Papers

Erdős, Odlyzko and Sárközy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$; their conjecture remains open even under the GRH. We prove that its analogue holds, unconditionally and in the wider range $q^{\Theta}$ for any fixed $\Theta > 3/4$, in Granville's random model of the primes. Almost surely, every reduced class modulo every sufficiently large modulus is representable, and the number of representations is asymptotic to its expected order. The choice of the model is decisive; the corresponding statement in the random prime model of Banks, Ford and Tao fails with positive probability, an elementary local defect that Granville's discard step removes.

For an integer $b\neq0,1$ let $\theta$ be the unique real root of $f_b(x)=x^3-3bx-b^3$ and let $K_b=\mathbb{Q}(\theta)$. We exhibit an explicit set of $b$ of positive density for which $\eta_b=-1/(\theta-(b+1))$ is the fundamental unit of $K_b$, and we determine the set of $b$ for which $\eta_b$ is the square of a unit: it is parametrized by the Pell equation $D^2-3E^2=1$, hence infinite, and under an explicit mild condition on the field discriminant it accounts for all $b$ for which $\eta_b$ is not the fundamental unit. That condition fails for only four $b$ with $|b|\le3000$, and at one of them, $b=3$, the conclusion itself fails. In the order $\mathbb{Z}[\eta_b]$ generated by $\eta_b$, by contrast, $\eta_b$ is the fundamental unit for every $b$, so these exceptions are a phenomenon of the maximal order. As an application we construct infinitely many biquadratic fields whose $3$-class field tower has length greater than $1$.

A phase that reaches a fault-tolerant processor in additive pieces can be remembered until the last piece arrives, or executed on arrival: the first option costs classical memory carried across rounds, the second costs magic states committed before the phase is known. We determine the exchange rate $\alpha$, committed $T$ gates per bit of memory forgone, for ancilla-free coordinatewise Clifford+$T$ compilation. The Ramanujan bound gives $\alpha\ge2$ with explicit constants, the square-root barrier of the spectral method. An elementary determinant method, using that quaternion numerators lie on spheres in both real embeddings of $\mathbb{Q}(\sqrt2)$, counts words near any rotation coset below that barrier. With a height dichotomy for the sphere sections it produces and a fibration over rational projections, it gives every $\alpha<5/2$ unconditionally and uniformly over frames, by an exact SMT check of its continuum case analysis; $5/2$ is where this method stops. At Clifford-framed cosets the volume law holds up to subexponential factors: all but a vanishing fraction of $z$-rotations need $T$-count $(3-o(1))\log_2(1/\varepsilon)$, and processes whose committed pieces are near Clifford-framed $z$-rotations, including per-rotation pipelines, have $\alpha\ge3-o(1)$, which a fractional-passthrough family attains under the Ross-Selinger typical-cost hypothesis. Under an equidistribution conjecture supported by exhaustive enumeration to $T$-count 22, $\alpha=3$ in general and memory should be shed in whole rotations. The bounds hold even when the phases cancel to the identity; side information enters through a conditional entropy; probabilistic mixing halves the costs but not the rate. With clean ancillas and a phase-gradient catalyst, table lookups batched over coordinates drive the rate to $O(1/\log\log(1/\varepsilon))$, so the constant-rate law is specific to coordinatewise synthesis.

Let $f\colon\mathbb{N}\to\mathbb{U}$ be a non-pretentious multiplicative function in the sense of Granville and Soundararajan. We study the convergence of the autocorrelation averages of $f$. In particular, we investigate the relationship between the local pretentious behaviour of $f$ on intervals of the form $[x^{\varepsilon},x]$ and its deviation from $1$ on the primes. As an application, we exhibit a class of non-pretentious multiplicative functions for which Elliott's conjecture in its original formulation holds, following up on a result by Klurman, Mangerel and Teräväinen.

We determine the algebra generated by the normalized parity-twisted crank moments and describe it explicitly as a proper subalgebra of the quasi-modular forms on $\Gamma_0(2)$ closed under $D=q\frac{d}{dq}$. We derive congruences from differential identities and construct a prime-detecting expression involving only the normalized twisted second moment, its products, and its derivatives. Its coefficient of $q^n$ vanishes if and only if $n$ is prime, for every $n\ge2$. We prove that its highest weight, eight, is minimal among prime-detecting elements of this algebra. Finally, we identify the same algebra as the one generated by MacMahon's functions $B_k$, obtaining a corresponding prime-detecting expression in $B_1$.

