Papers

We give algorithms to explicitly compute local Arthur packets and their intersections for even orthogonal groups $G_n$ over $p$-adic fields using extended multi-segments. In addition, we provide an algorithm which determines whether a representation of $G_n$ lies in a local Arthur packet. As applications, these results are used to identify distinguished parameters, prove the closure ordering conjecture, the enhanced Shahidi conjecture, the weak local Arthur packet conjecture, and the good parity unitary dual conjecture for $G_n$.

In this paper, we prove that $\mathbb{Z}$ is first-order definable in the ring of integers $\mathbb{Z}^{\text{ab}}$ of the maximal abelian extension $\mathbb{Q}^{\text{ab}}$ of $\mathbb{Q}$, which implies that the first-order theory of $\mathbb{Z}^{\text{ab}}$ is undecidable. More generally, writing $i = \sqrt{-1}$ and $\mathbb{Q}^{\text{tr}}$ for the field of all totally real numbers, we prove new definability and undecidability results for rings of integers of subfields of $\mathbb{Q}^{\text{tr}}(i)$, focusing especially on fields which contain infinitely many roots of unity. The key ingredient for these results is that there is a parameter-free positive-existential formula which defines the roots of unity $\mu(\mathcal{O}_L)$ inside $\mathcal{O}_L$ for every field $L\subseteq \mathbb{Q}^{\text{tr}}(i)$.

Let $N_g$ be the number of Ekedahl-Oort strata intersecting the supersingular locus in the Siegel modular variety. We prove that $N_g\sim 2^{g-1}$ as $g\to\infty$.

Let $X$ be an algebraic K3 surface defined over a global field $K$ of characteristic $0$ or $p \geq 3$, assumed ordinary in the case where $K$ is a global function field. We provide an exact formula for the density of primes $p$ of $K$ at which the geometric Neron-Severi group of the reduction $X_p$ has a given rank in terms of the Galois representation attached to $X$. As an extension, we prove that there are infinitely many reductions of $X$ with an element of the geometric Neron-Severi group representing a fixed square class under the intersection pairing and prove results about the density of such places. In particular, we show that if $X$ corresponds to a sufficiently generic moduli point, a density zero set of places of $K$ satisfy this condition.

We quantify the entropy-decrement proof of the logarithmically averaged two-point Chowla theorem, uniformly in the affine forms and the averaging window. For L_i(n)=a_i n+b_i with positive integral slopes and nonzero determinant Delta=a_1 b_2-a_2 b_1, put h=max(a_1,a_2,|b_1|,|b_2|,|Delta|,2). For either the Liouville or Mobius function in each position and 0<epsilon<=1/10, the correlation sum over x/omega<n<=x, weighted by 1/n, has absolute value at most epsilon log omega whenever x>=omega>=exp_4(C h epsilon^(-2)), where exp_4 denotes four iterated exponentials. A translation lemma separates the common translation from the tower: for lambda(n+b)lambda(n+b+k), with k nonzero, cancellation is uniform when |k|=o(log_4 omega) and |b|=omega^(o(1)), where log_4 is the fourth iterated logarithm. The same thresholds give logarithmic equidistribution of the four Liouville sign pairs. The constant C is effective but depends on uncomputed constants in two analytic inputs. Separately, writing ell=log log x, we prove fully explicit bounds ell/3-(4/3)log ell-7 <= N_lambda(x) <= ell-log ell+2.53/ell for ell>=64, where N_lambda is the non-pretentiousness level in Tao's hypothesis. Under GRH the lower bound improves to ell-log ell-log 2-2/ell; hence N_lambda(x)=ell-log ell+O(1). The arguments use an explicit zero-free region, Harnack's inequality, a finite resonator, and a comparison of smoothed Euler products. These results concern logarithmic averages, not natural-density two-point Chowla correlations.

