Papers

Let $v$ be a finite place of a rational function field with finite constant field. We establish an Ogus-type $v$-adic Chowla-Selberg formula for $t$-motives with "complex multiplication" by subfields of Carlitz cyclotomic function fields. For this purpose, we first introduce the $v$-adic crystalline action of the Weil group on the de Rham spaces of $t$-motives via the de Rham-crystalline comparison isomorphism of Hartl-Kim, and use it to give a "$v$-isocrystal" period interpretation of special values of Thakur's $v$-adic geometric gamma function. Our Ogus-type formula is then based on the isogeny theorem for CM $t$-motives, together with a recipe for constructing "abelian CM types" given in the $\infty$-adic setting over function fields.

Inspired by the polymath Abraham Ibn Ezra (1089-1167) who approximated {\pi} by 20/9 times the square-root of 2, and contemporary scholar, RSJ Reddy, who gave (14-sqrt(2))/4, we point out that both of these approximations are optimal in some sense, and we generate many other ones that are even more optimal, albeit not as nice.

We reconstruct the generalized hexagon of order $(2,8)$ from an abstract even rank-$26$ lattice of determinant $3$ and minimum $4$, and conversely reconstruct the lattice from the hexagon. Harmonic theta identities determine the $819$ shortest vectors in a non-zero discriminant class and their association scheme, and show that every non-empty positive-norm shell of the lattice and its dual is a spherical $5$-design. In the converse direction, the rank-$26$ idempotent gives the lattice, with saturation proved by a short dual-coset argument. A positive-definite Niemeier construction proves existence and uniqueness of the lattice and hence of the hexagon. The third moment of a projective $3$-design in the Cayley plane reconstructs the traceless Albert product. This upgrades angle-preserving bijections to elements of $F_4(\mathbb R)$ and proves geometric uniqueness of the tight $819$-point projective $5$-design. Finally, line deletion gives a rootless index-four sublattice of $N(A_1^{24})$ with a Golay trio and an oriented gluing. The gluing data form eight root-sign orbits, on which $L_3(2)\cong L_2(7)$ acts as on $\mathbf P^1(\mathbf{F}_7)$; two explicit Golay permutations prove transitivity. This proves uniqueness and gives the automorphism-group order. The group is identified afterward as ${}^3D_4(2):3$, with a central factor $C_2$ for the full lattice group. The reconstruction and deletion arguments parallel the length-$26$ binary code construction.

We establish a comprehensive algebraic, geometric, and physical classification of the wave closure space generated from primitive plane wave sequences on the discrete spatial lattice $\mathbb{Z}$. Resolving lattice degeneracies via unwrapped phase spaces, we prove that the linear wave group $\mathcal{G}$ under pointwise product, inversion, and root extraction is an infinite divisible abelian torsion group isomorphic to $(\mathbb{Q}/\mathbb{Z}) \times (\mathbb{Q}/\mathbb{Z})$, with canonical decomposition into Prüfer $p$-groups and maximal cyclotomic value field $\mathbb{Q}^{\mathrm{ab}}$. Extending to coordinate powers yields the polynomial phase group $\mathcal{G}_{\mathrm{poly}}$, classified by integer-valued polynomials $\operatorname{Int}(\mathbb{Z})$. Adjoining addition yields the group algebra $\mathbb{C}[\mathcal{G}_{\mathrm{poly}}]$, for which we establish the Permutation-Symmetric Phasor Superposition Theorem, factoring superpositions into collective barycentric carriers and closed Born probability envelopes. Using algebraic sieves over roots of unity, we construct idempotent projectors and complementary Not-sieve notch filters, achieving exact algebraic synthesis of both momentum and localized Kronecker position bases. Generalizing to non-abelian quaternions $\mathcal{V}_\mathbb{H}$, we prove a polar decomposition into scalar envelopes and $\mathrm{SU}(2)$ spinor rotors. Identifying coordinate advance with time, biquaternion determinants intrinsically yield Minkowski spacetime $s^2 = c^2 t^2 - \|\mathbf{x}\|^2$ and the Lorentz group $\mathrm{SO}^+(1,3)$. We demonstrate emergent vacuum zero-point jitter, topological selection of rational frequencies, and correspondences with discrete qudits.