Let $p$ be an odd prime number. Based on recent work of Zhang, we prove that for a finite set $S$ of primes of $\Q$ containing $\infty$ but not $p$, any continuous odd representation $G_{\Q,S} \to \GL_2(A)$ over a complete Noetherian local ring $A$ with finite residue field of characteristic $p$ has finite image, where $G_{\Q,S}$ denotes the Galois group of the maximal extension of $\Q$ unramified outside $S$. This proves the two-dimensional odd case of Boston's strengthening of the unramified Fontaine–Mazur conjecture over $\Q$. Furthermore, we show that for an imaginary quadratic field $K$, any continuous conjugate self-dual two-dimensional $p$-adic representation of its pro-$p$ Galois group $G_{K,S}(p)$ has finite image, provided the primes in $S$ satisfy a modest condition. As an application, we resolve the sufficiency direction of McLeman's $(3,3)$-conjecture on $p$-class field towers for $p>3$. Namely, we prove that if $K$ is an imaginary quadratic field with $p$-class rank two and the Galois group $G_{K,\varnothing}(p)$ of the maximal unramified $p$-extension of $K$ has Zassenhaus type $(3,3)$, then the $p$-class field tower of $K$ is finite. For $p=3$, the results of Ahlqvist and Pink give the same finiteness conclusion in ten of the thirteen possible cases for the fourth Zassenhaus quotient.

We study the de Bruijn–Newman family $H_\tau(z)$, with heat-deformation parameter $\tau$ and $H_0(z)=\tfrac18,\xi(\tfrac12+\tfrac{iz}{2})$. The moving zeros considered here are therefore zeros of $H_\tau$, not directly of $\zeta(s)$. Retaining the collective interaction term in the zero dynamics, we prove that for a simple nonreal zero $z=x+iy$ of maximal imaginary height, [ \frac{d}{d\tau}y^2\le -2-4y^2\mathcal G_\tau(z), ] where $\mathcal G_\tau\ge0$ is a projected interaction sum. A second exact inequality bounds $\mathcal G_\tau$ below by an off-zero logarithmic derivative $-\operatorname{Im}(H_\tau'/H_\tau)(x+i\eta)$, providing a direct interface with the effective Polymath approximation $H_\tau=B_\tau F_\tau$. At the frontier $X=6000000185827$, $\tau_0=129/800$, and $y_0^2=87677/2500000$, a directed Cauchy certificate gives $\mathcal G_\tau>3/2$ for $\tau_0\le\tau\le0.178$, $x\ge X$, and $|y|\le y_0$. Combining this contraction with certified upper and lower zero-free barriers, a two-envelope argument reduces the maximal nonreal height to $0.08$ by time $0.1747532546428610755\ldots$; the classical de Bruijn contraction then completes the landing. Relative to the publicly replayable but not yet peer-reviewed $0.1787854$ base certificate, this yields the audit-relative bound [ \Lambda\le0.1779532546428610755\ldots . ] The collective contraction, logarithmic-derivative bridge, and new high-$x$ certificates are independent contributions.

The index of a given monic irreducible polynomial $f(x)\in\mathbb Z[x]$ having a root $\theta $ is the index of $\mathbb Z[\theta]$ in the ring of algebraic integers of $\mathbb Q(\theta).$ Further, $f(x)$ is monogenic if its index is $1$. In this article, we prove that the index of $f(x)$ always divides the index of $f(g(x)),$ where $f(x), g(x)\in \mathbb Z[x]$. Additionally, we give the precise power of the index of $f(x)$ dividing the index of $f(g(x))$. Also, we provide a necessary condition for the monogenity of $f(g(x))$. As an application of our results, we prove that the $n$-fold composition $f^n(x)=(f\circ f\cdots\circ f)(x)$ of a polynomial $f(x)$ is monogenic for all $n\in\mathbb{N}$ if and only if the sequence $\langle I_n\rangle$ is convergent, where $I_n$ is the index of $f^n(x)$. Moreover, we show that for a given $n\in\mathbb{N}$, if $I_n$ is $m^{\text{th}}$ power free integer, then the $n$-tower $\mathbb Q\subseteq K_1 \subseteq\cdots \subseteq K_{n-1} \subseteq K_n,$ defined by $n$ iterates of a polynomial $f(x)\in\mathbb Z[x]$ contains at least $\max\{\lfloor n-\log_{\deg f}m \rfloor,0\}$ monogenic number fields.

