Papers

We make explicit and formalize a result of the author on squarefree numbers in short intervals, showing that for $0 < \varepsilon\le 1/90935 $, $X\ge \exp(10^{27}/\varepsilon^2)$, $H = X^{1/5 - 2/90935 + \varepsilon}$, we have that \[ \biggl|\sum_{X\le n\le X + H } \mu(n)^2 - \frac{6}{\pi^2}H\biggr| \le \frac{10^{450}}{\varepsilon} H X^{-\varepsilon/10^{25}}. \] This article gives an account of what went into making the exponent explicit. The Github repository linked contains the formalization in Lean 4 as well as an account of what went into the largely automated formalization.

An \emph{$n$-divisor set} is a finite set of positive integers containing a multiple of every integer from $1$ through $n$. Umans and Wang proposed, as the arithmetic-progression version of their Strong $(\alpha,\beta)$-Divisor Conjecture, an $n$-divisor arithmetic progression having at most $n^{2\beta}$ terms, each of magnitude at most $\exp(n^\alpha)$. We prove unconditionally that an $n$-divisor arithmetic progression of height $H$ with $\log H=o(\sqrt n)$ must have length \[ L\ge \left(\sqrt{\frac{8}{27}}-o(1)\right) \frac{n^{3/4}}{\sqrt{\log n}}. \] Consequently, the arithmetic-progression version is false whenever $\alpha<1/2$ and $\beta<3/8$. In particular, it is false at the proposed point $(\alpha,\beta)=(1/3,1/3)$, even if both bounds are relaxed by $n^{o(1)}$ at the exponent level. The proof uses primes in a fixed band below $\sqrt n$ to turn semiprime divisibility into a finite incidence structure. An elementary bounded-degree linear-space estimate then gives the result. This theorem concerns the one-dimensional arithmetic-progression version only; it does not disprove the higher-rank Strong Divisor Conjecture.

We axiomatize the differential-comparison continuation method of [Zf] into a general machine for proving Mahler measure identities at CM points: a modular parametrization with rigorous tail bounds, a single-valued holomorphic Mahler differential on the complement of the critical image, a propagation lemma replacing all monodromy and universal-cover arguments, interval-arithmetic path certification, and an exact three-track CM evaluation. Applied to Samart's family $n_4(s)=4\mathrm m(x^4+y^4+z^4+1+s^{1/4}xyz)$, the machine proves twelve further entries of his 2015 table [Sa15, Table 6], including the first identities in this family proved at non-degenerate CM points of class number two (attached to $\mathbb Q(\sqrt{-6})$ and $\mathbb Q(\sqrt{-15})$) and of class number four (attached to $\mathbb Q(\sqrt{-21})$, with four newforms and the genus group $(\mathbb Z/2)^2$). New ingredients include exact twist levels via grossencharacter conductors (Hecke's theorem), with the interval-locked Fricke ratio as an exact root-number sign lock; a twist-level trap ($g_{16}\otimes\chi_8$ has level $64$, not $32$); a genus-character decomposition of the lattice sums in class number two; and the discovery that Samart's applicability boundary $\{\mathrm{Im}\,\tau=1/\sqrt2\}$ is not the topological boundary of the continuation region. Every identity is certified by an exact algebraic track, rigorous final-identity interval locks (half-widths $4.6\times10^{-53}$ to $5.1\times10^{-52}$), and independent 50–60-digit cross-checks; all scripts are public. The evaluation track also yields six new identities at Heegner points ($D=11,19,27,43,67,163$), stated as conjectures with 60-digit evidence; we close with an umbrella conjecture organizing the proved and conjectured identities, and a precise analysis of the five remaining open entries.

