Papers

We construct a big Witt theory associated with coefficient-multiplicative normalized $\sigma$-twisted Lubin-Tate formal groups. The resulting $F$-big Witt ring extends Hazewinkel's ramified $q$-typical Witt theory and recovers the usual big Witt ring in the multiplicative case. We also introduce Fleck operators defined from iterated Coleman traces and prove Lubin-Tate analogues of the Fleck-Sun-Wan congruences for both positive and negative powers. By relating the Lubin-Tate Coleman theory to the $F$-big Witt theory, we obtain a Coleman $F$-norm whose integrality yields congruences among the corresponding Fleck coefficients.

We consider the problem of finding lower bounds for integer linear combinations of $\sqrt{a_1},\ldots,\sqrt{a_K}$, where $a_1,\ldots,a_K$ are positive integers such that their square roots are linearly independent over the rationals. We use a probabilistic approach and prove that for $K\geq 8$ and nonzero integers $m_1,\ldots,m_K$, $$ \bigg|\sum_{n\leq K} m_n\sqrt{a_n}\bigg| > e^{\frac{2^{K-1}-1}{K} - \frac{1}{2}} \bigg(\max_{n\leq K}|m_n|\sqrt{a_n}\cdot \sqrt{K}\bigg)^{-(2^{K-1}-1)}. $$ This inequality improves the dependence on $K$ in the classical product bound.

Positive integer solutions to $x^2+y^2+z^2- x y z =D$ correspond to the important Markov (Markoff) numbers when $D=0$. From a given solution triple, three more are found with Vieta involutions, making an infinite tree of solutions. Starting instead with three real numbers greater than $2$ gives lengths of closed geodesics in a punctured torus. In this paper we study these tree structures for any real $D$. A continuous function, originally related to a norm on homology, encodes all the numbers on each of these trees. The properties of this function are developed here in general, with a self-contained exposition, showing that the usual $D=0$ case is part of a bigger picture. Encoding function graphs are shown to be convex for $D<4$, straight lines for $D=4$ and concave for $D>4$. Among other consequences are generalizations to all $D$ of: estimates for counting numbers on these trees, descriptions of the geometry of the corresponding lattice curves, uniqueness conditions, and identities of McShane and Hines.

Let $R$ be non-commutative ring with unity. In this paper, we prove that a given matrix in $M_n(R)$ is a sum of $p$-th powers in $M_n(R)$, where $n < p$ if and only if its trace can be written as a sum of $p$-th powers in $M_n(R)$, for $n = 2,3,4$ and $p = 5$ and $n = 4,5,6$ for $p = 7$. This paper extends the results of S. A. Katre and Kshipra Wadikar, $(p=3)$ and Garge $(n \geq p), (p = 5, 7)$, and S. A. Katre and Deepa Krishnamurthi $(n \geq p)$.

Coven and Meyerowitz gave two cyclotomic conditions, (T1) and (T2), which characterize integer tiles whose cardinalities have at most two distinct prime factors. We prove that the same characterization holds without this restriction. The proof uses a reduction in Chinese remainder coordinates: slicing by the lowest base-p digit produces sets with a common tiling complement in a group of order smaller by a factor of p. This reduction preserves the cyclotomic data needed for an induction on the exponents in (T2).

We classify the modular curves $X_0(N)$ that admit a smooth embedding in $\mathbb P^1 \times \mathbb P^1$ and obtain a partial classification for the Shimura curves $X_0^D(N)$. We also prove that, for $D>1$, a Shimura curve $X_0^D(N)$ admits a smooth plane model if and only if its genus is at most one.

