Papers

In this second chapter of the $\textit{Arithmetic of critical values}$ series (ACV), we study certain double covers $E_f\to\mathbb{P}^1$ whose branch locus coincides with that of a quartic polynomial $f$. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves $E_f$ admit a $3$-isogeny. Our methods are Galois-theoretic, and lead us to a thorough analysis of the Galois closure of $f:\mathbb{P}^1\to\mathbb{P}^1$. We exploit its rich geometry to prove a Selmer companionship theorem for the family $E_f$, allowing us to exhibit elements in certain Tate-Shafarevich groups which are visible in an abelian surface. We also give some dynamical and Diophantine applications of our constructions, as well as new examples of Jacobians isogenous to a power of an elliptic curve.

We study supersingular isogeny graphs of rank-two Drinfeld modules over $A=\mathbb{F}_q[T]$. For distinct finite primes $\mathfrak p$ and $\mathfrak q$ of $A$, we prove that the graph in characteristic $\mathfrak p$, with edges given by cyclic $\mathfrak q$-isogenies, is connected. The proof combines Gekeler's ideal-class correspondence with strong approximation to realize the graph as a quotient of the Bruhat–Tits tree of the homothety classes of $A_\mathfrak q$-lattices in $(\mathbb{F}_q(T))_\mathfrak q^2$. We also introduce the completeness number $E(\mathfrak p)$, the least integer such that the graph is complete for every prime $\mathfrak q\neq\mathfrak p$ with $\operatorname{deg}\mathfrak q\geq E(\mathfrak p)$, and derive explicit upper bounds using the Ramanujan–Petersson bound for the eigenvalues of operators associated to Brandt matrices over function fields. In particular, if $d=\operatorname{deg}\mathfrak p$, then $E(\mathfrak p)\leq 2d+2$, with sharper bounds depending on $q$ and the parity of $d$. This confirms a conjecture of Micheli and Papikian that the graph becomes complete once $\operatorname{deg}\mathfrak q$ is sufficiently large relative to $\operatorname{deg}\mathfrak p$. We further obtain parity-dependent lower bounds and prove that $E(\mathfrak p)\geq d+\log_q d-K$ for an absolute constant $K>0$ and sufficiently large odd $d$, thereby refuting the previously suggested bound $E(\mathfrak p)\leq d+1$. Finally, we prove a criterion yielding an algorithm to compute $E(\mathfrak p)$.

We compute the homological Euler characteristics of $\mathrm{SL}_4(\mathbb{Z})$ and $\mathrm{GL}_4(\mathbb{Z})$ with coefficients in arbitrary irreducible rational highest-weight representations. Applying Wall's formula, we combine the orbifold Euler characteristics of centralizers of torsion elements with traces computed using the Jacobi-Trudi identity to derive explicit formulas and rational generating functions. Consequently, these Euler characteristics are quasi-polynomial functions of the highest-weight parameters, of total degree at most two. We further derive degreewise vanishing results and parity-sensitive lower bounds for the dimensions of cohomology groups. The two extensions of an $\mathrm{SL}_4(\mathbb{Z})$-coefficient system to $\mathrm{GL}_4(\mathbb{Z})$ yield sharper bounds, which grow linearly or quadratically in explicit infinite families. For symmetric powers, combining our formulas with Horozov's calculation of the determinant-twisted summand yields exact identities and lower bounds for the untwisted summand, together with a conjectural degreewise description of its cohomology.

Adapting the analytic method of Gross and Zagier, Roskam proved a prime-factorization formula for the norm of the difference of two level-two Weber singular moduli. Independently, Yang and Yin obtained an equivalent formula using Borcherds lifts. More precisely, the formula concerns the norm of \[ \omega_{2}\left(\frac{-1+\sqrt{d_{1}}}{2}\right) - \omega_{2}\left(\frac{-1+\sqrt{d_{2}}}{2}\right) \] for coprime negative fundamental quadratic discriminants $d_{1},d_{2}\equiv1\pmod 8$, where \[ \omega_{2}(\tau) = 2^{12}\frac{\eta(2\tau)^{24}}{\eta(\tau)^{24}}, \] and $\eta(\tau)$ denotes the Dedekind eta function. In this work, we revisit this formula from the perspective of arithmetic intersection theory and give a new proof.