We study the problem of finding tight explicit bounds on sums over zeros of $L$-functions near the $1$-line with a variety of zero-free regions and zero-density estimates. As an application to demonstrate the techniques, we apply them to the case of the Riemann zeta function and the prime number theorem to obtain new best estimates on the error terms in the prime number theorem. We show that for $x\geqslant21$ we have \begin{multline*} |\psi(x)-x| < 0.239\, x \exp\bigl(-0.1982767\,(1+\log\log\log(x)/(15\log\log(x)))\log(x)^{3/5}\log\log(x)^{-1/5}\bigr). \end{multline*} Combined with the zero-free half-plane $\Re(s)>7/8$ announced by OpenAI in October 2026, the same methods give $|\psi(x)-x| < 0.04934\,x^{3/4}\log(x)^2 + 0.02636425\,x^{7/8}$ for $x\geqslant 21$.

We study the variance of the number of lattice points in smoothed narrow sectors, where the average is taken over the position of the sectors. There are three regimes corresponding to specific ranges for the open angle of the sectors and in each we obtain an asymptotic formula for the variance.

Let $n\geq 1$ and let $\ell$ be an odd prime. Let $G=\mathrm{PGSp}_{2n}(\mathbb{F}_\ell)$ or $\mathrm{GSp}_{2n}(\mathbb{F}_\ell)$, and fix a faithful transitive permutation representation $\pi:G\longrightarrow S_d$. We study degree-$d$ number fields whose Galois closures have Galois group $G$ and whose associated permutation representation is $\pi$. For $\sigma\in S_d$, write $\operatorname{ind}(\sigma)$ for its permutation index, namely $\operatorname{ind}(\sigma) = d-\#\{\text{orbits of $\sigma$ on $\{1,\ldots,d\}$}\}$. If $\tau$ denotes a symplectic transvection, or its image in the projective symplectic group, we prove that the number of such fields with absolute discriminant at most $X$ is bounded below by a constant multiple of $X^{1/(2n\,\operatorname{ind}(\pi(\tau)))}$. For the natural vector and projective actions, the exponents we obtain are asymptotically $1/(4n)$ of those predicted by the weak form of Malle's conjecture as $\ell\to\infty$. The fields are constructed from the mod-$\ell$ Galois representations attached to the Jacobians of a one-parameter family of hyperelliptic curves. The proof combines large symplectic monodromy for this family with a squarefree sieve.

In this paper, we study the unramified Iwasawa module over the cyclotomic $\mathbb{Z}_2$-extension of the real quadratic field $\mathbb{Q}(\sqrt{p_1p_2p_3p_4})$, where $p_1, p_2, p_3$, and $p_4$ are distinct odd prime numbers. We give a criterion for the finiteness of an unramified Iwasawa module of a number field. The criterion is based on the nonexistence of a $2$-group with a certain prescribed quotient. Using this criterion, we construct an infinite family of such real quadratic fields with unramified Iwasawa module of type $\mathbb{Z}/4\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ and $2$-class group of type $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$. This gives the first example of an infinite family of real quadratic fields whose ideal class group has $2$-rank three and whose cyclotomic $\mathbb{Z}_2$-extension is totally ramified and satisfies Greenberg's conjecture.

For $r \ge 0$, let $\mathcal P_r=\{\binom{r}{0},\binom{r}{1},\ldots,\binom{r}{\lfloor r/2\rfloor}\}$ be the set of distinct entries in row $r$ of Pascal's triangle. We study the least possible least common multiple of $n$ entries chosen from one row, with the row itself also free: $a(n)=\min_{r\ge0,\ S\subseteq\mathcal P_r,\ |S|=n}\operatorname{lcm}(S)$. Whereas the least common multiple of an entire row is given by a classical identity of Farhi, allowing both the row and the selected coefficients to vary creates a different optimization problem. We first recast the fixed-row problem exactly as a weighted prime-power exclusion problem, which explains how omitting a few coefficients can remove expensive prime-power contributions. Our main asymptotic result is $\log a(n)=2n+O\!\left(n\exp\!\left(-c(\log n)^{3/5}(\log\log n)^{-1/5}\right)\right)$ for some absolute $c>0$, so $a(n)^{1/n}\to e^2$. A two-band refinement further shows that every optimal row satisfies $r_n=2n+O\!\left(n\exp\!\left(-c(\log n)^{3/5}(\log\log n)^{-1/5}\right)\right)$, and that the minimum prefix defect of an optimal support is $o(n)$.

