Papers

Let $A=\{a_{0}, a_{1}, \ldots, a_{k-1}\}$ be a set of $k>7$ integers such that $0=a_{0}<a_1<\cdots<a_{k-1}$ and $\gcd(A)=1$. The set $2^{\wedge}A=\{a+b: a, b\in A, a\neq b\}$ is called the restricted sumsets of $A$. Freiman-Lev conjecture is a well-known conjecture which related to restricted sumsets [V.F. Lev, Restricted set addition in groups, I. The classical setting, J. London Math. Soc. 62(2000), 27-40]. Up to now, Freiman-Lev conjecture is still open for all $a_{k-2}\geqslant 2k-4$ and $a_{k-1}\geqslant 2k-2$. In this paper, we complete the proof of the Freiman-Lev conjecture by resolving this final and most challenging case.

In this paper, we prove the mass equidistribution theorem restricted to vertical geodesic segments for holomorphic Hecke newforms of large square-free level. We utilize the effective proof of QUE in the level aspect. Moreover, we study this $L^2$ mass in the full geodesics and relate it to second moments of twisted $L$-functions.

In this work, we construct Galois representations with large image that are not GL r -odd (or, equivalently, not regular at infinity). More precisely, for every prime p {\v e} 3, every integer r {\v e} 2, and every integer a P rp1 'p'1q r q{2, r '2s satisfying a ” r pmod 2q, we construct a continuous Galois representation $\rho$\,: G Q __ GL r pQ p q whose image is commensurable with GL r pZ p q and satisfies |trp$\rho$pcqq| ” a, where c denotes a complex conjugation. We also obtain analogous results for the orthogonal group O r pQ p q.

We develop a direct analytic theory of stuffle-regularized multitangent functions and prove their reduction to finite linear combinations of monotangent functions, without using mould calculus. We first establish an asymptotic comparison between one-sided truncated multiple Hurwitz zeta functions and their stuffle regularizations at the natural parameter \(T_N-H(s)\), where \(T_N\) is the harmonic truncation and \(H(s)\) is the harmonic-number function. Applying this comparison to symmetric multitangent truncations yields meromorphic, \(1\)-periodic regularized multitangent functions. An explicit partial-fraction decomposition of each summand, combined with asymptotic estimates for moving truncation ranges, gives formulas for the reduction coefficients in terms of stuffle-regularized multiple zeta values. The constant term and the coefficient of the monotangent \(\mathcal T(1;s)\) are determined from the limits as \(\operatorname{Im}s\to\pm\infty\): both vanish whenever the index contains an entry greater than \(1\), whereas the exceptional indices \(\{1\}^r\) are evaluated through a sine-quotient generating function. As a consequence, we obtain a family of relations among regularized multiple zeta values.

We examine the equation $y^{a}=\sum\limits_{i=1}^{k}F_{n_{i}}$ for positive integers $y,a\geq 2$ and $k\geq3$. This equation can be expressed as a problem in terms of the Zeckendorf representations of integers. Using bounds on linear forms in logarithms and Baker-Davenport reduction methods, we are able to completely solve the equation for $ y\leq 25000$ when $k=3$, for $y\leq 1000$ when $k= 4$, for $y\leq 40$ when $k= 5$, and for $y\leq 3$ when $k=6$.

By combining the recent results of Pierce and Xu on the higher-dimensional Burgess bound with the classical one-dimensional Burgess bound, we prove that for any sufficiently large prime $p$, there exists a Pythagorean triple $(a_p,b_p,c_p)$ with $a_p,b_p,c_p\in\mathbb{Z}\cap(0,p)$ such that $a_pb_p/2$ is a primitive root modulo $p$. This confirms a conjecture of Z.-W. Sun for all sufficiently large primes.

Let $G=C_5\times C_5$, and let $S(G)$ denote the set of integer values of its group determinant. Previous work determines the values in $S(G)$ coprime to $5$ and proves that every $5$-divisible value is divisible by $5^8$. We prove the converse inclusion $5^8\mathbb{Z}\subseteq S(G)$. Consequently, $S(G)=\{m\in\mathbb{Z}:m\equiv\pm1\text{ or }\pm7\pmod{25}\}\cup5^8\mathbb{Z}$. The proof uses a general shift criterion and three explicit polynomials whose group determinants are $5^8$, $2\cdot5^8$, and $5^9$. Together with the known classification for $C_{25}$, this completes the Taussky–Todd integer group determinant problem for all groups of order $25$.