The study of monogenic polynomials is a classical problem in algebraic number theory. Existing criteria for deciding whether a polynomial is monogenic typically rely on discriminant computations together with methods such as Dedekind's criterion, Newton polygons, or valuation-theoretic techniques. In this paper, we develop a general local criterion for the $p$-maximality of orders generated by roots of arbitrary monic irreducible polynomials. As an application, we apply this to irreducible polynomials of the type $f(X)=X^n+A(BX+1)^m,$ where $1\le m<n$, $\gcd(n,mB)=1$, and $A,B\in\mathbb{Z}\setminus\{0\}$. We show that $f$ is monogenic if and only if both $A~\text{and}~n^n+(-1)^{n+m}B^n(n-m)^{\,n-m}m^mA$ are square-free. This provides a new proof of the main theorem of \cite{KK}, thereby proving Lenny Jones' conjecture \cite[Conjecture 4.1]{LJ}. Furthermore, we obtain explicit infinite families of irreducible non-monogenic polynomials, including trinomial, quadrinomial, and power-compositional families.

We develop Picard Farey symbols for finite-index subgroups of \(G=\PSL_2(\ZZ[i])\) from the Gaussian Farey tessellation of \(\HH^3\) by ideal octahedra. The ambient geometry naturally involves two different index-two extensions of \(G\): the orientation-preserving group \(\PGL_2(\ZZ[i])\) and a reflection extension. This distinction explains the two \(G\)-orbits of Farey faces, which merge into one \(\PGL_2\)-orbit, and the classical extended modular tessellation appears as a totally geodesic slice. A Picard Farey symbol records finitely many Gaussian-rational octahedral occurrences, their local stabilizers, and cooriented face-pairing transformations. We prove that such a symbol reconstructs the finite Picard-cell action and hence the subgroup; moreover, when subgroup membership is decidable, the symbol can be found by a terminating finite double-coset procedure. For torsion-free subgroups we obtain whole-octahedron fundamental domains. The marked symbol also supplies finite orbit relations for the established Gaussian Steinberg edge–face presentation, with its corresponding Bianchi modular-symbol relations. When the natural modular-to-Picard coset map is bijective, a side-connected Kulkarni special polygon lifts, using the same modular representatives, to a Picard fundamental polyhedron whose intersection with the invariant real plane is exactly that polygon; the level-$(2+i)$/level-$5$ family is worked out explicitly. Comparison examples show that octahedral boundary minimization, absolute geometric face minimization, and preservation of such a modular section are distinct problems. Canonical reduction and effective Hecke reduction remain open.

We investigate the problem of which multisets of integer entries yield integer eigenvalues in every square matrix arrangement? Combining module reduction with a dyadic criterion for complete splitting of cubic polynomials, we first show that if the multiset has at least one zero entry, the only possibilities have either at most one nonzero entry or exactly two whose product is a perfect square. We then apply block embedding to extend it to higher dimensions, by obtaining a linear threshold on the number of zeros. For any multiset close to a nonzero constant, we use rank reduction to derive an exact criterion for one exceptional entry, thus yielding nonconstant examples in infinitely many dimensions, and to also derive an exact reduction for two distinct exceptional entries in dimension three. The general fully nonzero three-dimensional case remains open.