We prove Steinness over $\breve{\mathbf{Q}}_p$ for the basic local Shimura tower attached to $G=\mathrm{GL}_n$, $\mu=(1,1,0^{n-2})$, with basic Newton slope $2/n$, for every odd $n\ge3$, at every finite level and for every prime $p$. The proof proceeds by constructing global analytic functions from the crystalline period map, proving compactness of their sublevel sets via a perfectoid normalization of Tate lattices, and showing that the resulting function map to affine space is finite. The key geometric inputs are the Fargues–Fontaine realization of the universal cover, a normalized determinant on the rank-two locus, and an integral PEL realization over $W(\overline{\mathbf{F}}_p)$ used to produce bounded global Hodge generators. The integral constructions include the prime $p=2$.

Given a polynomial \(g\) of degree \(\delta>0\) over a finite field, we study monic polynomials \(f\) of degree \(n\) that can be written as \(f=h+g^k\), where \(h\) is a monic irreducible polynomial of degree \(n\) and \(k\in\mathbb{N}\) with \(\delta k<n\). Following Erdős, we show that some \(f\) admit at least \(c\log n\) such representations. If \(\delta\rho_g<1\), where \(\rho_g=\varphi_q(g)/|g|\), then a positive proportion of degree-\(n\) polynomials admits at least two such representations. More generally, for sums of powers \(g^{\lfloor k_i^{r_i}\rfloor}\), we prove that a positive proportion of polynomials of degree \(n\) is representable whenever \(\sum_{i=1}^{t}r_i^{-1}\ge 1\), while certain residue classes modulo some irreducible polynomial contain no representable polynomial.

We give a unified arithmetic and geometric treatment of the Hall-Janko simple group and the icosian Leech lattice. Let $\mathcal{I}$ be the standard icosian maximal order over $K=\mathbb{Q}(\sqrt5)$. Following the mass-formula and Venkov framework for the Niemeier and Leech lattices, we classify positive definite unimodular Hermitian $\mathcal{I}$-lattices of quaternionic rank $3$. Hashimoto's mass formula, the Siegel-Weil average, and Gundlach's structure theorem determine the scalar theta series of every nonsplit class. A two-place harmonic separation of the Leech minimal shell proves that its three Hermitian components are quaternionic projective $5$-designs. For the norm-$2$ shell, integrality and projective moments recover the complete angle scheme and its $525$ orthogonal frames. The finite code of one frame then saturates the residual mass, giving exactly the split and Tits classes and an independent computation of the order of $\operatorname{Aut}_{\mathcal{I}}(L_{\mathrm T})\cong2.J_2$. Finally, a quaternionic line system with the prescribed five projective inner products has at most $315$ points, with equality precisely for the Hall-Janko configuration up to $P\operatorname{Sp}(3)$. Equality in Hoggar's bound supplies the design property; Cohen-Tits uniqueness, golden-angle fission, and the centered cubic moment give geometric rigidity.

Let $m\geq 1$ be a fixed integer, $a$ an integer satisfying $(a,m)=1$, and $z\geq 1$ a real parameter. Denote by $\omega_{z}(n;m,a)$ the number of distinct prime divisors $p$ of $n$ satisfying $p\equiv a\, (m)$ and $p\leq z$. We study an asymptotic behaviour of $\sum_{n\leq x}\left(\omega_{z}(n;m,a)-\frac{1}{\varphi(m)}\log\log z\right)^{k}$ as $x\to\infty$ for a wide range of positive integer $k\geq 2$, where $\varphi(\cdot)$ is the Euler function. Following a method of Granville and Soundararajan we lead an asymptotic formula for the above. Also, we investigate $\sum_{n\leq x}\left(\omega(n;m,a)-\frac{1}{\varphi(m)}\log\log x\right)^{k}$, where $\omega(n;m,a)$ denotes the number of distinct prime divisors $p$ of $n$ such that $p\equiv a\, (m)$.