In \cite{ABL21}, Autissier, Bonolis and Lamzouri obtained uniform estimates for the distribution function of the maximum of partial sums of a class of families of $m$-periodic complex-valued functions under certain conditions. For example, these conditions are verified by Kloosterman or Birch sums. In this article, under the same assumptions, we obtain an improved estimate for the tail of the distribution of the maximum of partial sums in these families, which gives for the first time an asymptotic formula for the logarithm of the distribution function, in a large uniform range. Furthermore, we prove a "structure theorem" for the maximum of partial sums of a family of complex $m$-periodic functions in our class, which shows that universally over this class, most of the partial sums with large norm are close to their imaginary part, and the maximum is attained around $m/2$. This is in sharp constrast with the results of Bober-Golmaker-Granville-Koukoulopoulos \cite{BGGK18}, Lamzouri \cite{Lam24} and Lamzouri-Nath \cite{LN24} for the distribution of the maximum of character sums in various families of Dirichlet characters.

Let $S_k$ be the space of cusp forms of weight $k$ for the full modular group ${\rm SL}_2(\mathbb{Z})$. In this paper, we will exhibit an explicit rational spanning subset of $S_k$ constructed from Eisenstein series at prime levels. The main ingredient of our proof is to show a result on determining cusp forms by noncentral quadratic-twist $L$-values.

Let $K$ be any quadratic number field, and let $O_K$ be the ring of algebraic integers in $K$. In 1975 J. Denef proved that Hilbert's Tenth Problem over $O_K$ has a negative solution. In this paper we establish the following undecidability result: There is no algorithm to decide whether an arbitrarily given polynomial equation $P(z_1,\ldots,z_{16})=0$ (with integer coefficients and 16 unknowns) has solutions over $O_K$. Moreover, when $K$ is a real quadratic field, we show that $15$ unknowns suffice for undecidability.

The $\ell$-adic Tate module of an abelian surface with quaternionic multiplication (QM) decomposes as two copies of a two-dimensional $\mathbb{Q}$-rational representation. To this day, there is no criterion to distinguish these representations from those attached to elliptic curves. For this reason, QM abelian surfaces are usually called fake elliptic curves. In this paper we characterize QM abelian surfaces defined over an imaginary quadratic field $K$. Namely, we consider an abelian surface $A/K$ without potential CM and whose $L$-function is a square. Under a reasonable conjecture, we show that $A$ has QM if and only if it has residual image contained in a non-split Cartan group modulo at least two primes. The characterization is unconditional for all indefinite quaternion discriminants up to 33. The proof is based on previous work of Siksek and Michaud-Jacobs on quadratic points on non-split Cartan modular curves. In particular, we prove that all quadratic points on $X_{ns}(6)$, $X_{ns}(10)$ and $X_{ns}(15)$ are non-exceptional.

Let $\psi(n)$ and $\phi(n)$ denote Dedekind's arithmetic function and Euler's totient function, respectively. We study the distribution of the compositions $\phi(\psi(n))$ and $\psi(\psi(n))$. In particular, we obtain quantitative upper bounds for the exceptional set associated with $\phi(\psi(n))$, thereby refining a density result of Sándor. We also prove that, for every fixed $c>0$, the set of integers $n\leq x$ satisfying $\psi(\psi(n))\leq cn$ has asymptotic density zero. Our method adapts sieve ideas used by Dixit and Bhattacharjee to compositions involving Dedekind's arithmetic function.

For positive integers $r, m$ and $N$, every $r$-coloring of $\{1, \dots, N\}$ contains a monochromatic solution to $x_1+\dots+x_{m+1}=y_1+\dots+y_m$ provided that $N \ge 3^r (r!)^{1/m}$, which is qualitatively optimal when $m$ is logarithmic in $r$.

Let $N_n^{\mathrm{tor}}(X)$ be the number of isomorphism classes of $n$-dimensional algebraic tori over $\mathbb{Q}$ whose Artin conductor is bounded by $X$. We prove that there is an absolute constant $C>0$ such that, for every positive integer $n \ge 2$, $N_n^{\mathrm{tor}}(X)\ll_n X^{\exp(C(\log n)^2)}$. The proofs of the main results were developed through an iterative dialogue with ChatGPT 5.6 Pro.

