Papers

We study non-archimedean Monge-Ampère measures of toric metrics on closed subvarieties of toric varieties, after reducing, if necessary, to the minimal torus-orbit closure containing the subvariety. The measure vanishes on the toric boundary and is supported on the tropical skeleton. Our main result explicitly describes their local structure in terms of the tropical germ of the subvariety and the tropical Green function of the metric. More precisely, the tropicalized Monge-Ampère measure is the corresponding polyhedral Monge-Ampère measure, which extends the tropical intersection product, while the original measure is recovered locally by tropical pullback from explicit tropical and convex data. This provides a toric counterpart to analogous results for subvarieties of abelian varieties, with new local phenomena arising from tropicalization.

Let $f$ be an extended Rademacher random multiplicative function (RMF). We show that, for every fixed deterministic function $V(x)$ tending to infinity, almost surely there are arbitrarily large $x$ for which \[ \sum_{n\leq x}f(n) \geq \frac{\sqrt{x}(\log\log x)^{1/4}}{V(x)}. \] The corresponding negative fluctuation holds as well. In particular, this gives an affirmative answer to Erdős Problem~\#1144. As a byproduct, our result has a direct corollary giving new almost sure lower bounds $\log\log x$ on the number of sign changes of partial sums up to $x$ for all sufficiently large $x$.

We determine all the groups that appear as the torsion group of an elliptic curve over a quintic number field. Apart from the groups that already occur infinitely often, which were determined by Derickx and Sutherland, exactly three groups occur: $\mathbb{Z}/28\mathbb{Z}$, $\mathbb{Z}/30\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$. Up to isomorphism of the pair $(K,E)$, the first and the third are each realised by a single elliptic curve and the second by two curves, which are $2$-isogenous over a common quintic field. The curves realising $\mathbb{Z}/28\mathbb{Z}$ and $\mathbb{Z}/30\mathbb{Z}$ were found by van Hoeij, while the group $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/18\mathbb{Z}$ is new, and $5$ is the smallest degree in which a non-cyclic sporadic torsion group occurs. The methods improve on those developed by Derickx and Najman, and are based on Hecke sieves and the arithmetic of cuspidal divisor classes.

We introduce a projective-kernel framework for the study of replicable functions in modular function theory. The main point is that the same two-point kernel simultaneously encodes the differential projective geometry of a modular function and the Faber–Grunsky data governing its replicability. This makes it possible to translate between Schwarzian invariants, coefficient identities, and modular correspondences within a single structure. The kernel yields a reconstruction theorem showing that the ordinary Schwarzian determines the normalized Grunsky matrix and hence the underlying Laurent expansion. When Norton's replicability relations are imposed, the projective kernel produces strong arithmetic restrictions on the possible cusp data. In the degree-one case this leads to a precise classification: replicability and complete replicability are equivalent to solvability of the projective monodromy, while the icosahedral case is excluded by an explicit Grunsky obstruction. The same principle extends to the arithmetic Hecke triangle groups. The projective-kernel viewpoint also isolates the remaining global difficulty in Norton's Hauptmodul conjecture and provides a natural setting in which replicability, projective monodromy, and modular differential invariants can be studied together.

We extend Gauss composition and Gauss genus theory over $\mathbf{Z}$ to $\mathbb{F}_{2^n}[T]$, a polynomial ring over a finite field $\mathbb{F}_{2^n}$ of characteristic 2. We find new invariants of binary quadratic forms over $\mathbb{F}_{2^n}[T]$ by using Arf invariant and introduce new definitions of proper equivalence and direct composition, and prove that the direct composition makes the set of proper equivalence classes of binary quadratic forms with the same invariants into a finite Abelian group, which is isomorphic to a Picard group of a corresponding extension ring of $\mathbb{F}_{2^n}[T]$. Building on this, we develop genus theory in characteristic 2 and prove that the kernel of the generalized Gauss's map is the subgroup of all squares in the class group.

For integers $m\geq3$ and $n\geq1$, an $m$-gonal form is called tight $\mathcal{T}(n)$-universal if it represents exactly the positive integers $\mathcal{T}(n)=\{ n, n+1, n+2, \ldots \}$. In this paper, we study the minimal criterion set $\mathrm{CS}(m,n)$ for tight $\mathcal{T}(n)$-universality. Our main result determines $\mathrm{CS}(m,n)$ for all $n \geq8$, $3 \leq m \leq \left\lfloor \frac{3n+1}{2} \right\rfloor$. More precisely, \[ \mathrm{CS}(m,n)= \begin{cases} \{ n, n+1, \ldots, 2n-1 \}, & m=5,\newline \{ n, n+1, \ldots, 2n \}, & m\neq5. \end{cases} \] We also establish the corresponding tight $\mathcal{T}(n)$-universality results and show that the upper bound on $m$ is optimal.