Let $\mathcal{N}$ be the set of non-Brjuno real numbers. We determine the exact critical exponent of $\mathcal{N}$ for both logarithmic Hausdorff measure and logarithmic capacity. For the gauges $h_\delta(r)=(\log(1/r))^{-\delta}$, the critical exponent is $2$: the $h_\delta$-Hausdorff measure is infinite locally for $0<\delta\leq2$ and vanishes globally for $\delta>2$, while the logarithmic capacity is positive exactly for orders $0<s\leq2$. In particular, $\dim_{\mathrm{cap}}\mathcal{N}=2$.

We develop a methodology for designing infinite products of rational blocks whose exponents are polynomials in the index $k$. Matching power sums of the slot constants through order $n$ forces the Type-$N$ product with binomial exponent $\binom{n+k-2}{n-1}$ to converge to a ratio of Vignéras multiple gamma values $\Gamma_n$; an analogue finder lifts any Type-1 evaluation to every higher type, and integer combinations of binomial exponents then realise arbitrary integer-valued polynomial exponents, yielding explicit products with exponents $k$, $k^2$, $k^3$, ... for constants such as $\pi/2$, $\sqrt{2}$, $e^{2K/\pi}$, and rational multiples of $\pi^{M!}$ (Part I). As the main application (Part II) we prove that for every positive integer $n$, a finite multiple-gamma template $\mathcal{S}_n$ with generalised Eulerian weights $T(n,k)$ (OEIS A225118) evaluates the Duke-Imamoğlu expression $\mathcal{D}_n = \beta'(-n) + (\log 4)\,\beta(-n)$. For odd $n$ this yields a convergent Wallis-Eulerian product for $e^{\beta'(-n)}$; for even $n$ the raw product diverges. The proof expands the template through the multiple-gamma functional equation, evaluates the quarter-integer coefficients in closed form, and identifies the resulting Eulerian-binomial sums with Duke's polynomials $P_{n+1,\ell}$.

We discuss the Babylonian division of trapezoids and the construction of such trapezoids with rational sides and transversals from Pythagorean triples.

We give a detailed account of the recent work of Hickman and Srivastava on counting integral points in a $ \delta $-neighbourhood of a dilate $ q \mathcal{M} $ of a non-degenerate curve $ \mathcal{M} \subset \mathbb{R}^{n} $, $ n \ge 3 $. Our aim is expository: we simplify the proof, provide additional details, refine the statement of the theorem and provide calculations from a more fundamental perspective. We also formulate a new conjecture based on recent work of Chen, Seeger, Srivastava and Technau.

In this paper, we study the generalized $L$-function considered by Berndt (1975). We introduce the generalized Apostol-Bernoulli polynomials and the generalized Apostol-Bernoulli functions, and establish some properties for them, including the Fourier series for these functions. We show that the values of Berndt's generalized $L$-function at integers are explicitly evaluated in terms of the generalized Apostol-Bernoulli functions.

Let $p$ be an odd prime and let $K/\mathbb{Q}_p$ be a finite unramified extension. From a finite presentation by roots of unity of order prime to $p$, we construct a localized Iwasawa pseudomeasure on $\mathbb{Z}_p^\times$. Although the pseudomeasure depends on the chosen presentation, its image modulo bounded measures depends only on the associated Bloch class: it is the Frobenius-depleted Coleman regulator of that class multiplied by a universal half-shifted zeta principal part. Consequently, every nonexceptional weight component is bounded, while the exceptional component has at most a simple pole with explicitly determined residue. For $p>3$ and $\mathbb{Z}_p$-valued coefficients, vanishing of the principal part is equivalent to vanishing of the corresponding class in $K_3(K;\mathbb{Z}_p)$. Cyclotomic refinements preserve the Bloch class and act on the associated pseudomeasures by explicit Iwasawa multipliers. Normalized finite linear combinations of refinements interpolate arbitrary finite jets of the bounded weight-space data, subject only to the normalization at the exceptional point. We also establish a half-shifted complex Mellin factorization, compare the construction with the GSWZ germ family, and derive, at simple degree-one places above primes $p>3$, a finite-polylogarithm criterion for the local $K_3$-class of the knot $5_2$.

