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Robin Zhang

Publications and source records attributed to Robin Zhang.

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On maximal Dynkin friezes

The maximal entries of Dynkin friezes over the positive integers have recently been determined for all finite Dynkin types except $B_n$ and $D_n$. In this note, we explicitly construct large positive integral points on affine cluster varieties of type $B_n$ (resp. $D_n$), giving rise to friezes of types $B_n$ (resp. $D_n$) over the positive integers with largest entries $F_{n+1} F_{n+2} - 1$ (resp. $F_n F_{n+1} - 1$) where $F_k$ is the $k$-th Fibonacci number. We conjecture that these are the maximal possible entries for their respective Dynkin types.

math.CO

Scalable Training of Mixture-of-Experts Models with Megatron Core

Scaling Mixture-of-Experts (MoE) training introduces systems challenges absent in dense models. Because each token activates only a subset of experts, this sparsity allows total parameters to grow much faster than per-token computation, creating coupled constraints across memory, communication, and computation. Optimizing one dimension often shifts pressure to another, demanding co-design across the full system stack. We address these challenges for MoE training through integrated optimizations spanning memory (fine-grained recomputation, offloading, etc.), communication (optimized dispatchers, overlapping, etc.), and computation (Grouped GEMM, fusions, CUDA Graphs, etc.). The framework also provides Parallel Folding for flexible multi-dimensional parallelism, low-precision training support for FP8 and NVFP4, and efficient long-context training. On NVIDIA GB300 and GB200, it achieves 1,233/1,048 TFLOPS/GPU for DeepSeek-V3-685B and 974/919 TFLOPS/GPU for Qwen3-235B. As a performant, scalable, and production-ready open-source solution, it has been used across academia and industry for training MoE models ranging from billions to trillions of parameters on clusters scaling up to thousands of GPUs. This report explains how these techniques work, their trade-offs, and their interactions at the systems level, providing practical guidance for scaling MoE models with Megatron Core.

cs.DC

Uniform bounds on periodic points of polynomials with good reduction

We establish effective bounds on the number of periodic points of degree-$d$ polynomials $\phi$ defined over $p$-adic fields and number fields, under a mild reduction hypothesis that is satisfied by all unicritical polynomials $X^d + c$ with $c$ integral at some prime dividing $d$. As a consequence, we verify the uniform boundedness conjecture for this class of polynomials over number fields $K$, giving the explicit uniform bound $\#\mathrm{Per}_K(\phi) \leq d^{[K:\mathbb{Q}]}$.

math.NT

Towards the $p$-adic derived Hecke algebra for weight one forms

This note outlines an approach to defining $p$-adic Shimura classes and $p$-adic derived Hecke operators on the completed cohomology of modular curves from upcoming work by the author. After reviewing the modulo-$p$ constructions of Harris and Venkatesh, we formulate a conjecture relating the action of $p$-adic derived Hecke operators on cusp forms of weight $1$ and level $\Gamma_1(N)$ to the $p$-adic logarithm of the Stark unit for the corresponding adjoint Deligne-Serre representation. This new $p$-adic conjecture can be viewed as complementary to the Harris-Venkatesh conjecture.

math.NT

A positive Siegel theorem: Dynkin friezes and positive Mordell-Schinzel

We determine the number of positive integral points on $n$-dimensional affine varieties associated to arbitrary $n \times n$ generalized Cartan matrices. An application to the theory of cluster algebras and combinatorics is the resolution of the Fontaine-Plamondon conjecture, which says that there are exactly $4400$ and $26952$ positive integral friezes of type $E_7$ and $E_8$ respectively. An application to number theory refines and generalizes theorems of Mohanty, Mordell, and Schinzel to the positive integers and higher dimensions by exhibiting examples of Diophantine equations $xyz = G(x, y)$ and $xyzw = G(x, y, z)$ of every degree greater than $3$ with infinitely many positive integral solutions.