We establish an asymptotic formula for the first moment of the central values of quadratic Dirichlet $L$-functions in the even hyperelliptic ensemble $\mathcal{H}_{2g+2}$ over $\mathbb{F}_q[x]$, for every fixed odd prime power $q$. Working with the completed $L$-function yields an exact two-term central-value formula with truncation levels $g$ and $g-1$; combining the two truncations before the square-dual contour shifts leaves the three cubic points $z^3 = q^{-4}$ as the only secondary poles, and they contribute the secondary term $q^{2g/3+2}(a_g g + b_g)$ with an error $O_\varepsilon(q^{g(1+\varepsilon)/2})$. The coefficients are real and $m \mapsto (a_m, b_m)$ has minimal period exactly three, so the coefficient of $g$ at the order $q^{2g/3}$ depends on $g$ modulo 3, in contrast with the earlier even-degree formulas; the difference is confirmed numerically.

We study the zeros of the quasimodular forms $G_{\{2\}^n}$ defined by iterated sums. We first show that, for every $n>0$, $G_{\{2\}^n}$ has exactly $n$ simple zeros on each of the vertical half-lines $\Real(\tau)=0$ and $\Real(\tau)=1/2$, and that the zeros for consecutive values of $n$ satisfy an interlacing property. The proof is based on an expression of $G_{\{2\}^n}$ in terms of the $n$-th derivative of $\eta^3$ and on the theory of bell-shaped functions, rather than on Rankin–Swinnerton-Dyer method. We also determine the asymptotic behavior of these zeros as $n\to\infty$. In addition, we prove that all zeros of $G_{\{2\}^n}$ are simple and that $G_{\{2\}^n}$ has infinitely many $SL_2(\ZZ)$-inequivalent zeros. We further show that quasimodular forms of maximal depth have no zeros at CM points. In particular, none of the zeros of $G_{\{2\}^n}$ are CM points. Finally, in the special case $G_{2,2}$, we show that each Ford circle contains exactly two distinct simple zeros.

Sárközy's theorem says that if $A \subset \mathbb{Z}$ has positive upper asymptotic density, then there are distinct $a_1, a_2 \in A$ and $n \in \mathbb{Z}$ such that $a_1-a_2 = n^2$. The same is true if $n^2$ is replaced by $F(n)$ for any polynomial $F \in \mathbb{Z}[x]$ with constant term zero. Green proved an $\mathbb{F}_q[t]$-analog of Sárközy's theorem with strong quantitative bounds, but required a technical condition on the number of roots of the polynomial $F \in \mathbb{F}_q[x]$. This condition was recently removed by Li and Sauermann. In this paper, we generalize Green's argument to accommodate equations in more variables in $\mathbb{F}_q[t]$, while pointing out that the technical condition can be removed by means of a simple observation.

For a positive integer $h$, a subset $A = \{a_1, \dots, a_k\}$ of an additive abelian group $G$ is called an $S_h$-set of size $k$ if all sums of $h$ distinct elements in $S$ are distinct. For fixed positive integers $h$ and $k$, let $v_h(k)$ denote the order of the smallest abelian group containing an $S_h$-set of size $k$. A lower bound for $v_2(k)$ is known. In this paper, we establish a lower bound for $v_3(k)$. Using our argument for $h=2$, we recover the known bound for $v_2(k)$.

We study the generalized Flint Hills series $\mathcal{F}_{s,t}(x)=\sum_{n\ge1} n^{-s}|\sin(\pi nx)|^{-t}$ for $s>0$ and $t>1$. An explicit comparison with a series over continued-fraction denominators yields the Hausdorff dimension $\min\{1,2t/(s+t)\}$ of its divergence set. At each critical exponent $\tau=1+s/t>2$ we construct numbers of irrationality exponent $\tau$ realizing both convergence and divergence; the convergent examples establish Meiburg's conjecture in the range $t>1$. Both parts of the critical fibre have Hausdorff dimension $2/\tau$. For $s>t$ and $h_\kappa(r)=r^{2/\tau}(\log(1/r))^\kappa$ we prove that the divergence set has zero $h_\kappa$-measure for $\kappa<-1$ and infinite measure for $\kappa\ge-1$. The divergent part of the critical fibre satisfies the same law, whereas its convergent part has infinite measure for every $\kappa$. The key estimate selects a rapidly growing subsequence of convergent denominators and gives a double-logarithmic bound on the approximation error. The convergence of the classical Flint Hills series $\sum_{n\ge1}(n^3\sin^2 n)^{-1}$ remains undecided.

