)$, and with the aid of empirical data we estimate that the ratio $$\frac{\phi(N)-2}{|\mathfrak{F}(N)|}$$ is approximately about $c*\log \log N$. Here, $\mathfrak{F}(N)$ is the set of unique $r$ appearing among all possible pairs $(r,f)$, $\phi(.)$ is the Euler's Totient function, and $c$ is a constant equal to 2 for prime $N$ and oscillates much for composite $N$. As a separate and independent case, we study the same problem of computing $(r,f)$ with $r>N$. We present a procedure to find such an $r$, which requires finding the least prime in an arithmetic progression.

We prove four explicit continued fraction representations for the Lerch transcendent $\Phi(z, s, M+1)$, where $\Re M>0$ and $(z,s)\in\{(-1,1),(-1,2),(1,2),(1,3)\}$. All of the continued fractions have unit partial numerators, while partial denominators depend on the parameter $M$. The proofs combine equivalent transformations of continued fractions, generalized hypergeometric functions, three-term recurrences, and asymptotic analysis of minimal solutions. For $z=1$, the corresponding representations give continued fractions for the values of the Hurwitz zeta function $\zeta(2, M+1)$ and $\zeta(3, M+1)$.

The $a$-number of a function field $F$ is the dimension of the kernel of the Cartier operator acting on the holomorphic differentials of $F$. We compute the action of the Cartier operator on the holomorphic differentials on the Hermitian function field $\mathcal{H}$, function fields of Fermat type $\mathcal{F}_{n,m}$ and the Hurwitz function field $\mathcal{H}_n$. From this, we obtain various results about the $a$-number of these function fields, including a generalisation of the results on the Fermat function field from Kodama and Washio.

We formulate a conjecture predicting the existence of exotic isomorphisms between Igusa stacks associated to different Shimura data whose underlying groups are pure inner forms of each other, and prove it in many cases of interest. In combination with the spectral action of Fargues–Scholze, this allows us to relate the cohomology groups of the two Shimura varieties, giving a geometric incarnation of the Jacquet–Langlands correspondence. An important ingredient in our proofs is the work of Xiao–Zhu on exotic Hecke correspondences between the perfect special fibers of different Shimura varieties, which we reinterpret in terms of exotic isomorphisms of perfect Igusa stacks.

As a natural generalization of the classical hyper-Kloosterman family studied by D. Haessig and S. Sperber, we study the $k$-th symmetric power $L$-functions attached to a weighted hyper-Kloosterman family $$Kl_{n,m}(t;x_{1},\cdot\cdot\cdot,x_{n})=x_{1}^{m}+x_{2}\cdot\cdot\cdot+x_{n}+\frac{t}{x_{1}x_{2}\cdot\cdot\cdot x_{n}}.$$ Under suitable conditions, we determine the bounds of degrees of these $L$-functions and prove that their $q$-adic Newton polygons admit uniform lower bounds.