We give a local proof of Langlands's Second Main Lemma for local epsilon factors over nonarchimedean local fields, in mixed and equal characteristic. The lemma compares local constants of characters of two intermediate fields in a bicyclic extension. It is one of the identities used in Langlands's construction of epsilon factors of local Weil representations. The proof builds on the work of Dwork, Langlands, and Lakkis. Langlands attributes the Second Main Lemma to Dwork, but Dwork did not publish a complete proof. Lakkis gave a detailed local treatment of the Second Main Lemma. For odd prime degree, his argument proves the required equality. In degree two, however, it proves only that the equality holds up to sign. The new point in the present paper is the wild dyadic case, i.e. biquadratic extensions over a nonarchimedean local field of residue characteristic two. For a biquadratic extension the First Main Lemma determines the square of the required equality, leaving a possible sign. We determine this sign by comparing the finite sums occurring in Lamprecht's formula. This completes the proof of the Second Main Lemma, including the equal-characteristic case. OpenAI ChatGPT was used extensively in the development of this proof. A substantial part of the argument consists of long local calculations, which are in principle accessible by standard methods, but would have required a very large amount of time. ChatGPT was therefore used to accelerate this technical work: to check calculations and compare them with the arguments of Dwork, Langlands, and Lakkis, and detect inconsistencies in intermediate versions of the proof. The author checked the final mathematical arguments and assumes responsibility for the results.

For an odd prime $p$, let $A_j$ be the $\omega^j$-eigenspace of the $p$-primary class group of $\mathbb Q(\zeta_p)$. Fix an even integer $d\ge4$, put $N=(p-1)/d$, and let $U_d=(\mathbb Z/d\mathbb Z)^\times$. For a relative density-one set of primes $p\equiv d+1\pmod{2d}$, we prove that the odd block $\bigoplus_{a\in U_d}A_{aN}$ has order at most $p^{\varphi(d)/2-1}$, and reflection shows that the even block $\bigoplus_{a\in U_d}A_{p-aN}$ has $p$-rank at most $\varphi(d)/2-1$. For each $d\in\{4,6\}$, both blocks vanish for a relative density-one set of primes $p\equiv d+1\pmod{2d}$; in particular, the even components vanish in accordance with Vandiver's conjecture. For each $d\in\{8,10,12\}$, the odd block has order at most $p$ for a relative density-one set of primes in the same progression, so every summand $A_{aN}$, $a\in U_d$, is cyclic, in accordance with Iwasawa's cyclicity conjecture. The proof uses an atomless limiting law for products of Dirichlet $L$-values, integrality of generalized Bernoulli norms, the relative class-number formula and reflection. A separate exact computation proves $A_{34}=0$ for every odd prime $p$; a single-file PARI/GP program reproduces the calculation.

Let $G$ be a finite group with $t$ conjugacy classes, and let $S_N$ be the symmetric group. Let $Z_t(N)$ be the number of zeros in the character table of the wreath product $G\wr S_N$. We prove \begin{equation*} Z_t(N)\ge \frac{2p_t(N)^{2}}{\log \frac{N}{t}}\left(1+O\left(\frac{\log \log \frac{N}{t}}{\log \frac{N}{t}}\right)\right), \end{equation*} where $p_t(N)$ is the number of $t$-multipartitions of $N$.

Let $b, p\in {\mathbb Z}$ with $b\ge 2$ and $p\ge 3$ a prime. If $b^{p-1}\equiv 1 \pmod{p^2}$, then $p$ is called a {\em generalized Wieferich prime base $b$}, or more succinctly, a {\em base-$b$ Wieferich prime}. When $b=2$, $p$ is also known simply as a Wieferich prime. We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots,\theta^{\deg(f)-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. Recently, necessary and sufficient conditions for the monogenicity of the trinomials $x^{2n}+bx^n+b$ were given that included base-$b$ Wieferich prime congruences, and also congruences involving a certain Lucas sequence. In this article, we prove a similar result for a different class of trinomials that does not rely on any congruence condition involving a Lucas sequence. Furthermore, we extend this result to a related class of $N$-nomials, for any $N\ge 4$.

We prove that, for any positive function $\Psi(t)\to\infty$, almost all positive integers $n$ satisfying $4\nmid n$ and $n\not\equiv7\pmod8$ can be written as $n=x^2+y^2+z^2$, where $x,y,z\in\mathbb{Z}$ and $1\leqslant z<\sqrt{\log n}\,\Psi(n)$. This improves Wooley's bound, and we show that the result is best possible. Our approach combines Hooley's weighted representation function for sums of two squares with Green's $W$-trick. We also give a necessary and sufficient condition, in terms of covering systems, for almost all odd integers to be represented as the sum of a squarefree number and a power of two, answering a question of Granville and Soundararajan.