Boyd's 1998 tables of conjectural identities $m(P)=r\,|L'(E,0)|$ between Mahler measures of two-variable polynomials and $L$-values of elliptic curves begin with the smallest possible conductor, $N=11$. The third conductor-$11$ identity, (C3), concerns the polynomial $S_0=y^2+(x^2+1)y+x^3$, which vanishes on the unit torus, and asserts that a signed split integral $I_{\mathrm{split}}$ of $\log|y|$ around the branch cut equals $b_{11}=L'(E_{11},0)$. We prove (C3). The proof identifies the split integral with a regulator integral along Samart's signed open chain $\tilde\gamma$, closed by a small-branch arc $\beta_0$ into a closed anti-invariant cycle $C'$, and proves its homology class is $2\gamma^-$ with $\gamma^-$ generating $H_1(E,\mathbb{Z})^-$: the period ratio is a-priori integral, and ball arithmetic (Arb) pins it to $2$. A direct regulator computation via Brunault's proved Siegel-unit formula—the symbol $\{x,y\}$ being a pair of modular units on $X_1(11)$, so Bloch's diamond theorem is not needed—then yields $I_{\mathrm{split}}=b_{11}$; the sign is certified in interval arithmetic, and the identity agrees with the numerical value to $366$ digits. For the family $S_k=y^2+(x^2+kx+1)y+x^3$ we determine exactly the torus intersections and the modular-unit/tempered cases; further Boyd-type evaluations are recorded as conjectures. At $k=-1$ (conductor $53$) the mechanism provably fails; an appendix treats Samart's conductor-$17$ analogue conditionally. All computations are reproducible from the accompanying code; every certification step is carried out within interval arithmetic.

We prove universality of limiting local eigenvalue statistics for random matrices over $\mathbb{Z}_p$. In previous work of the author and Van Peski (arXiv:2601.06283), the limiting eigenvalue correlation functions of additive Haar random matrices were studied in arbitrary finite extensions of $\mathbb{Q}_p$. The same Haar random matrix model plays a central role in the Ellenberg-Jain-Venkatesh heuristic for zeros of $p$-adic $L$-functions. We show that its limiting local eigenvalue statistics are unchanged for a broad class of random matrices with independent entries satisfying a mild non-concentration condition. Thus the random matrix predictions underlying the Ellenberg-Jain-Venkatesh heuristic are not artifacts of the particular Haar ensemble, but instead reflect universal limiting eigenvalue statistics. In this sense, our results provide additional theoretical support for the robustness of their random matrix heuristic. Our proof is based on a new framework, which we call the resultant distribution method. The method recovers limiting laws and root statistics of $p$-adic polynomials from the distributions of their resultant valuations against fixed test polynomials, together with suitable degree estimates. As a second application, we consider random $p$-adic polynomials with independent coefficients satisfying a mild non-concentration condition. Caruso (arXiv:2110.03942) determined the joint root correlation functions of the Haar coefficient model over finite extensions of $\mathbb{Q}_p$. We prove that, for roots of absolute value one, these limiting correlation functions are universal and persist for a broad class of independent coefficient distributions.

In this article, we study the good reduction of generalized Kummer surfaces, which are K3 surfaces obtained as minimal resolutions of quotients of abelian surfaces by finite groups. In particular, we establish a criterion for good reduction when the abelian surface has non-supersingular reduction and the group is cyclic. This extends the result of Lazda and Skorobogatov on Kummer surfaces.

Let $\alpha_1,\ldots,\alpha_r$ be independent Haar-random fractional Laurent series over $\mathbb{F}_q$, and let $L_r(n)$ be the least coefficient length of a polynomial denominator that simultaneously cancels the first $n$ negative coefficients. We prove that the minimal kernel is a line and that the residual vectors revealed immediately after the stopping times $T_n=n+L_r(n)-1$ are iid uniform on $\mathbb{F}_q^r$. Hence the jump indicators of $L_r(n)$ are iid Bernoulli variables with parameter $1-q^{-r}$; conditionally on a jump, the residual direction is uniform on $\mathbb P^{r-1}(\mathbb{F}_q)$. We also give an exact kernel-growth clock for positive jump sizes and a geometric tail bound uniform in the depth; for two series the jump is decided at the first or second kernel-growth epoch with probabilities $q^{-1}$ and $1-q^{-1}$. The marked renewal law yields exact binomial and fluctuation laws in the depth variable and the density of newly attained minimal denominator lengths $\frac{1-q^{-r}}{r}$ in the coefficient-length variable. For $r=1$ this is the classical iid partial-quotient degree law in the depth coordinate, for which we give an exact dictionary. The new probabilistic content is the simultaneous common-denominator law for $r\ge2$. We also establish exact profile-correspondence and record-duality formulas with joint linear complexity.