Let $u_j $ traverse an orthonormal basis of Hecke–Maass forms for $\mathrm{SL}_2 (\mathbb {Z}) $ with Hecke eigenvalues $\lambda_j (n)$ and Laplace eigenvalue $1/4+t_j^2$. In this paper, we consider the short-interval variant of the twisted spectral large sieve inequality of Luo for $ \lambda_j (n) n^{it_j} $ on the range $t_j \leqslant T$ and prove that the `Eisenstein–Kloosterman' cancellation discovered by Luo is effective on the interval $ T < t_j \leqslant T + M $ as long as $\sqrt{T} < M \leqslant T$. Moreover, our approach yields an improvement of the large sieve inequality of Luo.

We introduce a notion of pseudo-spherical representations for any Brylinski-Deligne (BD) cover of a split torus over a $p$-adic field. In particular, we do not assume that the cover is tame. We enumerate the pseudo-spherical representations and compute their dimension. When the torus is the maximal split torus of a connected split reductive group $G$, we enumerate a subset of distinguished pseudo-spherical representations, under mild assumptions on the BD datum. We expect these distinguished representations to provide the torus input for a Borel-Casselman Theorem for BD covers of $G$. We have verified this expectation for certain non-tame double covers in a separate work. While investigating the pseudo-spherical representations, we also prove that the restriction of irreducible genuine representations of covering tori to their maximal compact subgroup is multiplicity-free. From this we see that the irreducible genuine representations of this maximal compact subgroup are types in the sense of Bushnell-Kutzko. We conclude by describing the Hecke algebras attached to these types.

Motivated by a theorem of A. Schmidt on the part of the complex Lagrange spectrum below $2$, we study the restricted Lagrange spectrum $L_{\frac{1}{2}+i\mathbb{R}}$ arising from the approximation of complex numbers of the form $\frac{1}{2}+i\alpha$, $\alpha\in\mathbb{R}\setminus\mathbb{Q}$ by Gaussian rationals $p/q$ with $p,q\in\mathbb{Z}[i]$, $q\neq 0$. We show that this spectrum admits a description in terms of a dynamical spectrum associated with a real horseshoe. As a consequence, we obtain several fractal properties of $L_{\frac{1}{2}+i\mathbb{R}}$, such as continuity of the dimension function $t\mapsto\dim_H(L_{\frac{1}{2}+i\mathbb{R}}\cap(-\infty,t))$. We also prove that the set of complex numbers $z$ satisfying \begin{equation*} \left\lvert z-\frac{p}{q}\right\rvert\geq\frac{1}{2|q|^2}, \quad\text{for all } p,q\in\mathbb{Z}[i], q\neq 0, \end{equation*} is uncountable. In fact, we show that this inequality holds for every complex number of the form $z=\frac{1}{2}(1+i\theta)$ where $\theta\in\mathbb{R}\setminus\mathbb{Q}$ is a root of one of Schmidt's $C$-minimal forms.

We study the self-intersections of billiard trajectories in a square with small pockets of size $\varepsilon$ removed from its four corners. In particular, we establish an asymptotic formula for the $r$-th moment of the number of self-intersection points for each positive integer $r$, as well as the distribution of such points.

We prove Kato's main conjecture for modular forms at nonordinary primes for weights in the Fontaine-Laffaille range. The key ingredients in the proof are a reformulation of the conjecture in terms of signed Selmer groups due to Lei-Loeffler-Zerbes, certain $p$-adic families of Rankin-Eisenstein classes arising from the work of Lei-Loeffler-Zerbes and Kings-Loeffler-Zerbes, and the lower bound divisibility in an Iwasawa-Greenberg main conjecture for Rankin-Selberg $p$-adic $L$-functions obtained in our earlier work \cite{CLW}.