Denote by $\omega(n)$ the number of distinct prime divisors of the natural number $n$. In 2007, Granville and Soundararajan gave a quite new method to compute the higher moments $\sum_{n\leq x}(\omega(n)-\log\log x)^{k}$, for a wide range of integers $k\geq 2$. In this notes, we shall apply the method for $\omega_{z}(n)$ which denotes the number of distinct prime divisors $\leq z$ of $n$. Especially, for odd integers $k\geq 3$, we lead asymptotic formulas for $\sum_{n\leq x}(\omega_{z}(n)-\log\log z)^{k}$, under certain restrictions on $k$, $z$, and $x$. Also, we refer to a framework of the approach.

Ozaki proved that every finite $p$-group occurs as the Galois group of a maximal unramified $p$-extension of a number field. Hajir, Maire and Ramakrishna made this theorem effective, obtaining a base-field degree of order $|G|$. In this article, we reduce that degree by taking the Frattini structure of $G$ into account. More precisely, for an odd prime $p$ and a finite $p$-group $G$ of order $p^n$, we prove \[ \tau_p(G) \leq p^{\ell_\Phi(G)+ \left\lceil\log_p\left(\binom{n+2}{2}+1\right)\right\rceil}. \] Here $\tau_p(G)$ is the least degree of a number field realizing $G$ as its $p$-class tower group, and $\ell_\Phi(G)$ is the iterated Frattini length of $G$. Thus, for groups of bounded Frattini length, the order-scale bound $p^n$ is replaced by the quadratic bound $O_p(n^2)$. For $E_m=(\Z/p\Z)^m$, we further prove the sharp estimate $\tau_p(E_m)\asymp_p m^2$.

We provide explicit bounds on the number of zeros for $\zeta$ in short intervals near the edge of the critical strip. Ultimately we show that even in short intervals a positive proportion of zeros will not be near the edge of the critical strip. For example with $h>100$ and $t>10^{100}$ the proportion of zeros of $\zeta$ with imaginary part in $[t-h,t+h]$ which have real part in $[\frac{1}{16},\frac{15}{16}]$ is at least $13\%$ and a positive proportion is already obtained at $t>10^{43}$.

We find all ring of integers, R, and finite groups G, such that RG has almost trivial units. Moreover, for such a group ring we determine the Swan subgroup of its locally free classgroup

We study genus two curves whose Jacobians have complex multiplication by a biquadratic CM field. In this setting the Jacobian decomposes as a product of two elliptic curves with complex multiplication by orders in the same imaginary quadratic field. We derive upper bounds for the primes of stable bad reduction of such curves in terms of the discriminants of the CM orders. To do so, we give an explicit description of principally polarized abelian surfaces with CM by a biquadratic field via small CM cycles on an orthogonal Shimura variety of signature (3,2). We use it to prove an arithmetic intersection formula between such CM cycles with Humbert surfaces in terms of coefficients of incoherent Eisenstein series and representation numbers of ternary positive definite quadratic forms. Finally, employing the arithmetic properties of higher Green functions, we show that primes of stable bad reduction always exist.

Given an $m$-dimensional weight $\mathbf{r}$ and an $n$-dimensional weight $\mathbf{s}$, we prove that the set of $(\mathbf{r};\mathbf{s})$-badly approximable $m\times n$ matrices is hyperplane absolute winning on $\mathbb{R}^{m\times n}$. This fully answers a question \cite[Question 8.2 (iii)]{Kl} of D. Kleinbock in 1998.

The elementary symmetric partition map $\pre_s$, introduced by Ballantine, Beck, and Merca, sends an integer partition to the summands in the evaluation of the $s$-th elementary symmetric polynomial at its parts. By encoding partition parts as prime-exponent valuation vectors, we connect $\pre_s$ to Leo Moser's additive Multiset Recovery Problem (1957) and prove that $\pre_s$ is unconditionally injective on partitions of length $n$ whenever $n$ lies outside the Moser root set $\mathcal{Z}_s$, with no size restrictions. Furthermore, under the equal-size constraint $|\lambda| = |\mu| = N$, we prove that $\pre_4$ is injective at the isolated singular length $n = 12$, and that every fiber of $\pre_3$ on $\Part_6(N)$ has cardinality at most $2$, completely excluding both triplets and quartets.