Let $d$ be the smallest positive integer, not a multiple of $3$, for which there exists an algebraic number $\al$ of degree $d$ over $\mathbb{Q}$ whose three algebraic conjugates add to zero. We prove that $d=20$. This is derived from the following result: for any linear relation $\sum_{j=1}^d a_j \al_j=0$ with coefficients $a_j\in\mathbb{Z}$ among the conjugates $\al_j$ of an algebraic number of degree $d=p^m$, where $p$ is a prime number, $m \geq 1$, the sum $\sum_{j=1}a_j$ is divisible by $p$. If $d=2p^m$, $p\geq 3$ and $\sum_{j=1}^d|a_d| < p$, then $\sum_{j=1}a_j$ is an even number.

Let $E$ be a CM number field, $F$ a totally real subfield (e.g. $\mathbb Q$), and let $\mathfrak a$ be a non-zero fractional ideal of $E$. Endowed with the Hermitian trace form $h_{E/F}(x,y)=\mathrm{Tr}_{E/F}(x\overline{y})$, the ideal $\mathfrak a$ defines an ideal lattice over $\mathcal O_F$. In this paper, we give explicit formulas for the Jordan decomposition of this lattice at a prime ideal $\mathfrak p\subset\mathcal O_F$, in terms of the prime ideal factorization of $\mathfrak a$ in $E$. Following the approach of Erez, Morales, and Perlis, we reduce the computation to the local behavior at the primes above $\mathfrak p$. Our results provide local invariants for the isometry relation between ideal lattices, with potential applications to the study of structured lattices arising in arithmetic and cryptography.

We consider the Erdős-Moser equation $1^k+2^k+\cdots+(m-1)^k=m^k$ in arithmetic progressions. We prove among other things that when $k=2$, for any solution to exist, the above sum in arithmetic progression must consist of two or four terms. In either case, there are infinitely many solutions that can be completely characterized.

In the present paper, we introduce formulas for the logarithm function of abelian, uniformizable Anderson modules. Combining our formulas with a motivic map introduced recently by the second author, we relate special values of dual Goss $L$-functions of Drinfeld modules to the rigid analytic trivialization of their corresponding $t$-comotives. We apply these formulas to relate certain $L$-values to special values of matrix valued modular forms, generalizing the vector valued Drinfeld modular forms of Pellarin and Pellarin and Perkins in the rank two setting.

We introduce a notion of an admissible pair, and prove a variation to Pollock's conjectures on icosahedral and dodecahedral numbers.

We give a sieving algorithm for finding pairs of primes with small multiplicative orders modulo each other. This problem is a necessary condition for obtaining constructions of $2$-cycles of pairing-friendly curves, which have found use in cryptographic applications. Our database of examples suggests that, except for a well-known infinite family of such primes, instances become increasingly rare as the size of the primes increase. This leads to some interesting open questions for which we hope our database prompts further investigation.

We study long-time Strichartz estimates for the one-dimensional Schrödinger equation with quasi-periodic initial data. For two-frequency data with an algebraic frequency ratio, we observe that the behavior of the linear Schrödinger evolution changes depending on the algebraic degree of the ratio. Making use of this observation, we improve the Strichartz estimates on long time intervals. We also prove an endpoint $L^4$ Strichartz estimate. Our proofs use Roth-type Diophantine inequalities and Vinogradov-type mean value estimates for the Parsell–Vinogradov systems.

We introduce and study a finite simple graph of algebraic origin: the Vieta graph on the solution set over $\mathbb{F}_p$ to a symmetric, multivariate equation which is quadratic in each variable. This construction is a broad generalization of the Markoff graph over $\mathbb{F}_p$, extensively studied in the recent literature. We give a systematic approach, partly based on quadratic character sums, to the following basic counting questions: how many vertices does a Vieta graph have, and what is the degree distribution? We focus on explicit counts, addressing the low-dimensional cases in three and four variables.

We study two-times monogenic quartic orders; i.e., those of the shape $\mathbb{Z}[\alpha] = \mathbb{Z}[\beta]$, with algebraic integers $\alpha$ and $\beta$ not $\mathbb{Z}$-equivalent. Two specific types, describing possible algebraic relation among monogenizers of two-times monogenic orders were defined by Bérczes, Evetrse, Győry, who proved under certain conditions on the Galois group of the normal closure of a given number field $K$, that there can be only finitely many two-times monogenic $\mathbb{Z}$-orders in the ring of integers $K$ which are not of these specific two types. In this article, we prove this fact for all quartic number fields.