After recalling basic formulas concerning modular forms of half-integral weight, Gegenbauer polynomials, and Rankin–Cohen brackets, we give four versions of the Shimura lift and several applications to Hilbert modular forms associated to real quadratic fields. We claim no originality, but wanted to collect all the formulas in a single paper.

The Cardy-Smirnov function of a four-marked planar domain encodes the conjectural scaling limit of the crossing probability for critical percolation. Cardy predicted a closed formula for this limit on the rectangle in the cases of both site and bond percolation, and Smirnov proved the formula together with conformal invariance for critical site percolation on the triangular lattice. Kleban and Zagier later showed that on the rectangle the function admits a modular interpretation: it is determined by a modular functional equation together with a mild analytic ansatz. We carry out the analogue on the $\pi/3$ parallelogram. We derive a closed conformal-map formula for the Cardy-Smirnov function as an incomplete beta integral, prove a closed modular formula realizing it as an integral of the modular form $\eta(\tau)^2\eta(3\tau)^2$, and establish a uniqueness theorem showing that a single functional equation, together with a $q$-expansion ansatz and a nondegeneracy condition, determines the function uniquely.

Let $M_d(X)$ denote the average number of rational preperiodic points among degree-$d$ polynomials over $\mathbb{Q}$ with vanishing $z^{d-1}$ coefficient, constant term $1$, and height at most $X$. We prove that, for every integer $d\ge2$, $M_d(X) \sim \gamma_d X^{-1}$ for an explicit constant $\gamma_d>0$. For $d\ge4$, the error term is $O_{d,\varepsilon}(X^{-2+\varepsilon})$ for every $\varepsilon>0$; for $d=3$ the error term is $O(X^{-3/2})$; and for $d=2$ the error term is $O(X^{-3/2}\log X)$.

In this article we generalise a classical Kolyvagin's result on the vanishing of the $p$-part of the Shafarevich–Tate group of an elliptic curve (defined over $\mathbb{Q}$) over an imaginary quadratic field $K$ to modular abelian varieties of $\mathrm{GL}_2$-type (defined over a totally real number field $F$) and a CM field $K/F$. Combining this result with the abstract Iwasawa-theoretical argument of [MN19], we show that a similar vanishing holds over the layers of a suitably defined anticyclotomic multi-$\mathbb{Z}_p$-extension of $K$.

In this paper we give a geometric description of the behavior of analytic intertwining operators between parabolically induced representations of $\mathrm{GL}_n(\mathrm{F})$, where $\mathrm{F}$ is a local non-archimedean field, in terms of hyperbolic localization functors of Braden. As a consequence, we can show that the image of an intertwining operator is always semi-simple and give a lower bound on the order of its pole, which is conjectured to be an equality as well as an upper bound in terms of the singular support of certain perverse sheaves. Moreover, we are able to reduce the conjecture of Lapid and Mínguez on the shape of irreducible subrepresentations of induced representations to a computation of characteristic cycles. The theory of mixed Hodge modules plays a crucial role in the proofs.

Let $E/K$ be an elliptic curve over a number field $K$, and let $[C]\in H^1(K,E(\overline{K}))$ be a homogeneous space under $E$. Suppose that $C$ has period $n$ and index $d$. O'Neil asked whether one can always choose a lift of $[C]$ to $H^1(K,E[n])$ whose period-index obstruction has order exactly $d/n$. We give a negative answer to this question by constructing an explicit family of examples.Taking $K=\mathbb{Q}(\zeta_8)$, we prove that there exist infinitely many pairwise non-isomorphic elliptic curves $E/K$ such that each $E$ admits infinitely many homogeneous spaces $[C]\in H^1(K,E(\overline{K}))$ with period $8$ and index $16$, but the period-index obstruction of every lift of $[C]$ to $H^1(K,E[8])$ has order exactly $8$.

