arXiv subjects

Lynnelle Ye

Publications and source records attributed to Lynnelle Ye.

12 recordsLinked to original sources

arXiv2023

Bounding Lifts of Markoff Triples mod $p$

In 2016, Bourgain, Gamburd, and Sarnak proved that Strong Approximation holds for the Markoff surface in most cases. That is, the modulo $p$ solutions to the equation $X_1^2+X_2^2+X_3^2=3X_1X_2X_3$ are covered by the integer solutions for most primes $p$. In this paper, we provide upper bounds on lifts of mod $p$ points of the Markoff surface by analyzing the growth along paths in the Markoff mod $p$ graphs. Our first upper bound follows the algorithm given in the paper of Bourgain, Gamburd, and Sarnak, which constructs a path of possibly long length but where points grow relatively slowly. Our second bound considers paths in these graphs of short length but possibly large growth. We then provide numerical evidence and heuristic arguments for how these bounds might be improved.

math.NT↗

arXiv2020

A modular proof of the properness of the Coleman-Mazur eigencurve

We give a new proof of the properness of the Coleman-Mazur eigencurve. The question of whether the eigencurve satisfies the valuative criterion for properness was first asked by Coleman and Mazur in 1998 and settled by Diao and Liu in 2016 using deep, powerful Hodge- and Galois- theoretic machinery. Our proof is short and explicit and uses no Galois theory. Instead we adapt an earlier method of Buzzard and Calegari based on elementary properties of overconvergent modular forms. To facilitate this, we extend Pilloni's geometric construction of overconvergent forms of arbitrary weight farther into the supersingular locus. Along the way, we show that the Hecke operator $U_p$ is injective on spaces of forms of large overconvergence radius of any analytic weight.

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arXiv2020

Slopes in eigenvarieties for definite unitary groups

We generalize bounds of Liu-Wan-Xiao for slopes in eigencurves for definite unitary groups of rank $2$ to slopes in eigenvarieties for definite unitary groups of any rank. We show that for a definite unitary group of rank $n$, the Newton polygon of the characteristic power series of the $U_p$ Hecke operator has exact growth rate $x^{1+\frac2{n(n-1)}}$, times a constant proportional to the distance of the weight from the boundary of weight space. The proof goes through the classification of forms associated to principal series representations. We also give a consequence for the geometry of these eigenvarieties over the boundary of weight space.

math.NT↗

arXiv2018

Extensions of Vector Bundles on the Fargues-Fontaine Curve

We completely classify the possible extensions between semistable vector bundles on the Fargues-Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder-Narasimhan polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze's language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the euclidean geometry of Harder-Narasimhan polygons.

math.NT↗

arXiv2014

Benford's Law for Coefficients of Newforms

Let $f(z)=\sum_{n=1}^\infty λ_f(n)e^{2πi n z}\in S_{k}^{new}(Γ_0(N))$ be a normalized Hecke eigenform of even weight $k\geq2$ on $Γ_0(N)$ without complex multiplication. Let $\mathbb{P}$ denote the set of all primes. We prove that the sequence $\{λ_f(p)\}_{p\in\mathbb{P}}$ does not satisfy Benford's Law in any base $b\geq2$. However, given a base $b\geq2$ and a string of digits $S$ in base $b$, the set \[ A_{λ_f}(b,S):=\{\text{$p$ prime : the first digits of $λ_f(p)$ in base $b$ are given by $S$}\} \] has logarithmic density equal to $\log_b(1+S^{-1})$. Thus $\{λ_f(p)\}_{p\in\mathbb{P}}$ follows Benford's Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.

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arXiv2014

An explicit approach to the Ahlgren-Ono conjecture

Let $p(n)$ be the partition function. Ahlgren and Ono conjectured that every arithmetic progression contains infinitely many integers $N$ for which $p(N)$ is not congruent to $0\pmod{3}$. Radu proved this conjecture in 2010 using work of Deligne and Rapoport. In this note, we give a simpler proof of Ahlgren and Ono's conjecture in the special case where the modulus of the arithmetic progression is a power of $3$ by applying a method of Boylan and Ono and using work of Bellaïche and Khare generalizing Serre's results on the local nilpotency of the Hecke algebra.

math.NT↗

arXiv2014

Congruences of concave composition functions

Concave compositions are ordered partitions whose parts are decreasing towards a central part. We study the distribution modulo $a$ of the number of concave compositions. Let $c(n)$ be the number of concave compositions of $n$ having even length. It is easy to see that $c(n)$ is even for all $n\geq1$. Refining this fact, we prove that $$\#\{n 2$ and at least two distinct values of $r\in\{0,1,\dotsc,a-1\}$, $$\#\{n \frac{\log_2\log_3 X}{a}.$$ We obtain similar results for concave compositions of odd length.