Racinet's double shuffle Lie algebra $\mathfrak{dm}_0$ encodes the double shuffle relations of multiple zeta values. We study two analogues of $\mathfrak{dm}_0$ for multiple Eisenstein series and $q$-analogues of multiple zeta values, namely the space $\operatorname{BARI}_{\operatorname{swap},\mathrm{il}}$ of swap-invariant alternil bimoulds with the uri bracket of Kühn and Schneps, and Burmester's balanced double shuffle space $\mathfrak{bm}_0$. Both were conjectured to be Lie algebras. We prove both conjectures and show that $\mathfrak{bm}_0$ is isomorphic to the Lie algebra of finite-depth polynomial elements of $\operatorname{BARI}_{\operatorname{swap},\mathrm{il}}$.

We prove the Tate conjecture in every codimension on every power of an abelian variety of dimension at most four over a finite field. For geometrically simple fourfolds, we use Broe's theorem and the classification of Frobenius relations. For dihedral pairs of abelian surfaces, we construct algebraic classes using families of abelian threefolds, the Gross–Schoen height formula, and polarization contractions. We also prove the Hodge conjecture for every power of a complex CM abelian fourfold, using Markman's theorem on Weil classes and Milne's criterion. The finite field results imply standard conjecture~D and the expected pole orders of the zeta function on every power.

Let $R_N(m)$ count unordered partitions of $m$ into reduced positive fractions whose numerators and denominators are at most $N$, excluding integer parts. Uniformly for $\rho$ in compact positive intervals, $\log R_N(\lfloor\rho N\rfloor)=\sqrt{2\rho}\,N^{3/2}-\kappa(\rho)N^{3/2}/\log N+o(N^{3/2}/\log N)$. The positive, continuously differentiable function $\kappa$ is an explicit sum of lattice-distance integrals, with $\kappa(1)\approx 0.00264713$. For a uniform partition of $n$, all but $o_{\mathbb{P}}(n/\log n)$ prime-denominator blocks in $(n/2,n]$ have total 2 or 3, according to whether $p/n$ lies below or above an explicit threshold near 0.7522. For fixed $N$, we give the Ehrhart numerator and determine how its poles control quasipolynomial coefficients. Its residue distribution is asymmetric for $N\ge 4$, but agrees with an independent model in every moment below order $\lceil N/2\rceil$. We identify the first discrepancy and prove a Gaussian limit. We also obtain joint denominator and size laws under two sampling rules.

We determine how p-adic slopes of modular forms vary as their associated Galois representations modulo p vary over moduli of Galois representations. As a consequence, we show that the authors' ghost conjecture recently proven by Liu, Truong, Xiao, and Zhao is "stable under direct sums".

We develop a local-global theory for visibility of lattice points on families of parameterized curves. We introduce a notion of $p$-adic visibility and ask whether a lattice point is globally visible precisely when it is visible at every prime. We prove that this local-global principle holds for a broad class of families whose parametrizations are homogeneous with respect to positive weights. When the points on these curves fill the entire positive integer lattice, we compute the local and global densities of visible points and show that the global density is the product of the local densities. We then consider polynomial families of the form $y=qP(x)$, with $q\in\mathbb{Q}_{>0}$, and show that visibility can be detected prime by prime exactly when $P$ is a monomial. For non-monomial polynomials the local-global principle can fail, but the set of points where it fails has density zero; for separable polynomials we also obtain a quantitative bound for the number of non-visible points, improving the previously known bound. We further consider families whose lattice points lie on a proper lower-dimensional algebraic subset of the ambient space and show that their visibility densities can behave differently from those of the full lattice. Finally, we extend the theory from visibility from the origin to visibility from one lattice point to another and show that the corresponding local-global principle continues to hold for weighted homogeneous families.

We compute the Fourier expansions of the Kudla-Millson theta lifts of vector-valued Hilbert cusp forms of parallel weight. These expansions are with respect to 0-dimensional cusps of non-compact orthogonal Shimura varieties defined over any totally real number field, and computed using a mixed model of the adelic Weil representation under a partial Fourier transform. As a consequence, we obtain the injectivity of the theta lifts under some additional hypothesis on the variety. Furthermore, we show that the theta lifts are square-integrable harmonic differential forms, and deduce lower bounds for dimensions of cohomology groups of such varieties.