For \(n\ge2\), let \(A(n)=\pi(n)-\omega(n)\) and \(B(n)=\phi(n)-\pi(n)+\omega(n)\) denote the numbers of prime and nonprime totatives of \(n\), respectively, with \(1\) included in \(B(n)\). We construct an explicit injectively parametrized family \(\mathcal C_n\) of composite totatives by completing suitably restricted squarefree products of prime totatives with a larger prime totative that is the unique largest prime factor, making the parametrization recoverable. We prove \(\lvert\mathcal C_n\rvert/A(n)\ge (e^{-\gamma}-o(1))\log A(n)/\log\log A(n)\), realizing the sharp classical leading constant \(e^{-\gamma}\). For the same family, with the same defining parameters, we obtain an effective refinement with error of order \((\log\log A(n))^{-1/4}\) and absolute effectively computable constants. Classical minimal-order results and the prime number theorem show that \(e^{-\gamma}\) is optimal: no family contained in the nonprime totatives can satisfy a uniform lower bound at this scale with a larger leading constant. We also determine exact eventual thresholds. Let \(N_k\) and \(M_k\) be the least integers such that \(\phi(n)>k\pi(n)\) for every \(n\ge N_k\) and \(B(n)>kA(n)\) for every \(n\ge M_k\), respectively. We prove \(M_k\le N_{k+1}\) for every \(k\ge1\), determine \(N_1,\ldots,N_6\) and \(M_1,\ldots,M_5\) exactly, and obtain \(M_k=N_{k+1}\) for \(1\le k\le5\).

In a very recent, beautiful paper, Junyi Xie and Ziquan Yang gave a new proof of Faltings' isogeny theorem over number fields by reducing to Tate's theorem over finite fields. We show that the same ideas can be adapted to prove finiteness of an isogeny class over number fields from the finiteness of an isogeny class over finite fields. Hence we obtain a new proof of the Shafarevich conjecture over number fields. The proof in this paper was obtained through the use of GPT6 Astra Ultra.

We study the spherical statistics of all the points in a thin shell of a high-dimensional random lattice. The exact relations between lattice points make it unclear when predictions based only on spherical geometry should hold. We answer this question for several statistics, including the number of shell points, the balance of their directions, and the occurrence, repetition, and distribution of differences between them. We identify sharp thresholds as the shell radius grows and show that these statistics undergo several distinct phase transitions. A shell can already agree with one geometric prediction while still differing strongly from another. Our results hold for a single sampled lattice, with probability tending to one as the dimension grows. They include estimates that hold across a complete shell, bounds close to the transition thresholds, and extensions to randomly shifted lattices. A Lean formalization verifies the main results, assuming the classical formulas and probability model stated in the code.

We establish the first pointwise convergence result for ergodic averages with iterates along explicit and deterministic sequences of intermediate growth, that is, growing faster than any polynomial but slower than any exponential. In particular, we show that the sequence $(\lfloor \exp((\log n)^c)\rfloor)_{n\in\mathbb{Z}+}$, with $c\in(1,8/7)$, is universally $L^p$-good for every $p\in(1,\infty]$. This gives an affirmative answer to an open problem dating back to the mid 1980s and contributes to Bellow's program, initiated in the earlier part of the same decade, on the characterization of $L^p$-good sequences in pointwise ergodic theorems. The proof combines the so-called one-frequency circle method with a delicate application of Vinogradov's method for estimating exponential sums whose phases involve $\big(\lfloor \exp((\log n)^c)\rfloor\big)_{n\in\mathbb{Z}_+}$. An interesting feature of our analysis, reminiscent of estimates arising in the study of the zero-free region of the Riemann zeta function, is that the argument relies on the classical Vinogradov method, in the sense that it necessitates estimates on the number of solutions for the Vinogradov system of Diophantine equations with explicit dependence on the system's parameters.

For all prime powers $q\geq5$, we construct lattices $\mathcal{L}_q\subseteq\mathbb{Z}^q$ with kissing numbers \[ \tau(\mathcal{L}_q)\geq \left(\frac{1}{2\pi e^2}+o(1)\right)\sqrt{q}\,e^{2\sqrt{q}}. \] The same asymptotic bound holds on a set of integer dimensions of natural density $1$, and in every sufficiently large integer dimension $n$ with an additional factor $e^{-\tfrac{1}{2}n^{1/40}}$. The construction is an extension of a previous construction by Bennett-Peikert based on Reed-Solomon codes.