We study the average analytic rank in the family of $L$-functions $L(s, E_d)$ associated with the elliptic curves $E_d : y^2=x^3-dx$, as $d$ varies over fourth-power-free odd integers. Since this is a family of curves with complex multiplication, we have $L(s, E_d)=L(s - \frac12, \xi_d)$, where $\xi_d$ is a Hecke character over $\mathbb{Z}[i]$. Assuming the Generalized Riemann Hypothesis, we compute the one-level density of the low-lying zeros of this family for test functions whose Fourier transform is supported in $(-\frac35, \frac35)$. As a consequence, we obtain the upper bound $\frac{13}{6}$ for the average analytic rank $r(E_d)$ over the family. Under the additional assumption of a conjecture on the distribution of quartic Gauss sums at prime elements (a quartic analogue of Patterson's conjecture for cubic Gauss sums), we extend the admissible support to $(-1, 1)$ and improve the upper bound for the average analytic rank to $\frac32$. Both results imply that a positive proportion of twists satisfy $r(E_d) =1$, while the second also yields a positive proportion of twists with $r(E_d)=0$.

Let \(\pi_E:X_0(M)\to E\) be the strong Weil parametrization with Manin constant \(c_E\). We prove \(\deg \pi_E\mid c_E^{\Omega(N/M)}\deg g\) for every multiple \(N\) of \(M\) and every nonconstant morphism \(g:X_0(N)\to E'\) over \(\mathbb{Q}\), where \(E'\) is \(\mathbb{Q}\)-isogenous to \(E\) and \(\Omega\) counts prime factors with multiplicity. When \(c_E=1\), as is known for squarefree \(M\), the modular degree at level \(M\) therefore divides every such degree at every higher level. As an application of the divisibility theorem, we prove that no \(X_0(N)/\mathbb{Q}\) admits a morphism over \(\mathbb{Q}\) of positive odd degree at most \(1645\) to an elliptic curve of positive \(\mathbb{Q}\)-rank. For a fixed target \(E'\) and a generator \(u:E\to E'\), we also prove that if the Manin constant \(c_{u\circ\pi_E}=1\), the old homomorphisms induced by degeneracy maps form an integral basis of \(\operatorname{Hom}_{\mathbb{Q}}(J_0(N),E')\), and the old degree matrix determines the exact morphism degrees. The proofs bound denominators in the rational old basis. The divisibility and lattice results extend to compatible towers of intermediate modular curves, including the \(X_1\)-tower.

We define an enlargement of the class of exponential periods by applying the rapid decay cohomology theory of Hien to a larger class of integrable connections on varieties than those twisted by a regular function. The main and motivating examples are given by those associated to $E$-operators. This delivers a method to realise a class of absolutely convergent integrals involving $E$-functions as matrix elements of a period pairing, which we call $E$-periods. Furthermore, we generalise rapid decay cohomology to singular varieties and prove a version of Nori's basic lemma in this setting.

We study, under the Generalized Riemann Hypothesis (GRH), the moments of sums of Fourier coefficients of a fixed holomorphic Hecke eigenform twisted by the quadratic character $\chi_{8p}$, where $p$ ranges over odd primes. We establish the correct order of magnitude for the unsmoothed $m$-th moment for all real $m\geq 4$, and a sharp upper bound of order $XY^{m/2}\,\, (\log X)^{m(m-3)/2}\,\,$ for the smoothed $m$-th moment for all integers $m\geq 4$. A matching lower bound for all even integers $m\geq 4$ shows that this bound is optimal.

On imaginary quadratic fields, we establish the $\mathrm{GL}_2$ Bessel $\delta$-method and prove the hybrid Weyl-type subconvexity bound for $\mathrm{GL}_2 \times \mathrm{GL}_1$ twisted $L$-functions in the Archimedean aspect of the Hecke character on $\mathrm{GL}_1$.