We construct explicit examples of non-integral variations of $\mathbb{Q}$-Hodge structures. Our approach leverages Fenchel–Nielsen-type parameterizations, due to Kabaya and Maskit, of the Teichmüller component of relative character varieties. Additionally, we discuss various Diophantine results concerning the $\mathcal{O}_{K,S}$-integral points of such character varieties, and give a new proof of Beauville's classical theorem on families of elliptic curves. We conclude by collecting pathological behaviors of the Hodge locus of non-integral $\mathbb{Q}$VHS, in particular the failure of the Cattani–Deligne–Kaplan theorem and the André–Oort conjecture; our results indicate that for $\mathbb{Q}$VHS which are not $\mathbb{Z}$VHS, what can go wrong must go wrong.

Let $G$ be a connected reductive group over a non-archimedean local field. The main result of Fargues and Scholze [FS21] for the geometrization of the local Langlands correspondence is the construction of a “spectral action” on the category of $\ell$-adic sheaves on $\text{Bun}_{G}$, the stack of $G$-torsors on the Fargues-Fontaine curve. The goal of this paper is to prove a conjecture of Fargues which says that one can extend this construction to the larger stack $\text{Bun}_G^e$ of $G$-torsors on the Kaletha gerbe over the curve, as introduced by Fargues [Far22]. This “extended spectral action” allows for a version of the categorical local Langlands conjecture for an arbitrary connected reductive group $G$, and is the first such statement for those $G$ which are not extended pure inner forms of a quasi-split group, such as non-trivial inner forms of $\mathrm{SL}_{n}$. Finally, we prove this conjecture for tori, following the original argument of [Zou24] and, under the same assumptions as [Zou26] (including connected center), we reduce the “extended” version of the categorical conjecture to the one in Fargues–Scholze.

We develop a local-to-global differential framework for additive decomposition problems involving multiplicative subgroups of prime fields. Starting from Hanson–Petridis-type auxiliary polynomials, we use degree bounds, in the spirit of Stepanov's method, to lift local coefficient relations at their roots to global differential identities. This yields a unified treatment of \[ H=A+B,\qquad H\cup\{0\}=A-A,\qquad H=A\mathbin{\widehat{+}} A,\qquad H\cup\{0\}=A\mathbin{\widehat{+}} A, \] where $H$ is a proper multiplicative subgroup. This circle of problems is motivated by Sárközy's conjecture on the additive irreducibility of nonzero quadratic residues and its generalizations to multiplicative subgroups. Rudnev and Tyrrell recently classified all decompositions $H=A+B$, building on the approach introduced by Hanson–Petridis and further developed by Kalmynin. Our framework gives a new polynomial proof of the Rudnev–Tyrrell classification and substantially streamlines the existing proofs: it gives an independent proof of Kalmynin's equal-size theorem and reduces the classification to direct coefficient comparisons, avoiding the residue-theoretic input and more elaborate arithmetic analysis of earlier proofs. It also yields a streamlined proof of Kalmynin's resolution of a conjecture of Lev and Sonn on $H\cup\{0\}=A-A$. For the two restricted-sumset problems, we obtain complete classifications, substantially improving earlier results of Shkredov and Yip. We also establish some stability refinements.

We study automorphic functions for square-zero extensions $C$ of curves $\overline{C}$ over finite fields, a study initiated by Braverman-Kazhdan-Polishchuk in [BKP23]. More precisely, we study the cuspidality and Hecke-finiteness of the functions in the orbit decomposition introduced in loc. cit. for split connected reductive groups $G$, generalizing some of the results for $G=\mathrm{PGL}_2$. As a result, for $G=\mathrm{PGL}_3$, we prove a new case of a conjecture in [BK23] concerning the finite-dimensionality of the space of unramified Hecke-finite functions. We also introduce a formulation of support bounds for spherical cuspidal and Hecke-finite functions in terms of the Harder-Narasimhan stratification of $G$-bundles on the reduced curve. Using representation-theoretic constructions together with their geometric interpretations in terms of $G$-bundles on $C$ and certain twisted $G$-Higgs bundles on $\overline{C}$, we compute the optimal bounds in several cases and, in particular, determine the optimal bound for $G=\mathrm{PGL}_3$.