Let $p$ and $\ell$ be distinct odd primes. For a finite extension $F/\mathbb{Q}_p$, the local Langlands correspondence states that there is a canonical bijection between irreducible, smooth representations of $\text{GL}_n(F)$ and $n$-dimensional, $\Phi$-semisimple Weil–Deligne representations of the Weil group $W_F$. Given two $2$-dimensional, semisimple, continuous representations $\rho_1, \rho_2$ of $W_F$ with images in $\text{GL}_2(\mathcal{O}_K)$, where $K/\mathbb{Q}_\ell$ is a finite extension with maximal ideal $\lambda \subset \mathcal{O}_K$, a natural question to ask is when their mod $\lambda$ reductions $\overline{\rho}_1$ and $\overline{\rho}_2$ are isomorphic. In this paper, we give a complete characterization of when the reductions are isomorphic. For this, we first give a description of when the reductions of these representations are decomposable or irreducible, utilizing the correspondence between continuous $\ell$-adic representations of $W_F$ and $\ell$-adic Weil–Deligne representations. We conclude with some examples which arise in the modular method for solving generalized Fermat equations.

We develop a general method for uniform twisted metric Diophantine approximation with an arbitrary fixed real matrix. Using all successive minima of the associated diagonal lattice trajectory, we estimate lattice-point counts, accounting for clustering in every direction. Ratios of these counts give exact Hausdorff dimensions of sets of twisted $\psi$-Dirichlet vectors for a large class of functions $\psi$, including all except finitely many power functions {$T^{-\tau}$, $\tau>0$}. We obtain Hausdorff dimension formulae for the endpoint sets obtained by taking the intersection and union of the sets of twisted $c\psi$-Dirichlet vectors over $c>0$, and prove that taking the intersections causes no dimension drop. Furthermore, we obtain Hausdorff dimensions of level sets of the uniform Diophantine exponent. In addition, we prove zero-full laws for general Hausdorff measures of $\psi$-Dirichlet sets. Applications recover the one-dimensional Kim-Liao formulae, strengthen Aggarwal's escape-of-mass upper bound, reproduce the metric criteria of Moshchevitin and the second author and give certain extensions of the above statements.

We establish an exact zero-one law for twisted Diophantine approximation with an arbitrary fixed real matrix and a positive non-increasing approximation function satisfying dyadic regularity. The criterion is the convergence or divergence of a dyadic series whose summands are Khintchine-Groshev block volumes divided by homogeneous lattice-point counts. An equivalent formulation is using all successive minima of the associated diagonal lattice trajectory, capturing homogeneous clustering in every direction. We also obtain an almost-sure zero-infinity law for inhomogeneous $\psi$-Lagrange constants, separate Hausdorff-measure criteria, and dimension results. Within this regularity class, our theorem recovers the Kurzweil and Fuchs-Kim criteria. At the critical exponent, this series determines whether the set of badly approximable shifts has zero or full Lebesgue measure.

We prove the strong Artin conjecture for the largest solvable Artin representations in three dimensions, whose projective image is isomorphic to the affine special linear group $C_3^2\rtimes \mathrm{SL}(2, 3)$. It is also the last solvable case in three dimensions. This is achieved with a new case of base change and automorphic induction for non-Galois quartic extensions with no intermediate fields. We additionally deduce the strong Artin conjecture for all 6-dimensional primitive solvable Artin representations.

Pan and Zhang proved that a positive proportion of positive integers can be represented in the form $a^2+b^2+2^{c^2}$, where $(a,b,c)$ are nonnegative integers. In this note, we give a short proof of their result. The proof uses Romanoff's method and an elementary divisor argument to control the relevant second moment.

We estimate the number of integers up to $x$ whose $y$-friable component exceeds $z$. Uniformly for $\max\{2,\log z\}\leqslant y\leqslant z\leqslant x/2$, we determine the logarithm of this counting function up to an error $O(\log z/\log y)$. This sharpens an estimate arising from a problem in Ford's sieve methods lecture notes. The proof combines classical friable-number estimates with an elementary argument for the lower bound.