The flat DHS connection $\mathcal J_{\mathrm{DHS}}$ constructed in arXiv:2602.01461 is smooth on the configuration space of $n$ points on a fixed compact Riemann surface $\Sigma$ of arbitrary genus $h$, takes values in an infinite-dimensional Lie algebra $\hat{\mathfrak t}_{h,n}$ and is invariant under the modular group $\mathrm{Sp}(2h,\mathbb Z)$. This paper is the first in a series for a program whose goal is to extend the connection $\mathcal J_{\mathrm{DHS}}$ to a global flat connection on the Teichmüller space $\mathcal T_{h,n}$ valued in the Lie algebra of derivations of $\hat{\mathfrak t}_{h,n}$. Upon the choice of local coordinates adapted to the map $\mathcal T_{h,n}\to \mathcal T_h$, such a connection splits into three pieces: $\mathcal J_{\mathrm{DHS}}$, a piece $\mathcal L$ corresponding to holomorphic directions of $\mathcal T_h$ and a third piece corresponding to anti-holomorphic directions in $\mathcal T_h$. In this paper, we isolate the system of equations satisfied by $\mathcal L$ and obtain its solution locally and explicitly. The construction of a global extension of $\mathcal J_{\mathrm{DHS}}$ to $\mathcal T_{h,n}$ and the extension of the meromorphic connection of arXiv:1112.0864 to $\mathcal T_{h,n}$, are relegated to future publications in this series.

Recently, Jana and Kalita (Res. Number Theory $\textbf{8}~ (2022),$ Article $54$) obtained certain supercongruences motivated by some generalized Van Hamme type supercongruences, specifically for an integer $\ell \geq 2$ and an odd prime $p$ with $p \equiv -1 \pmod{\ell},$ \begin{align*} \displaystyle \sum_{n=0}^{\frac{p^v + 1}{\ell}} (-1)^n (2 \ell n - 1) (\ell^2 n^2 - \ell n + 1) \frac{(-\frac{1}{\ell})_n^3}{(1)_n^3} \equiv (-1)^{\frac{p^v + p + 2}{\ell}} p^{3v} \pmod{p^{3v+1}}, \end{align*} and \begin{align*} \displaystyle \sum_{n=0}^{\frac{p^v + 1}{\ell}} (2 \ell n - 1) (2 \ell^2 n^2 - 2 \ell n + 1) \frac{(-\frac{1}{\ell})_n^4}{(1)_n^4} \equiv - p^{4v} \pmod{p^{4v+1}}. \end{align*} Employing the $q$-telescoping technique, similar to the $q$-WZ method, we here establish some supercongruences involving certain $q$-shifted factorials. As particular cases, we provide $q$-analogues of the above supercongruences.

In the recent paper “Murmurations", Zubrilina showed that the Fourier coefficients of modular forms follow a certain murmuration pattern. The proof given in “Murmurations" proceeded by estimating the terms of the Skoruppa-Zagier trace formula. However, a few technical issues arise in the proof, including an omission of certain Chebyshev factors that arise in summations from the Skoruppa-Zagier trace formula. In this work, we address these issues to solidify the proof.

A binomial coefficient $\binom{n}{k}$ is good if all its prime divisors exceed $k$, and its deficiency is the number of $k$-smooth integers in $(n-k,n]$. A short constructive proof of the existence of infinitely many good binomial coefficients with deficiency exactly one is given, which provides an answer to a problem of Erdős, Lacampagne, and Selfridge.

We reformulate and discuss the conditions for the initial values in which a generalized Fibonacci sequence is eventually strictly monotonic. We then extend this exploration to the generalized \(k\)-bonacci sequence. We also give a lower bound on when an eventually strictly monotone generalized \(k\)-bonacci sequence becomes monotone.

