
Nicholas TriantafillouMassachusetts Institute of Technology | MIT · Department of Mathematics
Nicholas Triantafillou
About
11
Publications
600
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
40
Citations
Introduction
Skills and Expertise
Publications
Publications (11)
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...