For an elliptic curve $E$ defined over a field $K$ of characteristic $0$ with $\operatorname{End}_K \! E \cong \mathbb{Z}$, we classify which isogeny graphs $\mathcal{G}(E/K)$ can occur. We first show that $\mathcal{G}(E/K)$ decomposes as a weak Cartesian product of its $p$-primary isogeny graphs, one for each prime $p$, thereby reducing the problem to classifying $p$-primary isogeny graphs. We then show that each such graph is isomorphic, as an edge-weighted graph, to a member of an explicit family of edge-weighted graphs $\mathcal{H}_{p^k}^r$ and $\mathcal{H}_{p^{\infty,+}}^r$, every member of which occurs as a $p$-primary isogeny graph except for $\mathcal{H}_{2^k}^0$ for $k\ge 2$. The proof relies on a detailed study of the $p$-adic Galois representation attached to $E$, through which we identify each graph with a subgroup of $\operatorname*{GL}\nolimits_{2}(\mathbb{Z}_{p})$. More generally, we identify subgroups of $\operatorname*{GL}\nolimits_{2}(\widehat{\mathbb{Z}})$ for each possible isogeny graph and describe their corresponding modular curves, completing, in the genus $0$ case, the explicit parameterization of isogeny graphs via parameterized isogenous families of elliptic curves. We also introduce the $p$-blooming invariant $\mathfrak{I}_p(E/K)$, an isogeny class invariant determining the value of $r$ in the $p$-primary isogeny graph, and show that elliptic curves over fields with a real embedding attain the smallest possible value. As applications, we characterize the isogeny graphs of elliptic curves with potential complex multiplication; give an algorithm for determining the isogeny graph from the adelic Galois representation; recover the classification of rational isogeny graphs; and, under GRH, classify the isogeny graphs occurring over certain number fields.

We study the statement that every even number $n \ge 6$ is the sum of a prime and a member of a twin prime pair. It sits between the conjectures of Goldbach and Dubner and implies both the Goldbach and the twin prime conjectures. Our main result is conditional: if the number of twin primes up to $z$ is at least $c\,z/\log^2 z$ for all large $z$, a lower bound of the order predicted by Hardy and Littlewood, with nothing assumed about their distribution, then a positive proportion of the even numbers are so representable, with density at least an absolute multiple of $c$. The proof is Romanov's method with a Selberg sieve, and loses no factor of $\log\log$. Nothing in this direction can be unconditional, since representability on a set of positive density already implies the twin prime conjecture; we show further that a polylogarithmic bound on the least twin summand would force a power-type lower bound on the number of twin primes. We verify the statement exhaustively for all even numbers up to $10^{14}$, where the least twin summand never exceeds 23,029.

This note archives the status of Samart's Table-6 conjecture $n_4(81)=40M_7$, $M_7:=L'(g_7,0)$, where $g_7(\tau)=\eta(\tau)^3\eta(7\tau)^3$ is the newform of $S_3(\Gamma_0(7),\chi_{-7})$ and $n_4(s):=4m(x^4+y^4+z^4+1+s^{1/4}xyz)$. The conjecture is the cleanest of Samart's open interior-point entries (discriminant $-7$, class number $1$, a single $L$-value), and it was dropped as a theorem target in the companion paper, where the two $n_2$-family conjectures were proved. We record what is proved and precisely where those methods fail. First, a complete proof of the $L$-value side (P1): at the CM point $\tau_2=(7+\sqrt{-7})/4$ the Eisenstein–Kronecker expression underlying Samart's formula evaluates exactly to $\mathrm{EK}_4(\tau_2)=40M_7$, via lattice sums over the ring of integers of $\mathbb{Q}(\sqrt{-7})$ and the principal ideal $(\bar\varpi)$, with an exact cancellation of the parasitic $\zeta_K(2)$-terms. Second, two quantitative obstructions to the remaining half $n_4(81)=\mathrm{EK}_4(\tau_2)$: the critical image of the $n_4$-family is a two-dimensional astroid disc containing the parameter $c=3$ in its interior (in contrast to the one-dimensional slit $[0,64]$ of the $n_2$-family), so no continuation path can approach the CM point; and Samart's $U$-series converges on all of the upper half-plane but leaves the geometric sheet of the holomorphic Mahler measure everywhere below $\mathrm{Im}\,\tau=1/\sqrt{2}$, so the premise of the differential-comparison continuation fails. A 20-digit direct torus integration then decides the conjecture numerically: $n_4(81)-40M_7=+0.0586706795972872...$, five orders of magnitude above the integration error floor, so the identity as literally stated is refuted; a closed form for the true value $n_4(81)$ remains open and appears to require regulator/monodromy machinery.