math.NT

Mod $\ell$ gamma factors and a converse theorem for finite general linear groups

The local converse theorem for Rankin-Selberg gamma factors of $\mathrm{GL}_2(\mathbb{F}_q)$ proved by Piatetski-Shapiro over $\mathbb{C}$ no longer holds after reduction modulo $\ell \neq p$. To remedy this, we construct new $\mathrm{GL}_n \times \mathrm{GL}_m$ gamma factors valued in arbitrary $\mathbb{Z}[1/p, \zeta_p]$-algebras for Whittaker-type representations, show that they satisfy a functional equation, and then prove a $\mathrm{GL}_n \times \mathrm{GL}_{n-1}$ converse theorem for irreducible cuspidal representations. In the $\mathrm{GL}_2 \times \mathrm{GL}_1$ case, we define an alternative "new" gamma factor, which takes values in $k$ and satisfies a converse theorem that matches the converse theorem in characteristic $0$.

math.NT

The Harris-Venkatesh conjecture for derived Hecke operators II: a unified Stark conjecture

We prove a compatibility theorem between the Stark conjecture and the Harris-Venkatesh conjecture for imaginary dihedral modular forms of weight $1$. The key technical input is a general two-variable $\mathrm{PGL}_2$ Siegel-Weil formula that precisely gives the Laurent series coefficients of Eisenstein series as a dual theta lift of Eisenstein series. This two-variable Siegel-Weil formula is applied to Rankin-Selberg periods of imaginary dihedral optimal forms and a refinement of Stark's formula relating $L$-values with elliptic units, yielding compatibility between derived Hecke operators and adjoint $L$-values for Deligne-Serre representations. We formulate a general unified conjecture encompassing the Stark and Harris-Venkatesh conjectures and conceptually reinterpret the action of derived Hecke operators as modular Stark unit data at almost all primes.

math.NT

The Harris-Venkatesh conjecture for derived Hecke operators I: imaginary dihedral forms

The Harris-Venkatesh conjecture relates the action of derived Hecke operators on weight-one modular forms to algebraic units. We introduce the Harris-Venkatesh period, defined on a product of modular curves, and show that it specializes to the derived Hecke action on weight-one forms. We also define and construct optimal forms, two-variable modular forms that explicitly realize a theta lifting from the Jacquet-Langlands correspondence. Using a positive-characteristic multiplicity-one argument comparing the Harris-Venkatesh period on newforms and optimal forms, we prove the full Harris-Venkatesh conjecture for imaginary dihedral weight-one modular forms.

math.NT

The Harris-Venkatesh conjecture for derived Hecke operators III: local constants

The first two papers in this series prove the Harris-Venkatesh conjecture on derived Hecke operators and its refinement with the Stark conjecture for imaginary dihedral modular forms of weight $1$. This paper explicitly describes the constants appearing in the Harris-Venkatesh conjecture for dihedral modular forms by evaluating $\mathrm{GL}(2) \times \mathrm{GL}(2)$ Rankin-Selberg periods and zeta integrals on newforms and optimal forms. One consequence is a formula for the ratio between Petersson norms and adjoint $L$-values. These calculations also extend to exotic modular forms of odd level and to exotic modular forms with $2$-ordinary Deligne-Serre representation.

math.NT

Grupos ortogonales sobre cuerpos de caracter\'{i}stica positiva

This exposition examines the theory of orthogonal groups and their subgroups over fields of positive characteristic, which has recently been used as an important tool in the study of automorphic forms and Langlands functionality. We present the classification of orthogonal groups over a finite field using the theory of bilinear forms and quadratic forms in positive characteristic. Using the determinant and spinor norm when the characteristic of $F$ is odd and using the Dickson invariant when the characteristic of $F$ is even, we also look at special subgroups of the orthogonal group. -- -- Esta exposici\'{o}n examina la teor\'{i}a de los grupos ortogonales y sus subgrupos sobre cuerpos de caracter\'{i}stica positiva, que recientemente se han utilizado como una herramienta importante en el estudio de las formas autom\'{o}rficas y la funcionalidad de Langlands. Presentamos la clasificaci\'{o}n de grupos ortogonales sobre un cuerpo finito $F$ utilizando la teor\'{i}a de formas bilineales y formas cuadr\'{a}ticas en caracter\'{i}stica positiva. Usando el determinante y la norma del espinor cuando la caracter\'{i}stica de $F$ es impar y usando la invariante de Dickson cuando la caracter\'{i}stica de $F$ es par, tambi\'{e}n encontramos subgrupos especiales del grupo ortogonal.