We prove that Stark–Shintani ray class invariants (Stark units) associated to real quadratic fields are algebraic numbers. These invariants are given by special values of Faddeev's modular quantum dilogarithm, introduced by Garoufalidis–Kashaev–Zagier. Our main discovery is that special values of the modular quantum dilogarithm satisfy an explicit overdetermined system of polynomial equations, matching a variation on the defining equations of Andersen–Kashaev's notion of a quantum dilogarithm on a product of two cyclic groups. We give two and a half proofs that this system of equations defines a zero-dimensional variety. The simplest follow from an uncertainty principle for finite Fourier transform and $2$-adic valuation bounds. The last proof is more involved and shows finite quanatum dilogarithms can be used to categorify fusion rings introduced by Izumi, and the algebraicity of the special values then follows by Ocneanu's rigidity theorem. As a byproduct, we obtain an explicit infinite family of irrational near-group fusion categories. As a further application, we prove a family of quadratic relations for Stark units recently conjectured by Appleby, Flammia, and Kopp motivated by Zauner's conjecture about SIC-POVMs (complex equiangular lines).

Let $p$ be an odd prime and let $S\subseteq\mathbb{F}_p^2$ have cardinality divisible by $p$. We prove that, for all sufficiently large primes $p$, no such set has exactly four special directions, and obtain a conditional extension to larger numbers of special directions under an affine-independence assumption on the corresponding projection functions. A separate second-moment argument shows more generally that, for every fixed $k\ge4$, no subset of $\mathbb{F}_p^2$ has exactly $k$ special directions once $p$ is sufficiently large. In fact, the result holds uniformly for $k$ up to a positive constant times $\sqrt p/\log p$. In contrast, for multisets every prescribed collection of at most $p$ directions can occur as the set of special directions of a $\{0,1,2\}$-valued multiset on $\mathbb{F}_p^2$.

Let $G$ be a finite abelian group, and let $k$ be an infinite perfect field containing a primitive root of unity of order the exponent of $G$. We prove that the $G$-equivariant version of Gabber's presentation lemma proposed by Bachmann holds over $k$ if and only if $G$ is cyclic of $p$-power order, for some prime $p$.

Let $S$ be a free profinite group on a finite ordered basis $X$, and let $S^{(n,p)}$, $n=1,2,\ldots,$ denote its lower $p$-central filtration. There is a natural duality between $S^{(n,p)}/S^{(n+1,p)}$ and $H^2(S/S^{(n,p)},\mathbb{F}_p)$. These $\mathbb{F}_p$-linear spaces admit natural bases indexed by Lyndon words of length $\leq n$ in the alphabet $X$. These bases are known to be unitriangularly dual. We prove that they are much closer to being fully dual, by showing that the pairing between two basis elements vanishes unless the corresponding Lyndon words are permutations of one another. We further show that the value of the pairing is essentially independent of $n$.

Uniform distribution modulo 1 is a classical notion of pseudo-randomness for sequences in the unit interval, which is quantified in terms of the discrepancy. Sequences whose discrepancy is of the smallest possible asymptotic order are called low-discrepancy sequences. The Poissonian pair correlation is another notion of pseudo-randomness, which studies the distribution of the gaps between pairs of elements of the sequence on a local scale. It is known that Poissonian pair correlation implies uniform distribution mod 1, and that the opposite implication is not true in general. It has also been observed that classical examples of low-discrepancy sequences fail to have Poissonian pair correlation, and it has been speculated that the two properties might be irreconcilable due to the high degree of structural rigidity that is required for low-discrepancy behavior. As we prove in this paper, this is not the case: we construct an example of a low-discrepancy sequence with Poissonian pair correlation.