The Tamagawa factor w.r.t.\ the prime $p$ of an abelian variety $A$ over a non- archimedean local field $K$ is found to be congruent modulo $q - 1$ to the wavelet eigenvalues of a $p$-adic Laplacian integral operator on the $O_K$-rational points of its Néron model, where $q = p^f$ is the cardinality of the residue field of $K$. The method is to express the volume of the $K$-rational points of $A$ w.r.t.\ the canonical measure in terms of the local $L$-factor given by the Frobenius action on $\ell$-adic cohomolgy, and the Tamagawa factor; and then observe that this coincides with the Serre invariant of that compact $p$-adic analytic manifold modulo $q - 1$. A previous result by Á.M.\ Ledezma and the author on hearing Serre invariants then yields the asserted congruence.

The classical Gauss–Kuzmin distribution describes the asymptotic distribution of continued fraction digits. Geometrically, this may be interpreted as the equidistribution of faces of Klein sails in $\mathbb R^2$. In this paper, we establish a three-dimensional analogue of this phenomenon. We prove the equidistribution of local face structures in generic Klein sails in $\mathbb R^3$, thereby obtaining a higher-dimensional generalization of the Gauss–Kuzmin distribution. In addition, we resolve several open questions posed by Karpenkov~\cite{Ka17}. Our approach is based on homogeneous dynamics and is motivated from the work of Kontsevich and Suhov~\cite{KS99}. More precisely, we construct a cross-section for the diagonal flow on $\operatorname{SL}_3(\mathbb R)/\operatorname{SL}_3(\mathbb Z)$, such that visits to the cross-section encode the geometry of three-dimensional sails. The principal technical contribution is the proof that the associated cross-sectional measure is finite, which enables us to derive the limiting face statistics.

Let $\A(2n)$ denote the set of up–down alternating permutations of $\{1,2,\ldots,2n\}$, and let \[ E_{2n}(q)=\sum_{\sigma\in\A(2n)}q^{\operatorname{inv}(\sigma)}. \] Andrews and Foata proved that $E_{2n}(q)\equiv q^{2n(n-1)}\pmod{(1+q)^2}$, and Liu recently obtained the cubic refinement \[ E_{2n}(q)\equiv q^{2n(n-1)}-\binom n2(1+q)^2 \pmod{(1+q)^3}. \] Using the reciprocal generating function for the $q$-secant numbers, a third-order expansion of Gaussian coefficients at $q=-1$, finite differences, and Newton interpolation, we prove the fourth-order refinement \[ E_{2n}(q)\equiv q^{2n(n-1)}-\binom n2(1+q)^2 +\binom n2(2n^2-2n-3)(1+q)^3 \pmod{(1+q)^4}. \] More generally, the recurrence yields an effective procedure for computing the expansion modulo $(1+q)^K$ for any prescribed $K$. We then apply the same local-expansion strategy to the generalized $q$-Euler numbers $E_{pn\mid p}(q)$ of Sagan and Zhang. For every prime $p$, we prove uniform congruences modulo $[p]_q^3$ and $[p]_q^4$; the fourth-order term is governed by a central $q$-Wolstenholme-type quotient associated with ${2p\brack p}_q$. Thus the fourth-order secant congruence is the first case of a general higher-cyclotomic method.

Let $p$ be an odd prime number and $\zeta_p$ a primitive $p$-th root of unity. By computing special arithmetic elements modulo prime ideals, we have investigated the Iwasawa invariants and exceptional pairs $(p,\chi \omega_p^k)$ of $\mathbf{Q}(\sqrt{d},\zeta_p)$. Here $\chi$ denotes the Dirichlet character associated to $\mathbf{Q}(\sqrt{d})$, and $\omega_p$ the Teichmüller character at $p$. First, we determine $\varLambda$-isomorphism classes of Iwasawa modules, and describe structural differences between exceptional and non-exceptional pairs in terms of unramified extensions outside $p$. Next, we report 15 new exceptional pairs in the range $|d|<200$ (resp.~$|d|<10$) and $p <2,\!000,\!000$ (resp.~$p <30,\!000,\!000$), including $(p,k,d)=(28,\!679,\!999,\;2,\!284,\!521,-8)$ for which the $\chi \omega_p^{1-k}$-part of the Iwasawa $\lambda$-invariant is equal to two.

