Nicholas Triantafillou

Nicholas Triantafillou
Massachusetts Institute of Technology | MIT · Department of Mathematics

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11
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Introduction
Skills and Expertise

Publications

Publications (11)
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We study low-lying zeroes of $L$-functions and their $n$-level density, which relies on a smooth test function $\phi$ whose Fourier transform $\widehat\phi$ has compact support. Assuming the generalized Riemann hypothesis, we compute the $n^\text{th}$ centered moments of the $1$-level density of low-lying zeroes of $L$-functions associated with wei...
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Let K K be a number field with ring of integers O K \mathcal O_{K} . We prove that if 3 3 does not divide [ K : Q ] [K:\mathbb Q] and 3 3 splits completely in K K , then there are no exceptional units in K K . In other words, there are no x , y ∈ O K × x, y \in \mathcal O_{K}^{\times } with x + y = 1 x + y = 1 . Our elementary p p -adic proof is in...
Preprint
Given a smooth, proper, geometrically integral curve $X$ of genus $g$ with Jacobian $J$ over a number field $K$, Chabauty's method is a $p$-adic technique to bound #$ X(K)$ when $\text{rank} J(K) < g$. In this paper, we study limitations of a variant of this approach which we call 'Restriction of Scalars Chabauty' (RoS Chabauty). RoS Chabauty typic...
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We give an introductory account of two recent approaches towards an effective proof of the Mordell conjecture, due to Lawrence--Venkatesh and Kim. The latter method, which is usually called the method of Chabauty--Kim or non-abelian Chabauty in the literature, has the advantage that in some cases it has been turned into an effective method to deter...
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We describe an algorithm to compute the zeta function of a cyclic cover of the projective line over a finite field of characteristic $p$ that runs in time $p^{1/2 + o(1)}$. We confirm its practicality and effectiveness by reporting on the performance of our SageMath implementation on a range of examples. The algorithm relies on Gon\c{c}alves's gene...
Article
In this paper, we present efficient algorithms for computing the number of points and the order of the Jacobian group of a superelliptic curve over finite fields of prime order p. Our method employs the Hasse-Weil bounds in conjunction with the Hasse-Witt matrix for superelliptic curves, whose entries we express in terms of multinomial coefficients...
Chapter
A set A ⊆ [n] ∪ {0} is said to be a 2-additive basis for [n] if each j ∈ [n] can be written as j = x + y, x, y ∈ A, x ≤ y. If we pick each integer in [n] ∪ {0} independently with probability p = pn → 0, thus getting a random set A, what is the probability that we have obtained a 2-additive basis? We address this question when the target sum-set is...
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The limiting distribution of eigenvalues of N x N random matrices has many applications. One of the most studied ensembles are real symmetric matrices with independent entries iidrv; the limiting rescaled spectral measure (LRSM) $\widetilde{\mu}$ is the semi-circle. Studies have determined the LRSMs for many structured ensembles, such as Toeplitz a...
Article
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A subset A of {0,1,...,n} is said to be a 2-additive basis for {1,2,...,n} if each j in {1,2,...,n} can be written as j=x+y, x,y in A, x<=y. If we pick each integer in {0,1,...,n} independently with probability p=p_n tending to 0, thus getting a random set A, what is the probability that we have obtained a 2-additive basis? We address this question...
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An {\it omnimosaic} $O(n,k,a)$ is defined to be an $n\times n$ matrix, with entries from the set ${\cal A}=\{1,2,\...,a\}$, that contains, as a submatrix, each of the $a^{k^2}$ $k\times k$ matrices over ${\cal A}$. We provide constructions of omnimosaics and show that for fixed $a$ the smallest possible size $\omega(k,a)$ of an $O(n,k,a)$ omnimosai...

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