Let $\sigma\mathrm{Mex}(n)$ and $\sigma_e\mathrm{Mex}(n)$ denote the sum of minimal excludants and sum of even minimal excludants over all overpartitions, respectively. In this work, we study these two combinatorial objects from an arithmetic perspective. We prove that, for $n\geq 0$, $\sigma_e\mathrm{Mex}(n)\equiv\sigma\mathrm{Mex}(n)-\bar{p}_{\geq 2}(n)\pmod{2^2}$, where $\bar{p}_{\geq 2}(n)$ denotes the number of overpartitions of $n$ with all parts at least $2$, and obtain relation with basic hypergeometric series. Furthermore, we prove the asymptotic behavior of $\sigma\mathrm{Mex}(n)$, $\sigma_e\mathrm{Mex}(n)$, and $\bar{p}_{\geq 2}(n)$ as $n\to\infty$. In particular, we prove that $\sigma\mathrm{Mex}(n)\sim 2\,\sigma_e\mathrm{Mex}(n).$

For a matrix $\gamma\in\mathrm{SL}_2({\mathbb Z})$, we define \[ {\mathcal R}(\gamma)=a_1^2+a_2^2+a_3^2+a_4^2, \qquad \text{where} \ \gamma=\begin{pmatrix}a_1&a_2\\ a_3&a_4\end{pmatrix}, \] and let $S_{\mathrm{sq}}(X)$ count the number of matrices $\gamma$ with $\|\gamma\|_\infty = \max\{|a_1|, |a_2|,|a_3|,|a_4|\} \leq X$ and such that ${\mathcal R}(\gamma)$ is squarefree. We prove that \[ S_{\mathrm{sq}}(X) = {\mathfrak S}_{\mathcal R}^{\mathrm{sq}}N(X) +O(X^{19/10+o(1)}), \quad \text{as}\ X\to \infty, \] where $N(X)=\#\{\gamma\in{\mathrm{SL}}_2({\mathbb{Z}}):\|\gamma\|_\infty\leq X\}$ and ${\mathfrak{S}}_{\mathcal{R}}^{\mathrm{sq}}$ is an explicit positive Euler product of local $p^2$-densities. The proof combines the $\delta$-method for small moduli with a sum-of-two-squares estimate for large square divisors. This complements a result of J. B. Friedlander and H. Iwaniec (2009) on prime values of ${\mathcal R}(\gamma)$, which, however, is conditional on a very strong form of the Elliott–Halberstam conjecture. We also show that ${\mathcal R}(\gamma)$ is squarefree and has at most $9$ prime divisors for at least $cN(X)/\log X$ matrices $\gamma\in\mathrm{SL}_2({\mathbb Z})$ with $\|\gamma\|_\infty\le X$, where $c>0$ is an absolute constant.

Let $(a,b,c,d) = \left(2mn, 2mp, m^2 - n^2 - p^2, m^2 + n^2 + p^2\right)$ be a primitive Pythagorean quadruple and let $S=\langle a, b, c, d\rangle$ be the numerical semigroup generated by $a,b,c,$ and $d.$ For convenience, we also let $Q = n^2 + p^2, \delta = \gcd(n,p),$ and $n = \delta n_0$ for some $n_0 \in \mathbb{Z}$. In this paper, we determine the Frobenius number of $S$ and derive an explicit formula in terms of $m$, $n$, and $p$ with the assumption that $m\geq 2Q$ for $\delta = 1$ and $m \geq \frac{2Q}{\delta} - 1$ for the remaining cases. The proof is based on an explicit complete residue system modulo $2mn$, a normalization procedure for arbitrary semigroup elements, and a lift-orbit description of boundary representatives.

Recently, Garvan and Sarma studied sums of non-overlined parts in overpartitions and partitions without repeated odd parts. In this paper, we study the corresponding statistic for overlined parts. We define $\mathrm{OSOME}(n)$ as the sum of all odd overlined parts minus the sum of all even overlined parts in the overpartitions of $n$. We obtain a closed form for its generating function and use it to prove congruences by elementary $q$-series methods, the theory of half-integral weight modular forms, and known congruences for the overpartition function.

We give new conditions on allowable powers of a prime $p$ dividing an integer group determinant for the group $\mathbb{Z}_p^n$, and show these conditions are sharp for $n\leq 5$. The values coprime to $p$ for these groups are already known; determining which multiples of allowable powers of $p$ occur is much more complicated. We show that all multiples of sufficiently large powers of $p$ occur as integer group determinants of $\mathbb{Z}_p^n$. We also provide a complete characterization for the group $\mathbb{Z}_3^3$, where particular arithmetic conditions are required for multiples of certain powers of $3$.