Call a set $A$ of positive integers admissible if, whenever $a_1\cdots a_r=b_1\cdots b_s$ with $a_1,\dots,a_r$ distinct elements of $A$ and $b_1,\dots,b_s$ distinct elements of $A$, necessarily $r=s$. Erdős asked whether admissible sets can have density $1-\varepsilon$ for every $\varepsilon>0$, and whether $\{1,\dots,N\}$ always contains an admissible subset of size $(1-o(1))N$. For the variant in which repetitions are allowed both questions were answered negatively by Erdős, Ruzsa and Sárközy and by Granville and Soundararajan; for products of distinct elements, the first question was answered only recently (with density bound $7/8$), and the second has remained open. We show that every admissible $A\subseteq\{1,\dots,N\}$ satisfies $\sum_{a\in A}1/a\le\tfrac12\log N+(\log\log N+2)^2$, and that there is an absolute constant $\eta>0$ such that every admissible $A\subseteq\{1,\dots,N\}$ has $|A|<(1-\eta)N$ for all large $N$. Both questions therefore have negative answers. The proofs are elementary; the second rests on a coupling that replaces the largest divisor of an integer composed of small primes, which avoids the divisor-function losses inherent in counting quotients along a multiplication table. Both negative answers are formally verified in Lean 4 against the statements of the Formal Conjectures project.

Salikhov (2008), Zeilberger-Zudilin (2020) and Bai (2026) bound the irrationality measure of pi with the same shape of complex integral. This note gives one closed-form law mu(a0,a1,b) for the whole family, shows that it reproduces all three published records, proves that it is exactly scale-invariant, and gives evidence that Bai's parameter point is the global optimum of the family - a statement Bai explicitly declines to make about her own result. The admissible configurations are forced by a Machin condition: zeros at 0, +-w, +-wbar and poles at +-P give a linear form in 1 and pi if and only if P^2 - N(w) = 2P Im(w). The law depends only on the two ratios a0/a1 and b/a1, so the infinite three-parameter lattice is exactly two-dimensional. The 2-adic saving is exactly 5(2a1-b)/2 per n, in two regimes, and no further saving exists. The decay rate is governed by the min-max value of the steepest-descent method, the lowest level at which the two endpoints of the integral become connected within a sublevel set of the integrand; the value obtained from the least critical level is a lower bound for mu and serves as a screening. Among the sixteen admissible configurations with P < 100, exactly one carries a triple that yields a bound, namely Salikhov's (P = 5, w = 1+2i), and the least mu over all sixteen and over all primitive triples with a0, a1 <= 10 and b <= 22 is 7.103205334137 at (2,2,3). Within that configuration the minimum is 7.101862832356 at (1857,1857,2785), which is Bai's point, the vertex sitting exactly at Q = 2a/(3a-2b) = 3714. Each statement carries an explicit label, PROVED or VERIFIED or CONJECTURE, and the reservations of scope are stated in the note. An ancillary file re-derives every numerical claim from scratch, with no external dependency.

We expand on the concept of admissible pairs from \cite{Levin:GCD} and explore its relation with the main Diophantine arithmetic inequalities, with discriminant term and for points of bounded degree, that have been predicted by Vojta \cite{Vojta:1998}. In this context, among other new results, we prove a harder implication which is in the spirit of Vojta's approach to the abc Conjecture (from \cite{Vojta:1998}). As our main result, and application of our viewpoint here, we deduce for the case of certain general type nonsingular Deligne-Mumford stacks, with projective course moduli space, a form of the Bombieri-Lang Conjecture for $(D_0,S)$-integral points of bounded degree. A key input for this is a slicing theorem, for Deligne-Mumford stacks, that was obtained by Abramovich and Várilly-Alvarado, \cite{Abramovich:VarillyAlvarado:Pera:2017}, and building on earlier work of Kresch and Vistoli \cite{Kresch:Vistoli:2004}. Another important ingredient is an inequality of Silverman, from \cite{Silverman:1984}, which bounds the discriminant of points in projective space in terms of their heights. As an illustration of our results, we discuss them within the context of the interesting work of Abramovich and Harris \cite{Abramovich:Harris:1991} and others.

This short note is a concise yet complete overview, intended for students, researchers, and AIs, bringing together in a unified notation the closed-form and reciprocity identities arising from weighted mean squares of the Riemann zeta function and Dirichlet $L$-functions on the critical line. In addition, we prove a new reciprocity identity: the exponentially weighted counterpart of a formula of Lewis and Zagier for the primitive character of modulus $4$.