We establish several new series evaluations involving the central $q$-binomial coefficients, with the inspiration coming from earlier work by Vignat and one of the authors on the limiting case at $q\to 1$.

For coprime $1<a<b$, let $M_n^{a,b}(\mathbb{F}_q)$ be the set of commuting pairs of nilpotent $n\times n$ matrices over $\mathbb{F}_q$ with $X^a=Y^b$. Huang, Jiang, and Oblomkov assembled their orders as an Eulerian $q$-series $Z_{a,b}(q)$. They conjectured that it is an explicit product $P_{a,b}(q)$ involving Jacobi's theta function and Dedekind's eta-function, implying the threefold equality $$\underbrace{\prod_{n\ge1}(1-q^n)\cdot\Biggl(\sum_{n=0}^{\infty}\frac{|M_n^{a,b}(\mathbb{F}_q)|}{|\mathrm{GL}_n(\mathbb{F}_q)|}\Biggr)\Biggr|_{q\mapsto q^{-1}}}_{\text{point count}}\;=\;\underbrace{Z_{a,b}(q)}_{q\text{-series}}\;=\;\underbrace{P_{a,b}(q)}_{\text{theta quotient}}$$ If true, the point count on $X^a=Y^b$ is essentially a modular function on $\Gamma(a+b)$. The conjecture is layered in $a$, with an identity for each $b$. The $a=2$ layer is classical, including identities of Rogers–Ramanujan and Andrews–Gordon. For $a\geq3,$ nothing was known. We prove the $a=3$ layer in full: a new infinite family of Rogers–Ramanujan identities, and a geometric origin for Warnaar's products. AxiomProver verified these new identities in Lean assuming existing literature.

In 1986, Ivić and Tenenbaum introduced arithmetic functions with squarefull kernel, which are also called $s$-functions. Later, Erdős and Ivić gave an asymptotic estimate on the shifted convolution sums of $s$-functions. Recently, Bergelson and Richter studied the orbits along the prime Omega function in a uniquely ergodic topological dynamical system and established a new dynamical generalization of the prime number theorem (PNT). These orbits can be viewed as functions of invariant average under multiplications. In this paper, we show that both $s$-functions and their shifted convolutions are asymptotically uncorrelated to the orbits along the prime Omega function in a uniquely ergodic system. As a consequence, we obtain a refinement of the PNT via the local distribution of $s$-functions. Furthermore, several variants of these results are established as well.

Let $\rho$ be an absolutely irreducible 2-dimensional $p$-adic representation of the absolute Galois group of ${\mathbb Q}_p$, and let $\Pi(\rho)$ be the unitary Banach space representation of $G = {\rm GL}_2({\mathbb Q}_p)$ associated to $\rho$ by the $p$-adic Langlands correspondence. We deduce from results due to Colmez, Dospinescu, Paškūnas, Emerton, and others that if the locally analytic representation $\Pi(\rho)^\rm{la}$ has a subquotient isomorphic to a non-zero subquotient of a locally analytic principal series representation of $G$, then $\rho$ is trianguline.

The year 2026 marks the 150th anniversary of a remarkable achievement in mathematics: in 1876, the French mathematician Édouard Lucas showed that the 39-digit number $M_{127} := 2^{127} -1$ is prime. This stood for 75 years as the largest known prime, and it remains the largest prime number discovered and certified without the aid of a mechanical device. We review the history behind this discovery and give a modern proof of the closely related Lucas-Lehmer test, which is still the main engine behind the certification of large prime numbers.

We present several explicit classes of algebraic integers $\alpha>1$ such that $\lfloor \alpha^n\rfloor$ is composite for all but finitely many $n$.

Using non-$\mathbb{A}^1$-invariant motivic cohomology, we prove motivic refinements of certain known computations of the algebraic $K$-theory of singular rings, such as rings of the form $\mathbb{Z}/p^n$, $\mathbb{Z}[x]/(x^e)$, and $\mathscr{C}(X;\mathbb{C})$ for $X$ a compact Hausdorff space. These refinements are made possible by the use of integral $p$-adic Hodge theory, as a replacement for the standard use of trace methods in $K$-theory.