In this paper, we introduce several novel approaches utilizing the circle method to obtain the asymptotic formula for the number of integral points of bounded height lying on a hypersurface in a weighted projective space. Let $F(\mathbf{x} ; \mathbf{y})$ be a given weighted form of degree $d$ in variables $\mathbf{x} \in \mathbb{R}^{s_1}$ and $\mathbf{y} \in \mathbb{R}^{s_2}$, where variables $\mathbf{x}$ and $\mathbf{y}$ have weights $w_1$ and $w_2$ with $w_1<w_2,\left(w_1, w_2\right)=1$, and $d>w_1 w_2$. Write $$ R_F(P):=\#\left\{(\mathbf{x} ; \mathbf{y}) \in \mathbb{Z}^{s_1+s_2}: F(\mathbf{x} ; \mathbf{y})=0,|\mathbf{x}| \leq P^{w_1 / d},|\mathbf{y}| \leq P^{w_2 / d}\right\} . $$ In particular, we show that whenever $$ s_1+s_2-\sigma_F>\left(1+\frac{w_2}{w_1}\right) \frac{d}{w_1} 2^{d / w_1}, $$ where $\sigma_F$ is the dimension of the affine singular locus of $F$, the quantity $R_F(P)$ has the expected asymptotic formula, that is $$ R_F(P)=c P^{s_1 w_1 / d+s_2 w_2 / d-1}+o\left(P^{s_1 w_1 / d+s_2 w_2 / d-1}\right), $$ where $c$ is the product of local densities. Furthermore, the constant $c$ is positive whenever $F(\mathbf{x} ; \mathbf{y})=0$ has a nonsingular solution over $\mathbb{R}$ and $\mathbb{Q}_p$ for every prime $p$. As a corollary, we verify the integral Hasse principle for the quasi-smooth hypersurface defined by $F(\mathbf{x} ; \mathbf{y})=0$ in a weighted projective space of sufficiently large dimensions.

We exhibit new subgroups of rational torsion points in geometrically simple Jacobians of genus-two curves over $\mathbb Q$. The largest group, which has order 96 and invariants [2,2,2,12], is realized by curves of the form $y^2 = x(x-a^2)(x-b^2)(x-c^2)(x-u^2)(x-v^2)$ where $a,b,c,u,v$ are positive integers that satisfy $a^2 + b^2 + c^2 = u^2 + v^2$ and $a^4 + b^4 + c^4 = u^4 + v^4$. We also find realizations of the groups [2,2,20], [2,2,4,4], [2,2,2,8], [2,4,8], and [6,6]. Finally, we record, to the best of our knowledge, all known subgroups that arise in genus-two Jacobians over $\mathbb Q$, in the geometrically simple case and in general.

We generalize the homological sieve method, developed by Das, Lehmann, Tosteson, and the author, to study certain quartic del Pezzo fibrations, and we prove a version of Manin's conjecture over global function fields in these cases. Our proofs combine the $3$-dimensional positive-characteristic minimal model program, the geometry of the space of sections and the Abel–Jacobi mapping, and the homological sieve method. Our quartic del Pezzo surfaces have Picard rank $2$, and admit two birational morphisms to non-split quadric surfaces. In particular, they do not possess any conic fibrations.

We prove uniform exponential bounds for the compactly supported Betti numbers of spaces of morphisms from curves of fixed genus to projective varieties. For targets in a fixed projective space cut out by a prescribed number of equations of fixed degrees, the bound is exponential in the degree of the morphism and is independent of the ground field, the source curve, and the target. The proof constructs bounded-degree affine presentations involving only linearly many variables and equations, and then applies Katz's estimate. As an application, we establish a higher genus function field version of Manin's conjecture for split quartic del Pezzo surfaces, generalizing a recent theorem of Das–Lehmann–Tanimoto–Tosteson. The passage from $\mathbb{P}^1$ to arbitrary source curves requires uniform bounds for compactly supported Betti numbers over all curves of fixed genus, together with new geometric results on higher genus morphism spaces, including irreducibility and dimension estimates for the relevant incidence strata. These results, combined with the necessary control of the configuration space contributions, allow the homological sieve and virtual height zeta function argument to yield, over sufficiently large finite fields and within a slightly shrunken nef cone, the predicted asymptotic with the expected leading constant.