We define numerical adelic characteristic classes for vector bundles on projective varieties over number fields. The metric data are carried by the tautological quotient line bundle on the associated projective bundle. The construction uses the established intersection theory of adelic line bundles. Higher characteristic classes are multilinear intersection functionals. They are continuous for simultaneously controlled model sequences that are Cauchy for the supremum norm. We construct the resulting tautological numerical intersection algebra. We also prove controlled numerical Bogomolov–Gieseker inequalities. We first treat the case of a curve over $K$, where the numerical inequality is combined with the Deligne pairing and the adelic Hodge index theorem. We then pass to higher dimension, using Moriwaki's model-level dimension induction before taking the controlled Zhang limit. The curve theorem includes equality and uniform-gap criteria.

We prove finiteness and Poincaré duality for pro-étale $\mathbf{Q}_p$-local systems on proper $p$-adic rigid-analytic spaces in the framework of Banach–Colmez spaces and establish their optimal cohomological vanishing bounds. As a consequence, we also obtain ordinary finiteness and duality for their arithmetic pro-étale cohomology. We deduce these results from their analogs for perfect complexes on the Fargues–Fontaine curve, which in turn reduce to Poincaré duality for perfect complexes over period rings. We give a simple proof of the latter duality via the same diagrammatic argument as in our previous work for finite coefficients. Along the way, we establish optimal $v$-descent for perfect complexes over certain period rings on affinoid perfectoid spaces.

Restriction to the full cusp divisor of a modular curve defines a secondary invariant whose rational indeterminacy comes from a single global modular form. For every integral weakly holomorphic level-one modular form $h$ of weight divisible by four, we prove that the imported value $[h/2]$ vanishes at every nontrivial $\Gamma_0(N)$ level. The construction localizes coefficients before completion and rationalizes only after passing to homotopy groups. At odd prime level, we identify the full cusp spectrum as a product of two real Tate factors and construct a single holomorphic weight-two correction. This correction yields an actual integral global homotopy class and a rational global source class satisfying the equality required for joint annihilation. The equality transports to every odd composite level, while even levels follow by inverting two. At level three, the Mahowald–Rezk homotopy calculation leaves only the periodic $\nu$ family in stems $8K+3$. A cusp-residue homomorphism detects its secondary value, which has exact order two throughout the periodic family. As an application, the Bunke–Naumann secondary invariants of products in bidegrees $(8k+1,8k'+2)$ vanish after passage to every nontrivial level.

We prove the expected asymptotic in Waring's problem over $\mathbb F_q[T]$, with a power-saving error, whenever $n>2d$, the characteristic is greater than $(d-1)^2$, and $q$ satisfies an explicit lower bound. This range is sharp in general for the expected asymptotic uniformly in the target polynomial. Our main new input is an aggregate minor arc estimate: we count functionals according to the codimension of their associated singular loci and use intersection theory to bound the degrees of the resulting parameter spaces. In particular, if $n\ge (2+\varepsilon)d$, the required lower bound on $q$ is polynomial in $d$ of degree $2+4/\varepsilon$.

Let $q \equiv 1 \bmod 8$ be a prime power. We prove that $L\left(\tfrac{1}{2}, \chi\right) \neq 0$ for 100% of the imaginary quadratic Dirichlet characters of $\mathbb{F}_q(T)$.

For a vector of positive integers $\mathbf{b} = (b_1,\ldots,b_h)$ with $\gcd(b_1,\ldots,b_h) = 1$, we study sets $A \subseteq \mathbb{N}$ for which every sufficiently large integer has a bounded positive number of representations \[ n = b_1 x_1 + \cdots + b_h x_h \qquad (x_1,\ldots,x_h\in A). \] We prove that such a set exists for every binary vector $\mathbf{b} \neq (1,1)$, and for some general higher-dimensional families, including $\mathbf{b} = (u_1, p^d u_2, \ldots, p^{(h-1)d} u_h)$ where $p\nmid u_1\cdots u_h$.

Let $c\phi_k(n)$ denote the number of $k$-colored generalized Frobenius partitions of $n$. We prove that, for every $m\geq2$ and every $k\equiv2\pmod{2^m}$, \[ \sum_{n\geq0}c\phi_k(n)q^n\equiv\frac{\varphi(q)\,(q^2;q^2)_\infty}{(q;q)_\infty^2}\sum_{n\geq0}c\phi_{k/2}(n)q^{2n}\pmod{2^m}, \] where $\varphi(q)$ is the classical theta function. For $m=2$ this recovers a congruence of Chan, Wang, and Yang. Applied with $k=18$, we determine $c\phi_{18}(2n+1)$ modulo $16$ completely. In particular, we also prove \[ \sum_{n\geq0}c\phi_{18}(6n+1)q^n \equiv4\sum_{r\in\mathbb{Z}}q^{r(3r-1)/2}\pmod{16}, \] which proves the congruences $c\phi_{18}(30n+19)\equiv c\phi_{18}(30n+25)\equiv0\pmod{16}$ recently conjectured by Das, Nath, and Sarma (2026). It also yields further congruences modulo $16$ and a simple modulo-$8$ characterization that recovers and extends a recent congruence of those authors. As a second application we set $k=10$ and determine $c\phi_{10}(2n+1)$ modulo $8$.