Associated with the classical Kloosterman sums, we consider the quadratic twisted symmetric power moments. We investigate the cohomological properties and construct the motives over the field of the rationals underlying such moments. We determine the associated modular forms in the lower degree cases.

We formulate a conjectural generalization of the arithmetic Siegel-Weil formula, relating intersection multiplicities of cycles on unitary Rapoport-Zink spaces to central derivatives of local Whittaker functions. The new aspect of the conjecture is that it incorporates the action of certain Hecke correspondences on the Rapoport-Zink space. We verify the conjecture in a low-dimensional case. A significant portion of the paper is devoted to generalizing the classical theory of representation densities for local Hermitian spaces, as this is needed for explicit computation of the relevant Whittaker functions. We expect this generalization to be of independent interest.

We prove the dynamical uniform boundedness conjecture for unicritical polynomials $z^d + c$ over every number field $K$ when $d \geq 4$, and over every number field $K$ not containing $\mathbb{Q}(\sqrt{-3})$ when $d = 3$. Furthermore, our unconditional bounds on perodic points are effectively computable. In the remaining cases, we show that the dynamical uniform boundedness conjecture is implied by the boundedness of Mordell-Weil ranks over $K$ for a specific family of elliptic curves when $d = 3$ and for Jacobians of dynatomic curves when $d = 2$.

This is a mostly expository paper on the construction of theta functions on the Siegel and Hermitian half-spaces using the machinery of the Weil representation, especially theta functions weighted by polynomials. When the polynomial in question satisfies suitable homogeneity and harmonicity conditions, the resulting theta function is holomorphic. Non-holomorphic Siegel and Hermitian theta functions can be constructed in a similar way, using a more general class of polynomials.

Restricted forms of Lagrange's four-square theorem, in which the representing variables are required to satisfy additional arithmetic conditions, have been studied from several points of view. Among the refinements proposed by Zhi-Wei Sun is the conjecture that every nonnegative integer has a representation \[ N=x^2+y^2+z^2+w^2,\qquad x,y,z,w\in\mathbb Z_{\ge0}, \] for which \[ x\ \text{and}\ x+24y\ \text{are perfect squares}. \] We prove this conjecture for all sufficiently large integers. The proof starts from an elementary parametrization of the two square conditions and combines a semi-linear two-squares sieve with Poisson summation, finite-field cancellation, and switching.

We solve Serre's problem on the specialisation of Brauer group elements in all cases where the number of variables is sufficiently large.

In 2021, Chern and Tang introduced two families of two-parameter modular functions associated with the Rogers-Ramanujan continued fraction. They established recurrence relations that express the members of these families in terms of eta quotients, and used these expressions to obtain dissection formulas. Motivated by their work, we construct a three-parameter extension and derive the corresponding recursive eta quotient representations. Our result also has applications to dissection formulas. In particular, it is used in separate work to obtain $5$-dissection formulas for overpartitions with restricted odd differences.

Let $d,K,N\in \mathbb{N}$ with $K\geq 3$ and $d\geq 4K+4$. Let $\Delta\subset \mathbb{Z}^d$ be the vertex set of a nondegenerate $(K-1)$-simplex, and let $A\subseteq[N]^d$ contain no nontrivial similar copy of $\Delta$. We prove that \[ |A|\ll_{\Delta,d} N^d\exp\!\left(-c_{\Delta,d}\sqrt{\log N}\right) \] improving upon a polylogarithmic bound due to Magyar. We perform a density increment argument using the circle method, and we introduce a “cut operator” method to decouple the weighted exponential sum over the system of quadratic forms describing the simplex. Our proof combines ideas from graph theory, functional analysis, and the geometry of numbers. In the process, we apply Finner's fractional form of Hölder's inequality, the analytic large sieve, and Kim's mean value formula for primitive lattice flags.

We study archimedean smooth transfer for the Jacquet–Rallis relative trace formula comparison, in particular, identities between orbital integrals on GL(n) and its unitary forms. We work with Lie algebras and a (g,K)-module setting. Our main result states that for n=2, meaning GL(2) acting on gl(3), every polynomial type Schwartz function has a transfer to the unitary side and vice versa. Our proof relies on the Weil representation and the study of invariant differential operators. It also suggests certain structural properties of the (g,K)-modules in question which we formulate as conjectures.