The classical $A_2$-frame construction of the Coxeter–Todd lattice uses the hexacode as its glue. Here we replace the inert Eisenstein reduction at $2$ by the ramified Gaussian chain ring $S=\mathbb{Z}[i]/(1+i)^3$. For odd $\ell$, a one-generator cyclic graph code over $S$ lifts from an $A_2(4)^{2\ell}$-frame to an even strongly $6$-modular lattice $L_\ell$ of rank $4\ell$. In the real frame ordering the same glue is a one-generator negacyclic code. The cases $L_5$ and $L_7$ have minima $6$ and $8$ and give examples for the type-$6\mathrm{a}$ entries in dimensions $20$ and $28$, which are listed as open in the Nebe–Sloane catalogue. After completing the square, the minimum calculation becomes a four-symbol carry problem; for $L_7$ the key relation is the boundary map of a $7$-cycle. For $\ell=3$, $L_3$ has an index-$8$ overlattice isometric to the Coxeter–Todd lattice.

We study cyclotomic cosine evaluations beyond the fully symmetric setting by combining Galois-orbit structure with finite-frequency expansions. First, for truncation-compatible poly?nomial families over Q, we distinguish invariance of the evaluated value from invariance of the polynomial formula. Summability by a rational function of the level forces eventual rationality of the evaluated values. Conversely, every rational-valued sequence can be realized by a compatible family whose total degree is uniformly bounded by two, showing that value-level Galois invariance alone carries essentially no stable-range rigidity. We then introduce explicit orbit-structured pattern sums indexed by multiplicative configurations. For every fixed pattern, at each sufficiently large admissible level n, its cyclotomic cosine evaluation collapses to an affine function n times S0(r,h) minus 2 to the abs(h) power, where S0 is the zero-frequency coefficient of the associated Laurent polynomial. Consequently, every family uniformly generated from finitely many such patterns by a single polynomial recipe has eventually polynomial evaluation. Finally, decomposing the punctured index set into gcd-orbits yields an exact orbitwise formula in terms of Ramanujan sums. The semiprime and prime-power cases become transparent specializations of this general identity. These results identify both the obstruction to extending symmetric rigidity from value-level Galois invariance and a concrete non-symmetric class exhibiting uniform stable-range collapse

Let $k$ be a field of characteristic $p\ge0$. We prove that $m$ points and $n$ lines in $k^2$ determine at most $3(mn)^{2/3}+m+n+2mn/p$ incidences, the last term being omitted in characteristic zero. Over the prime field $\mathbb{F}_p$ the coefficient of $mn/p$ can be replaced by $1$. The proof uses the polynomial method, and for $m=n$ the bound is sharp up to an absolute constant over prime fields. As applications, over prime fields in which $-1$ is not a square we obtain the $L^2\to L^r$ extension estimate for the paraboloid in $\mathbb{F}_p^3$ for $r>10/3$. Over every odd prime field, we show that a two-source extractor construction of Bourgain has exponentially small error at every min-entropy rate greater than $1/3$. We also improve sum-product estimates for small sets in positive characteristic and obtain projection and Furstenberg estimates over prime fields. The incidence inequalities with exact constants have been formalized in Lean.

We prove that the generalized Fermat equation $x^3+y^5=z^7$ has no solution in nonzero coprime integers, and we make explicit how the proof extends the prior work cited below. Dahmen-Siksek had established the signed local descent and eliminated the three cases in which the associated degree-seven algebra is reducible. Putz had then proved that the remaining irreducible case can involve only seven septic fields: six pure fields and one exceptional field. The missing step was to exclude those seven fields. For the six pure fields we construct an explicit Fano resolvent, descend to a smooth plane quartic of genus three over $\mathbb{Q}(\sqrt{-7})$, and prove that its rational-parameter locus consists only of five points above the branch values. For the exceptional field we combine the modularity and conductor results of Pacetti-Villagra Torcomian with level lowering over $\mathbb{Q}(\sqrt{5})$ and a finite Hecke comparison at $29$. We also reconstruct the three reducible-sector arguments of Dahmen-Siksek as independently replayable calculations, including their database-free identification of the quadratic field, and add an intrinsic formal-group treatment at the ramified prime. The paper labels every major step as a literature input, reconstructed input, or new argument. All project-specific finite calculations are supplied as exact certificates, programs, inputs, and authenticated logs.