We construct random open covers of higher-rank locally symmetric spaces using a construction we call scaffolded Poisson processes. Let $X=G/K$ be a symmetric space of noncompact type and real rank at least $2$. We prove a general vanishing theorem for the normalized torsion in first homology along sequences of torsion-free lattices in $G$. In particular, if $G$ is simple, we get \[ \dfrac{\log |H_1(M_n;\mathbb{Z})_{\operatorname{tors}}|}{\mathrm{vol}(M_n)} \longrightarrow 0 \] for any sequence of distinct manifolds $M_n = \Gamma_n \backslash X$. This answers a question of Abért, Gelander, and Nikolov, and confirms the degree-one vanishing with trivial integral coefficients predicted by a conjecture of Bergeron and Venkatesh in the higher-rank setting. In addition, we get quantitative bounds with respect to the minimal injectivity radius for both the torsion in first homology and the minimal number of generators of $\Gamma$. Finally, we prove the analogous statements for affine buildings.

We establish Weyl-type subconvexity bounds in the level aspect for cuspidal automorphic representations of $\mathrm{GL}_2/F$ with simple supercuspidal ramification at a prime ideal $\mathfrak{q}$. More precisely, for the family $\mathcal{F}_t^\zeta(\mathfrak{q}^3;\omega)$ consisting of representations of conductor $\mathfrak{q}^3$, central character $\omega$, and prescribed simple supercuspidal local component, we prove the fourth moment estimate \begin{align*} \sum_{\substack{\pi \in \mathcal{F}_{t}^{\zeta}(\mathfrak{q}^3;\omega) \\ C_v(\pi) \leq \mathbf{C}_v,\ v \mid \infty}} |L(1/2,\pi)|^4 \ll_{F,\varepsilon} \mathbf C_\infty^{1+\varepsilon} N_F(\mathfrak{q})^{2+\varepsilon}. \end{align*} As a consequence, we deduce the Weyl-type bound \begin{align*} L(1/2,\pi) \ll_{F,\varepsilon} C_{\infty}(\pi)^{1/4+\varepsilon} C_{\mathrm{fin}}(\pi)^{1/6+\varepsilon}. \end{align*} In particular, this bound applies to a genuinely non-self-dual family of odd conductor exponent, beyond the reach of the cubic moment method.

In this article, we research the mean value of integral of exponential sum $S(\alpha)=\sum_{u\in I}e(\alpha u^k)$, where $I$ is a short interval whose length is $2N^{\theta},\theta<1$, and the broken arc $\mathfrak{m^*}$ is a subset of a following minor arcs \[ \mathfrak{m}=\bigcap_{j\leq k-1}\left\{\alpha:\forall q<(\log N)^A,h<q,(h,q)=1,\Big|\alpha-\frac{h}{q}\Big|>\frac{1}{qN^{(k-j-1/2)\theta}}\right\} \] which has measure at least $c>0$. By setting $m$ is a sufficiently large number, $N$ is be sufficiently large in terms of $m$. When $k>\log m$ and $\frac{\log k}{\log m}<1/2$ we set the following estimate: \[ \int_{\mathfrak{m^*}}\bigg|\sum\limits_{{N_1}<u<{N_2}}{e(zu^k)}\bigg|^{m}\mathrm{d}z\ll_cN^{\theta m(\frac{2k+1}{2k+2}+o(1))} \] We can find this estimate moving beyond the even-odd restriction of powers. To get this bound, we first set a strong estimate for almost $\alpha$ by Diophantine approximation and Vinogradov's main value theorem. Then, combining this result, we construct a refined Weyl differencing argument by partitioning the differences step into large and small range, which significantly outperforms the classical one. By the version of probability, we can calculate the multiplicity of each sum. Put them together and we can complete the proof.