We obtain an asymptotic formula for a cubic moment of self-dual $GL_2$ $L$-functions studied by Petrow and Young, with a small extra averaging over characters. Our asymptotic consists of the main term conjectured by Conrey–Farmer–Keating–Rubinstein–Snaith and a power-saving error term whose strength depends on the amount of extra averaging.

Let $K$ be an imaginary quadratic field and let $p \ge 5$ be a prime that is unramified in $K$. Let $\mathcal{A}_f/\mathbb{Q}$ be an abelian variety of $\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner hypothesis. In the case where $p$ is inert in $K$, we further assume that $\mathcal{A}_f$ is an elliptic curve. We develop an Euler-characteristic formula for signed Selmer groups over anticyclotomic $\mathbb{Z}_p$-extensions that applies when the corresponding Selmer modules have arbitrary $\Lambda$-rank. Assuming one inclusion in the signed Iwasawa main conjecture, we apply this formula to prove Kolyvagin's conjecture on the non-vanishing of the Kolyvagin system attached to Heegner points. Our results extend to the non-ordinary setting the results of Wei Zhang, Burungale–Castella–Grossi–Skinner, Castella–Sano and Kim in the ordinary case, and complement the works of Sweeting and Kim in the non-ordinary case under different hypotheses. We also study the effect of the exceptional zero phenomenon on the Iwasawa main conjecture in the inert case.

We develop a plus and minus Iwasawa theory for CM elliptic curves with potentially supersingular reduction at a prime $p\ge5$. We construct local points that satisfy suitable "jumping trace" relations using the Honda–Demchenko theory applied to height-two formal groups over local fields with small ramification degree. These local points allow us to study plus and minus Selmer groups of CM elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension, to construct the corresponding plus and minus $p$-adic $L$-functions, and to prove an Iwasawa main conjecture relating these objects. As an application, we obtain asymptotic growth formulae for the $p$-primary part of the Tate–Shafarevich groups.

We study nearest-neighbour distances and strict best returns of cyclic orbits on $(V\times G)/\Lambda$, where $V$ is a finite-dimensional real normed space, $G$ has a bi-invariant ultrametric, and $\Lambda$ is an arbitrary closed subgroup. For the quotient of the maximum product metric, the number of distinct nearest-neighbour distances in every finite orbit segment is at most $P(\overline V)+1$, where $P$ is an open-ball packing constant and $\overline V$ is obtained by factoring out the continuous real part of $\Lambda$. For inner-product norms the bound depends only on the span of the projected subgroup. This gives $2^d+1$ for the maximum norm on $\mathbb R^d$ and strong-kissing-number bounds for inner-product norms, extending the corresponding real and adelic distance bounds. Our second main result is an exact abelianization principle: every such cyclic orbit is isometric, with its integer indexing preserved, to a cyclic orbit on an abelian mixed quotient $(V\times U)/\Lambda_{\mathrm{ab}}$. The construction preserves the reduced and effective real factors. Thus noncommutativity and non-normality enlarge the class of ambient presentations but not the class of cyclic-orbit metrics. The record-count identity and its growth consequences require only an isometric embedding of a metric space. For mixed homogeneous spaces we prove packing recurrences and, using abelianization, the sharp three-step and five-step recurrences in effective Euclidean dimensions one and two, without a normalizer assumption. We also extend the purely real recurrence to noncompact quotients and give counterexamples showing why the stronger real recurrence can fail after an ultrametric factor is added.

We develop a projection and fibering method for sets of small combinatorial doubling in (not necessarily abelian) discrete groups. As an application, in the abelian case we prove that, if $A$ is finite, the ambient group has exponent $r$, and $|A+A|\leq K|A|$, then \[ |\langle A\rangle|\leq r^{(2+o(1))K}|A|. \] This answers a question of Ruzsa with an optimal leading coefficient, independent of Fox–Pham. The main ingredient is a discrete version of a fiber spillover argument. For sets in $2$-step nilpotent groups of exponent $r$, we also prove that $|A^3|\leq K|A|$ implies $|\langle A\rangle|\leq r^{(2+o_K(1))K}|A|$. The proof combines the abelian theorem with a weighted averaging of central fibers and commutators.