We prove that the irrationality exponent of $\zeta(3)$ satisfies $\mu(\zeta(3)) < 5.5138800$, improving the bound $5.513891$ obtained by Rhin and Viola in 2001, which has been the best known for twenty-five years. The proof is an application of their Theorem 5.1 at a new integral direction. That direction is not reached by enlarging the search range: its eight parameters are 52000 times theirs, shifted by at most 23. The mechanism is the one recently used by Bai for $\pi$. In particular the Rhin-Viola point, optimal in every region covered by enumeration, is not optimal.

Robin's inequality reformulates the Riemann hypothesis as a bound on the sum-of-divisors function $\sigma$. We introduce a finite-truncation version of Robin's inequality determined intrinsically by the arithmetic structure of $n$, \[ \frac{\sigma(n)}{n} < e^\gamma \sum_{j=0}^{\omega(n)} \frac{(\ln\ln\ln n)^j}{j!}, \qquad n>5040, \] where $\omega(n)$ denotes the number of distinct prime factors of $n$, and $\gamma$ is the Euler-Mascheroni constant. The resulting bound is pointwise stronger than Robin's inequality and we prove it unconditionally for integers with $\omega(n)\le 6$, primorials, odd integers and square-free integers. We also prove that this inequality is equivalent to Robin's inequality, and hence to the Riemann hypothesis. Consequently, the Riemann hypothesis is equivalent to the truncated inequality holding for all colossally abundant numbers. We further show that if the inequality fails, its minimal counterexample must be a superabundant number, placing any obstruction within a rigid extremal class of integers.

Using Soundararajan's resonance method, we obtain extreme values of the central derivatives \[ L^{(r)}(1/2,f\otimes\chi_d), \] where $f$ is a normalized newform and $\chi_d$ ranges over primitive real characters associated with negative fundamental discriminants. The twisted first moment estimate used in the proof may be of independent interest. Applying Gross–Zagier-type formulas, we obtain correspondingly large Néron–Tate heights of Heegner points on elliptic curves and large absolute values of Beilinson–Bloch heights of Heegner cycles attached to modular forms.

The Odlyzko-Poonen conjecture states that a monic polynomial of constant coefficient $1$ and with remaining coefficients chosen independently and uniformly from $\{0,1 \}$ is irreducible over the rational numbers with probability tending to one as the degree tends to infinity. We prove the Odlyzko-Poonen conjecture unconditionally.

A $\textit{magic-lined hypercube}$ (or, simply, $\textit{magic hypercube}$) of order $k$ and dimension $n$ is an arrangement of the numbers $1,\dots,k^n$ in a $k\times\cdots\times k$ ($n$-fold) grid such that every line of $k$ numbers parallel to a coordinate axis has the same magic sum. While such hypercubes exist for every order $k\ge 3$ and every dimension $n$, no magic-lined hypercube of order $2$ exists in any dimension $n\ge 2$. For hypercubes of order $2$, we thus relax the magic condition from lines to two-dimensional faces or planes. We call an arrangement of the numbers $1,\dots,2^n$ in a $2\times\cdots\times2$ ($n$-fold) grid magic-faced if every $2\times2$ face has the same magic sum. We prove that a magic-faced hypercube of order $2$ exists in every dimension $n\ge0$, and that it is unique up to a certain natural set of transformations of size $2^n(n+1)!$ when $n$ is even and $2^n n\cdot n!$ when $n$ is odd. As an application, we recover and generalize the classical $4\times4$ Khajuraho magic square, answer a question of Coxeter on the group acting on the $384$ “most-perfect" $4\times4$ magic squares, and extend the picture to higher dimensions. In particular, we prove that, in dimension $n$, these most-perfect objects form a single orbit under a certain natural action of the Weyl group $W(B_{2n})$.

We study geometric inverse problems for finite sets $A\subset \mathbb{Z}^d$ whose Fourier $L^1$ norm, or Wiener norm, \begin{align*} \|A\|:=\int_{\mathbb{T}^d}\Big|\sum_{a\in A}e(a\cdot t)\Big|dt \end{align*} is subpolynomial in $|A|$. In particular, we show a variety of geometric phenomena are incompatible with small Fourier $L^1$ norm. Our principal application concerns spherical Freiman models. Suppose $\|A\|=|A|^{o(1)}$ and $D\subseteq A$ is Freiman isomorphic to a set $S$ of lattice points on a sphere; then, $|D|/|A|$ must be polynomially small in $|A|$. Moreover, if $|D|/|A|$ is not too small, almost all of $S$ lies in a small number of spherical caps, and almost all caps have rich additive structure. The proof uses a new mechanism for studying additive structure across subsets of $A$, along with decoupling results of Bourgain-Demeter.