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arXiv2014

Strings of special primes in arithmetic progressions

The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves that there exist arbitrarily long strings of consecutive primes that lie in any arithmetic progression that contains infinitely many primes. Using the techniques of Shiu and Maier, this paper generalizes Shiu's Theorem to certain subsets of the primes such as primes of the form $\lfloor πn\rfloor$ and some of arithmetic density zero such as primes of the form $\lfloor n\log\log n\rfloor$.

math.NT↗

arXiv2014

Congruence properties of Borcherds product exponents

In his striking 1995 paper, Borcherds found an infinite product expansion for certain modular forms with CM divisors. In particular, this applies to the Hilbert class polynomial of discriminant $-d$ evaluated at the modular $j$-function. Among a number of powerful generalizations of Borcherds' work, Zagier made an analogous statement for twisted versions of this polynomial. He proves that the exponents of these product expansions, $A(n,d)$, are the coefficients of certain special half-integral weight modular forms. We study the congruence properties of $A(n,d)$ modulo a prime $\ell$ by relating it to a modular representation of the logarithmic derivative of the Hilbert class polynomial.

math.NT↗

arXiv2014

Bounding sums of the Möbius function over arithmetic progressions

Let $M(x)=\sum_{1\le n\le x}μ(n)$ where $μ$ is the Möbius function. It is well-known that the Riemann Hypothesis is equivalent to the assertion that $M(x)=O(x^{1/2+ε})$ for all $ε>0$. There has been much interest and progress in further bounding $M(x)$ under the assumption of the Riemann Hypothesis. In 2009, Soundararajan established the current best bound of \[ M(x)\ll\sqrt{x}\exp\left((\log x)^{1/2}(\log\log x)^c\right) \] (setting $c$ to $14$, though this can be reduced). Halupczok and Suger recently applied Soundararajan's method to bound more general sums of the Möbius function over arithmetic progressions, of the form \[ M(x;q,a)=\sum_{\substack{n\le x \\ n\equiv a\pmod{q}}}μ(n). \] They were able to show that assuming the Generalized Riemann Hypothesis, $M(x;q,a)$ satisfies \[ M(x;q,a)\ll_ε\sqrt{x}\exp\left((\log x)^{3/5}(\log\log x)^{16/5+ε}\right) \] for all $q\le\exp\left(\frac{\log 2}2\lfloor(\log x)^{3/5}(\log\log x)^{11/5}\rfloor\right)$, with $a$ such that $(a,q)=1$, and $ε>0$. In this paper, we improve Halupczok and Suger's work to obtain the same bound for $M(x;q,a)$ as Soundararajan's bound for $M(x)$ (with a $1/2$ in the exponent of $\log x$), with no size or divisibility restriction on the modulus $q$ and residue $a$.

math.NT↗

arXiv2012

The Proportion of Weierstrass Semigroups

We solve a problem of Komeda concerning the proportion of numerical semigroups which do not satisfy Buchweitz' necessary criterion for a semigroup to occur as the Weierstrass semigroup of a point on an algebraic curve. We also show that the family of semigroups known to be Weierstrass semigroups using a result of Eisenbud and Harris, has zero density in the set of all semigroups. In the process, we prove several more general results about the structure of a typical numerical semigroup.

math.CO↗

arXiv2011

Chomp on Graphs and Subsets

The game subset take-away begins with a simplicial complex Δ. Two players take turns removing any element of Δas well as all other elements which contain it, and the last player able to move wins. Graph Chomp is a special case of subset take-away played on a simplicial complex with only vertices and edges. The game has previously only been analyzed for complete graphs, forest graphs, and very small special cases of higher-dimensional simplicial complexes. We generalize a common method of reducing some game positions to simpler ones by symmetry and provide a complete analysis of complete n-partite graphs for arbitrary n and all bipartite graphs. Finally, we give partial results for odd-cycle pseudotrees, which are non-bipartite graphs with a single cycle.

math.CO↗