In a recent paper arXiv:2503.20727, Ichino and Prasanna proved a supercongruence for binomial coefficients and related it to representations of $\mathrm{GL}_2$ over $p$-adic rings. We prove a $q$-analog of this supercongruence using the $q$-Lucas theorem, from which a strengthening of the main result of loc. cit. can easily be deduced. Our proof also substantially simplifies the original argument of arXiv:2503.20727.

We study Verlinde sums $V_{n}(N,m)$, which for our purposes are finite sums of trigonometric functions $\{r_{x}\}$, associated to $\textrm{ SU}(N)$. In this article we prove the following results: (i) the polynomial $D_N(w):=\prod_x(w-r_x)$ is equal to, up to an explicit and computable factor, the iterated resultant of $Q(u)=u^m-1$ with an explicit elementary polynomial $W_N$ which is independent of $n$ and $m$; (ii) for fixed $N$ and $m$, $V_n(N,m)$ satisfies a linear recursion of order at most $\binom{m-1}{N-1}$, together with arguments showing that one actually has a recursion of length equal to the number of distinct roots of $D_{N}$; (iii) for fixed $N$ and $m$, a Binet-type closed form expressing $V_n(N,m)$ as an explicit finite sum of $n$-th powers of algebraic numbers which are rescaled reciprocals of the distinct roots of $D_N(w)$ is derived. Several explicit examples are provided for $N=2,3,4,5,6$ and $8$.

A positive integer $n$ is said to be strongly pseudoperfect if there is a subset $S$ of the positive divisors of $n$ such that $d \in S$ if and only if $n/d \in S$, and $\sum_{d \in S} d = 2n$. We construct new infinite families of strongly pseudoperfect numbers, and also prove new restrictions on when a positive integer $n$ can be strongly pseudoperfect.

We study a $p$-adic rigidity phenomenon suggested by rank two attractors. Near an ordinary special fiber of a Calabi–Yau family, the excellent Frobenius of Beukers and Vlasenko acts on the parameter disk. Achinger–Zdanowicz construct a canonical $W_2$-lift of the Frobenius twist, while Brantner–Taelman construct a canonical formal lift. We conjecture that the Dwork and Achinger–Zdanowicz first-order obstruction maps agree and that the fixed points of the excellent Frobenius are precisely the parameters of the Brantner–Taelman canonical formal lifts. Our main theorem gives a cohomological criterion in arbitrary dimension. Under its hypotheses, a Frobenius-stable rank two factor of the $p$-adic cohomology containing the holomorphic class forces that class to span a Cartier-stable line and gives supercongruences modulo $p^{2s}$ for every $s\geq1$. Moreover, if the hypotheses of the Beukers–Vlasenko excellent lift theorem hold, then the excellent Frobenius fixes the corresponding parameter. We apply the criterion to the Hulek–Verrill $A_4$ family at $t_*=-1/7$, a parameter predicted to be a rank two attractor. Assuming Dummigan's Conjecture~1.1, we prove that the excellent Frobenius fixes $t_*$, the supercongruences hold, and the fiber with parameter $t_*\bmod p^2$ is the Achinger–Zdanowicz canonical lift at every prime where the required Dwork ordinarity conditions are satisfied. A CM fiber of the mirror quartic K3 pencil gives a second application. We also prove that the $q$-power map on a split formal torus has the unit section as its unique periodic point in the identity residue disk over every finite extension of $W(\mathbb{F}_q)[1/p]$, where $q$ is the cardinality of the residue field.

Let $d \geq 5$, $0 < \gamma < d - 2$, and $\Omega_N$ be the binomial random subset of $Q_N = [-N,N]^d \cap \mathbb Z^d$ with retention probability $p_N = N^{-\gamma}$. We prove that, with failure probability of optimal exponential order, every subset $B \subseteq \Omega_N$ of fixed positive relative density realizes, at each scale $p_N^{-2/(d - 2)} \lesssim \lambda \lesssim N^2$, a squared distance of the form $q^2 \lambda$, where $q$ belongs to a fixed finite set depending only on the dimension and the density. The lower scale $p_N^{-2/(d - 2)}$ is sharp. As a consequence,the squared-distance set $D^2(B)$ of $B$ has maximal order $N^2$ and contains affine copies of every fixed finite subset of $\mathbb Z$. The main new input is a finite multidilate supersaturation theorem for dense subsets of $Q_N$, which, together with boundedness estimates for the associated spherical distance graphs down to the sharp scale, allows us to apply Schacht's transference theorem.