Let $E$ be a real quadratic field and let $U_E$ be the quasi-split unitary group in three variables associated with $E$. We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation $\pi_U$ of $U_E$ and the periods of its Rogawski stable base change to $\mathrm{GL}_3(E)$. This generalizes earlier works of Tilouine-Urban and Hida in the case of $\mathrm{GL}_2$. The divisibility we prove involves a new kind of automorphic periods for self-conjugate cuspidal automorphic representations of $\mathrm{GL}_3(E)$, defined within the middle degree of the cuspidal cohomology rather than the top or bottom degrees. Moreover, we prove an à la Hida adjoint $L$-value formula for $\mathrm{GL}_3(E)$, relating these newly defined middle-degree periods to the usual top and bottom automorphic periods. Finally, we also prove a similar formula for $U_E$, which is the first instance of such a result for quasi-split unitary groups.

We prove a $p$-adic divisibility between the automorphic periods of a cuspidal automorphic representation $\pi$ of $\mathrm{GL}_3(\mathbb{Q})$ and the periods of its Arthur-Clozel base change to a real quadratic field $E$. As a corollary, we establish a divisibility predicted by the Bloch-Kato conjecture for the adjoint motive of $\pi$ twisted by an even quadratic character. This generalizes earlier works of Tilouine-Urban and Hida in the case of classical modular forms. The divisibility we prove involves a new kind of automorphic periods for conjugate self-dual cuspidal automorphic representations of $\mathrm{GL}_3(E)$, defined within the middle degree of the cuspidal cohomology instead of the top or bottom degrees. Moreover, we prove an à la Hida adjoint $L$-value formula for $\mathrm{GL}_3(E)$ that relates these newly defined middle-degree periods to the usual top and bottom automorphic periods.

Appell functions are a large class of multi-variable quasi-elliptic functions, which are also instances of higher depth mock modular forms. A central aspect of these functions is their non-holomorphic modular completion. Our previous work developed Appell functions for a general positive definite lattice $\Lambda$, and expressed the non-holomorphic completion in terms of the generalized error functions $M_P$, which are integrals over a $P$-dimensional hyperplane in $\Lambda\otimes \mathbb{C}$. In this sequel, we express $M_P$ as a $P$-dimensional iterated integral of variables $w_j\in \mathbb{H}$, such that the completion of the Appell function takes the form of an iterated Eichler integral of modular forms. We apply this to the case of the root lattice of the $A_N$ Lie algebra, which is of particular interest because of their appearance in partition functions of topological twisted Yang-Mills theories.

Let $f(X)$ be a polynomial over a finite field, and let its defining function record the multiplicities of its roots. In this paper, we introduce the affine stabilizer group and the affine stabilizer group scheme of the defining function of $f(X)$. These objects characterize the affine symmetries of the roots of $f(X)$ together with their multiplicities. We establish structural descriptions of these affine stabilizers and relate them to the coefficient structure of $f(X)$. In particular, the affine stabilizer group determines quotient structures of the defining function, which yield canonical descriptions of the corresponding semi-invariant polynomial spaces. The resulting coefficient constraints provide explicit restrictions on the possible supports of polynomials with prescribed affine symmetries. The affine stabilizer group scheme further captures infinitesimal symmetries arising from multiple roots and gives refined coefficient constraints beyond those obtained from ordinary affine transformations. We apply this framework to generator polynomials of constacyclic codes over finite fields. The affine stabilizer structure of a generator polynomial induces subcodes defined by semi-invariant conditions and provides constraints on their dimensions and Hamming distances. In the special case of multiplicative symmetries, the construction specializes to the support-primitive decomposition of constacyclic codes.