math.GR

The $abcd$ conjecture, uniform boundedness, and dynamical systems

We survey Vojta's higher-dimensional generalizations of the $abc$ conjecture and Szpiro's conjecture as well as recent developments that apply them to various problems in arithmetic dynamics. In particular, the "$abcd$ conjecture" implies a dynamical analogue of a conjecture on the uniform boundedness of torsion points and a dynamical analogue of Lang's conjecture on lower bounds for canonical heights.

math.NT

Modular Gelfand pairs and multiplicity-free representations

We give a generalization of Gelfand's criterion on the commutativity of Hecke algebras for Gelfand pairs and multiplicity-free triples over algebraically closed fields of arbitrary characteristic. Using more lenient versions of projectivity and injectivity for modules, we prove a general multiplicity-freeness theorem for finitely-generated modules with commutative endomorphism rings. For representations of finite and profinite groups, Gelfand pairs over the complex numbers are therefore also Gelfand pairs over the algebraic closure of any finite field. Applications include the uniqueness of Whittaker models of modular Gelfand-Graev representations and the uniqueness of modular trilinear forms on irreducible representations of quaternion division algebras over local fields.

math.RT

A Galois-dynamics correspondence for unicritical polynomials

In an analogy with the Galois homothety property for torsion points of abelian varieties that was used in the proof of the Mordell-Lang conjecture, we describe a correspondence between the action of a Galois group and the dynamical action of a rational map. For nonlinear polynomials with rational coefficients, the irreducibility of the associated dynatomic polynomial serves as a convenient criterion, although we also verify that the correspondence occurs in several cases when the dynatomic polynomial is reducible. The work of Morton, Morton-Patel, and Vivaldi-Hatjispyros in the early 1990s connected the irreducibility and Galois-theoretic properties of dynatomic polynomials to rational periodic points; from the Galois-dynamics correspondence, we derive similar consequences for quadratic periodic points of unicritical polynomials. This is sufficient to deduce the non-existence of quadratic periodic points of quadratic polynomials with exact period 5 and 6, outside of a specified finite set from Morton and Krumm's work in explicit Hilbert irreducibility.

math.NT

Section rings of $\mathbb{Q}$-divisors on minimal rational surfaces

We give bounds on the degree of generation and relations of section rings associated to arbitrary $\mathbb{Q}$-divisors on projective spaces of all dimensions and Hirzebruch surfaces. For section rings of effective $\mathbb{Q}$-divisors on projective spaces, we find the best possible bound on the degrees of generators and relations.

math.AG

Spin canonical rings of log stacky curves

Consider modular forms arising from a finite-area quotient of the upper-half plane by a Fuchsian group. By the classical results of Kodaira-Spencer, this ring of modular forms may be viewed as the log spin canonical ring of a stacky curve. In this paper, we tightly bound the degrees of minimal generators and relations of log spin canonical rings. As a consequence, we obtain a tight bound on the degrees of minimal generators and relations for rings of modular forms of arbitrary integral weight.

math.AG

On quadratic periodic points of quadratic polynomials

Bounding the number of preperiodic points of quadratic polynomials with rational coefficients is one case of the Uniform Boundedness Conjecture in arithmetic dynamics. Here, we provide a general framework that may reduce finding periodic points of such polynomials over Galois extensions of $\mathbb{Q}$ to finding periodic points over the rationals. Furthermore, we present evidence that there are no such polynomials (up to linear conjugation) with periodic points of exact period 5 in quadratic fields by searching for points on an algebraic curve that classifies quadratic periodic points of exact period 5 and suggesting the application of the method of Chabauty and Coleman for further progress.

math.NT