Let $\chi$ be a Dirichlet character mod $D$ with $L(s,\chi)$ its associated $L$-function, and let $\psi(x,q,a)$ be, as usual, Chebyshev's prime-counting function for the primes of the arithmetic progression $a$ (mod $q$) with $(a,q)=1$. Let $\chi$ be a primitive character modulo $D$, and let $\nu>0$ be small. We prove that if $L(s,\chi)$ has a Siegel zero at $s=\beta=1-\frac{1}{\eta\log D}$ with $\eta>\eta_0(\nu)$ for some large $\eta_0(\nu)$, there exists a range of $x$ for which the asymptotic $$\psi(x,q,a)=\frac{\psi(x)}{\phi(q)}\left[1+O(\varepsilon_{\eta_0})-\chi\left(\frac{aD}{(q,D)}\right)\right]$$ holds for $q<x^{\frac{30}{59}-\nu}$. We also show slightly better bounds for $q$ if we take an average over a range of $q$, finding an Elliott-Halberstam-type result for $q\sim Q$ on the range $Q<x^{\frac{16}{31}-\nu}$. This improves on a 2003 result of Friedlander and Iwaniec that requires $q<x^{\frac{233}{462}}$ and builds on recent work of Sachpazis.

We prove that the Zhang-Kawazumi invariant $\varphi(X)$ of a compact and connected Riemann surface $X$ of genus $g\ge 2$ is strictly larger than \[\frac{g(g+2)-(2g+1)H_g}{g-1},\] where $H_g=\sum_{k=1}^g \frac{1}{k}$ denotes the $g$-th harmonic number. If $X$ is hyperelliptic, we give the stronger bound $\varphi(X)>\frac{g}{2}(H_g-1)$. The proof relies on a new expression of $\varphi(X)$ in terms of a quadratic form on the space of smooth Hermitian matrix-valued functions on $X$, evaluated at certain projector matrices. As an arithmetic application, we deduce new lower bounds for the self-intersection number $\omega_a^2$ of the admissible adelic metrized canonical bundle $\omega_a$ of a smooth projective curve of genus $g\ge 2$ over a number field and hence new uniform height bounds in the Bogomolov conjecture.

The study of heavenly abelian varieties is motivated by a question of Ihara. When an elliptic curve $E/K$ is heavenly at $\ell$, the extension $K(E[\ell^\infty])/K(\mu_\ell^\infty)$ is pro-$\ell$ and unramified away from $\ell$. These are the same arithmetic conditions as the fixed field of the kernel attached to pro-$\ell$ étale covers of the projective line over $K$ minus three points. In this paper we study heavenly elliptic curves defined over cubic number fields. Following the work on McLeman and Rasmussen in the quadratic case, we find that there is a wider range of possible behaviors for the trace of a balanced elliptic curve in the cubic case. In this setting, we introduce a further distinction between balanced and totally balanced curves. With this, we show that trace of the representation on the $\ell$-torsion for totally balanced elliptic curves is non-surjective. We also compute a bound on primes $\ell$ after which any heavenly elliptic curve defined over a cubic number field must be balanced. Finally we compare the trace behavior of totally balanced elliptic curves with CM elliptic curves.

The construction of a $p$-adic height pairing for a geometric $p$-adic representation of the absolute Galois group of a number field depends on a global $p$-adic logarithm and on local splittings of the Hodge filtrations at the primes above $p$. We study the dependence on these splittings for suitable two-dimensional symplectic self-dual representations, including self-dual twists of representations attached to even-weight newforms at non-ordinary primes not dividing the level. We express the difference between the height pairings associated with the two splittings determined by Frobenius explicitly in terms of local Bloch–Kato logarithms. As an application over $\mathbb{Q}$, we prove that at least one of the two cyclotomic $p$-adic height pairings is non-trivial under the additional assumptions that the Frobenius eigenvalues at $p$ are distinct and the localization map at $p$ from the Bloch–Kato Selmer group is non-zero.