We study the linearization of the random $p$-adic orthogonal matrix model. Let $A_n$ and $B_n$ be Haar-random $n\times n$ special orthogonal and alternating matrices over $\mathbb{Z}_p$, respectively. For an odd prime $p$, we prove that $\mathrm{cok}(A_n-I_n)$ and $\mathrm{cok}(B_n)$ have the same limiting distribution as $n \to \infty$ through integers of the same parity. The two parity-dependent limits are interchanged for orthogonal matrices of determinant $-1$, while the full orthogonal group gives their equal mixture. For $p=2$, linearization does not preserve the limiting cokernel distribution, and we explicitly determine the limiting distributions for the special orthogonal model in terms of the spinor norm, which controls the parity of the $2$-adic valuation of the order of the torsion subgroup.

The works of Rivoal, Zudilin, and Krattenthaler and Nesterenko give an explicit sequence $p_n/q_n$ ($p_n,q_n\in \mathbb{Z}, q_n>0$) of rational approximations to Catalan's constant $G=1-1/3^2+1/5^2-1/7^2+1/9^2-\cdots$ satisfying $\vert G-p_n/q_n\vert \leq 1/q_n^{0.52}$ for sufficiently large $n$. We build on their works to give an explicit sequence $p_n/q_n$ ($p_n,q_n\in \mathbb{Z}, q_n>0$) with $\vert G-p_n/q_n\vert \leq 1/{q}_n^{0.62}$ for sufficiently large $n$.

The construction of a mixed motive associated with Catalan's constant $G=\sum_{n=0}^\infty (-1)^n/(2n+1)^2$ in [arXiv:2510.20648] led to a supply of period integrals that evaluate to linear forms in 1 and $G$. Inside this supply, the boundary of the integration domain leads to a natural 3-parameter family of period integrals. We show that this family is closely related to the family of hypergeometric period integrals considered earlier by Rivoal, Zudilin and Nesterenko.

For $k\geq2$, let $\alpha_k$ be the supremum of the exponents $a$ for which almost every integer $n$ has $k$ distinct divisors in a multiplicative interval of relative length $(\log n)^{-a}$. Select each positive integer $i$ independently with probability $1/i$, forming a random set $\mathbf A$, and let $\beta_k$ be the supremum of the $c<1$ for which, with probability tending to one as $D\to\infty$, the set $\mathbf A\cap[D^c,D]$ has $k$ distinct subsets with the same sum. We prove that $\alpha_k=\beta_k/(1-\beta_k)$, resolving a conjecture of Ford, Green and Koukoulopoulos [Invent. Math. 232 (2023), 1027–1160]. We also prove that their weak and strict entropy thresholds coincide. The proof combines flag refinement and entropy concavity with an upper bound for approximate subset sums that is uniform in arbitrary translations. A model with independent geometric prime exponents then transfers this bound to divisors.

In this paper, we establish lower bounds for large values of the Hurwitz zeta function with rational parameter when the real part $\sigma \in [1/2,1]$. These results improve the earlier results of Ramachandra and Sankaranarayanan in 1989. On the critical line, our result recovers the corresponding lower bound of de la Bretèche and Tenenbaum (2019) for the Riemann zeta function, while for $1/2< \sigma \leq 1$, our lower bounds attain the same order as the corresponding lower bounds for large values of the Riemann zeta function. Our proofs are based on the resonance method.

For arithmetic quotients of Bruhat–Tits trees with finitely many cusps, we establish an explicit unitary correspondence between the spherical Eisenstein transform, with the Eisenstein series normalized by their constant terms, and the scattering transform of an associated Jacobi operator with finite core. Tracking the Haar measure, stabilizer weights, height coordinates, and cusp widths yields the Plancherel measure and shows that the absolutely continuous spectrum has multiplicity equal to the number of cusps. From a discrete Green identity we derive a matrix-valued Maass–Selberg formula for the Hermitian matrix $iS(\theta)^*\partial_\theta S(\theta)$, where $S(\theta)$ is the scattering matrix. Its trace is determined by $\det S(\theta)$, while the full matrix retains additional cusp-to-cusp information. After the corresponding change of normalization, the finite Schur complement obtained by eliminating the cusp rays agrees with the resonance matrix of Arends-Peterson-Weich. Using their resonance computations as input, we distinguish eigenvalues supported entirely in the finite core from poles of the scattering matrix. The Nagao and $\Gamma_0(T)$ quotients, together with a four-cusp quotient arising from an elliptic curve over $\mathbb F_3$, make the normalizations and matrix-valued conclusions explicit.