Let $\overline{\mathrm{OPT}}_m(n)$ denote the number of overpartition $m$-tuples of $n$ into odd parts. We prove that for every odd $m\ge1$ and every $i\ge3$, \[\sum_{n\ge0}\Bigl(\overline{\mathrm{OPT}}_m\bigl(2^in\bigr)-\overline{\mathrm{OPT}}_m\bigl(2^{i-1}n\bigr)\Bigr)q^n \equiv 2^{\,i+1}\sum_{k\ge0}q^{(2k+1)^2} \pmod{2^{\,i+2}} .\] Thus $\overline{\mathrm{OPT}}_m(2^in)\equiv \overline{\mathrm{OPT}}_m(2^{i-1}n)\pmod{2^{i+1}}$, with equality of $2$-adic valuations exactly at the odd squares. The proof is elementary and uniform in $m$: a single family of integer polynomials, given by a three-term recurrence, governs every $U$-operator identity involved, and a divisibility statement supplies one power of $2$ per iteration.

In this article, we describe a new maximal function field $G$ over the finite field $\mathbb{F}_{11^2}$ with $11^2$ elements and show that it cannot be obtained as a subfield of the Hermitian function field. This provides the first known $\mathbb{F}_{p^2}$-maximal function field that satisfies this property. Further, we compute the automorphism group of $G$. The new maximal function field was found using AI.

We investigate the mixed moments of products of irreducible $\mathrm{GL}(1)$ and $\mathrm{GL}(2)$ $L$-functions on the critical line, and establish sharp moment bounds for these quantities. In particular this establishes the order of these mixed moments in certain ranges. This determines unconditionally, for the first time, the order of magnitude of the first moment of cubic Dedekind zeta functions with negative discriminant. The proof of the lower bound introduces a new method for computing mixed moments, that is a mix of the classical method of Heath–Brown with the modern machinery of Harper, Heap, Radziwill, and Soundararajan.

We investigate the equal-characteristic case of the Moon–Taguchi bounded-conductor finiteness conjecture for mod $p$ Galois representations over global function fields. We first establish a conditional finiteness theorem: for any global function field $K$ of characteristic $p$ and any integer $n \ge 1$, there are only finitely many isomorphism classes of continuous, semisimple, everywhere unramified, and geometric representations $\rho: G_K \to \mathrm{GL}_n(\overline{\mathbf{F}}_p)$ that admit an everywhere unramified characteristic-zero lift. Furthermore, we prove that the conjecture fails in general without this lifting hypothesis. Concretely, we construct infinitely many global function fields $K$ of characteristic $p$ admitting a continuous, surjective, absolutely irreducible, everywhere unramified, and geometric representation $\rho_r: G_K \twoheadrightarrow \mathrm{SL}_2(\mathbf{F}_{p^r})$ for each integer $r \ge 4$. As a corollary, we construct an everywhere unramified Galois extension \(L/K\), regular over \(\mathbf F_p\), with Galois group \[ \mathrm{Gal}(L/K)\cong \prod_{r\geq4}\mathrm{PSL}_2(\mathbf F_{p^r}). \] Combined with known cross-characteristic finiteness theorems, this completely resolves the question posed by Moon and Taguchi in dimension two: such extensions exist over global function fields of characteristic \(p\), whereas they cannot exist over global function fields of characteristic different from \(p\).

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

We introduce two independent numerator exponents into the Zeilberger–Zudilin integral and specialize them to \[ A_1=A_2=\frac{1857}{2785}. \] The resulting integer linear forms in $1$ and $\pi$ prove \[ \mu(\pi)<7.101862832357. \] This lowers the Zeilberger–Zudilin upper bound $7.103205334137\ldots$ by more than $0.001342501780$; the difference between the unrounded bounds is $0.0013425017806509\ldots$, approximately $0.01890\%$. The same parameter point is a strict two-dimensional local minimizer of the explicit auxiliary upper-bound function in its admissible arithmetic chamber. This is a local statement about that function, not a claim that the point is a global optimizer among all constructions.