We study the prime geodesic theorem for congruence subgroups of indefinite quaternion orders. We prove that the geodesic analogue of the Chebotarev density theorem for these groups holds with exponent $25/36 + \varepsilon$. In particular, we deduce that the prime geodesic theorem holds with exponent $25/36 + \varepsilon$ for any congruence subgroup of any indefinite quaternion order over $\mathbb{Q}$. The idea of the proof is to reduce the problem to the split case, which has been handled previously by the author.

We develop a comprehensive theory of dualizable additive categories. We provide several equivalent characterizations, notably identifying them as separated Grothendieck prestable categories satisfying the $\mathrm{AB4}^*$ and $\mathrm{AB6}$ axioms. We establish a connection to almost mathematics by demonstrating that they arise precisely as the categories of connective almost modules over connective $\mathbb{E}_1$-rings. Furthermore, we prove that dualizable additive categories are generated by flat objects, and that the passage to flat objects yields an equivalence between dualizable additive categories and compactly assembled additive categories. As a primary application within analytic geometry, we characterize the category $\mathrm{Nuc}(R)_{\geq 0}$ of connective nuclear modules (in the sense of Clausen–Scholze) over an adic $\mathbb{E}_\infty$-ring $R$ via a universal property, identifying it as the additive rigidification of the category of connective complete $R$-modules. Finally, we construct the universal finitary stable localizing invariant for dualizable additive categories, the presentable stable category $\mathcal{M}\mathrm{ot}_{\mathrm{pst}}$ of prestable motives, and demonstrate that its unit corepresents nonconnective algebraic $K$-theory. We prove that the motives of small additive categories and those of dualizable additive categories generate the same presentable stable subcategory.

We prove a quantitative version of the convergence case of Khintchine's celebrated theorem in metric Diophantine approximation, but where the approximated points are restricted to lying on the parabola. A novel feature of our result is that unlike other results in literature, the approximating function is no longer required to be monotonic. This requires us to obtain explicit constants in classical number theoretic results, most notably in Burgess' bound for character sums in short intervals.

Let $D$ be a negative discriminant, and let $K=\mathbb{Q}(\sqrt{D})$. Let $\mathcal{Q}(D)$ denote the set of primitive positive definite binary quadratic forms over $\mathbb{Z}$ of discriminant $D$. We introduce the set of adelic framed forms \begin{equation*} \widehat{\mathcal{Q}}(D)= \left\{(Q,\,\gamma)\in \mathcal{Q}(D)\times\mathrm{SL}_2(\widehat{\mathbb{Z}})~|~ Q\left(\gamma\begin{bmatrix}1\\0\end{bmatrix}\right)\in \widehat{\mathbb{Z}}^\times\right\} \end{equation*} and its orbit space $\widehat{C}(D)$ under the natural action of $\mathrm{SL}_2(\mathbb{Z})$. We define an explicit adelic analogue of the Gauss-Dirichlet composition law on $\widehat{C}(D)$ and endow $\widehat{C}(D)$ with the quotient topology induced by the subspace topology on $\widehat{\mathcal{Q}}(D)$ inherited from the product topology on $\mathcal{Q}(D)\times\mathrm{SL}_2(\widehat{\mathbb{Z}})$, where $\mathcal{Q}(D)$ is discrete and $\mathrm{SL}_2(\widehat{\mathbb{Z}})$ has its profinite topology. We then prove that there is an isomorphism of topological groups \begin{equation*} \widehat{C}(D)\simeq\mathrm{Gal}\left(K^\mathrm{ab}(\mathfrak{t}^{1/\infty})/K(\mathfrak{t})\right), \end{equation*} where the Galois group is endowed with the Krull topology, $\mathfrak{t}$ is a positive transcendental real number, and $\mathfrak{t}^{1/\infty}=\{\sqrt[N]{\mathfrak{t}}~|~N\geq1\}$. Moreover, we identify an explicitly defined subgroup of $\widehat{C}(D)$ with $\mathrm{Gal}(K^\mathrm{ab}/K)$ and describe the corresponding Galois action on special values of modular functions. In this way, classical Gauss composition, finite-level form class groups, and Shimura reciprocity are brought together within a single adelic framework. Finally, we show that the abstract group structure of $\widehat{C}(D)$ uniquely determines the imaginary quadratic field $K$.