We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, improve on the weighted large sieve inequality. We compute in passing the $L^2$-norm of the Beurling-Selberg majorant of the characteristic function of an interval.

We modify the approach to the arithmetical form of the large sieve by relying on the Parseval identity rather than on an approximate Bessel inequality and as a consequence, we prove that sifted sets are either small or the attached Fourier polynomial is very regular at the origin. We also discuss the optimality of this approach and shall show in passing that $N\int_{-\delta}^\delta|S(\beta)|^2d\beta\ge(1-2/\sqrt{\delta N})|S(0)|^2$ for any trigonometric polynomial of length~$N$.

We develop Hachimori's study on the unramified Iwasawa modules using extensions with restricted $p$-ramification. Let $K$ be an algebraic number field, and $p$ a prime number which splits into two distinct primes $\mathfrak{p}$, $\mathfrak{p}'$ in $K$. Assume that $K$ and $p$ satisfies several (somewhat strict) conditions. Let $K_\infty /K$ be the cyclotomic $\mathbb{Z}_p$-extension. In the present paper, we give a method to study the structure of the unramified Iwasawa module $X (K_\infty)$ by using abelian $p$-extensions unramified outside $\mathfrak{p}$. We give a sufficient condition for $|X (K_\infty)|$ to be finite in terms of such extensions. We also give a similar criterion for $X (K_\infty)$ to be finitely generated over $\mathbb{Z}_p$. In the latter part of the present paper, we consider the case where $k = \mathbb{Q} (\sqrt{q})$ with an odd prime number $q$ and apply our results to the cyclotomic $\mathbb{Z}_2$-extension $k_\infty /k$. We give a necessary and sufficient condition for $X (k_\infty)$ to be cyclic over $\mathbb{Z}_2$, which is different from the former results given by either Mouhib-Movahhedi or Mizusawa-Mouhib. We also give several sufficient conditions for the validity of Greenberg's conjecture.

Let $[a_1(x),a_2(x),\ldots,a_n(x),\ldots]$ be the continued fraction expansion of irrational $x\in[0,1)$, and let $q_n(x)$ be the denominator of the $n$-th convergent. In this paper, we study how the growth rate of $a_{n+1}(x)$ on a prescribed logarithmic size interacts with its upper growth rate relative to $q_n(x)$. For $\psi:\mathbb N\to\mathbb{R}_{\ge0}$ satisfying $\psi(n)\to \infty$ and $\alpha, \beta\in [0, \infty]$, define the joint level set \[F_{\alpha,\beta}:=\Big\{x\in [0,1)\colon \liminf_{n\to\infty}\frac{\log (a_{n+1}(x))}{\psi(n)}=\alpha,\ \limsup_{n\to\infty}\frac{\log (a_{n+1}(x))}{\log q_n(x)}=\beta\Big\}. \] We determine the Hausdorff dimension of $F_{\alpha,\beta}$ for all values of $ \alpha$ and $\beta$. Our results is related to several earlier results on the metric theory of continued fractions, including those of Bugeaud [Math. Ann. {327} (2003)] on irrationality exponents, Wang–Wu [Adv. Math. {218} (2008)] on the growth of partial quotients, and Song–Tan–Zhang [Nonlinearity {37} (2024)] on the joint distribution of convergence and irrationality exponents.

We study the zero-run structure of the $(3,2)$-Raney numbers modulo a prime $p$. For every prime $p$, we determine the left-to-right maxima of the zero-run lengths, the complete zero-run spectrum, and the exact number of nonzero entries in $0\le n<p^m$. Interestingly, these results fall into three cases: $p=2$, $p=3$, and $p\geq5$, and the behaviors in these three cases are very different. For $p=2$, we characterize exactly the odd terms and determine the positions of the left-to-right maxima and the results involve Fibbinary integers, Fibonacci numbers, and Jacobsthal numbers. For $p=3$, we characterize exactly the nonzero terms and determine their residues. For $p\ge 5$, the zero runs are governed by a multiscale system of residue intervals modulo powers of $p$, from which both the record values and the complete zero-run spectrum are obtained.