Zobrazeno 1 - 10
of 187
pro vyhledávání: '"Rouse, Jeremy"'
Autor:
Daniels, Harris, Rouse, Jeremy
Let $E$ be an elliptic curve over a number field $L$ and for a finite set $S$ of primes, let $\rho_{E,S} : {\rm Gal}(\overline{L}/L) \to {\rm GL}_{2}(\mathbb{Z}_{S})$ be the $S$-adic Galois representation. If $L \cap \mathbb{Q}(\zeta_{n}) = \mathbb{Q
Externí odkaz:
http://arxiv.org/abs/2402.11049
Let $f(t_1,\ldots,t_n)$ be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function $D_f(X)$, the number of integers of absolute value up to $X$ represented by $f$. When $f$ is isotropic or $n$ is at least $3$, we s
Externí odkaz:
http://arxiv.org/abs/2304.07399
Autor:
Rouse, Jeremy, Thompson, Katherine
Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on the Peterss
Externí odkaz:
http://arxiv.org/abs/2206.00412
Autor:
Lang, Xiaoan, Rouse, Jeremy
We study the problem of determining, given an integer $k$, the rational solutions to $C_{k} : x^{3}z + x^{2} y^{2} + y^{3}z = kz^{4}$. For $k \ne 0$, the curve $C_{k}$ has genus $3$ and there are maps from $C_{k}$ to three elliptic curves $E_{1,k}$,
Externí odkaz:
http://arxiv.org/abs/2205.13442
Publikováno v:
Forum Math. Sigma 10 (2022), Paper No. e62, 63 pp
We discuss the $\ell$-adic case of Mazur's "Program B" over $\mathbb{Q}$, the problem of classifying the possible images of $\ell$-adic Galois representations attached to elliptic curves $E$ over $\mathbb{Q}$, equivalently, classifying the rational p
Externí odkaz:
http://arxiv.org/abs/2106.11141
Publikováno v:
In Journal of Number Theory March 2024 256:290-328
Autor:
Newton, Alexis, Rouse, Jeremy
We prove that $164634913$ is the smallest positive integer that is a sum of two rational sixth powers but not a sum of two integer sixth powers. If $C_{k}$ is the curve $x^{6} + y^{6} = k$, we use the existence of morphisms from $C_{k}$ to elliptic c
Externí odkaz:
http://arxiv.org/abs/2101.09390
Let $C$ be a curve defined over a number field $k$. We say a closed point $x\in C$ of degree $d$ is isolated if it does not belong to an infinite family of degree $d$ points parametrized by the projective line or a positive rank abelian subvariety of
Externí odkaz:
http://arxiv.org/abs/2006.14966
Autor:
Holley-Reid, John, Rouse, Jeremy
Publikováno v:
Involve 16 (2023) 727-735
Let $r_{k}(n)$ denote the number of representations of the integer $n$ as a sum of $k$ squares. In this paper, we give an asymptotic for $r_{k}(n)$ when $n$ grows linearly with $k$. As a special case, we find that \[ r_{n}(n) \sim \frac{B \cdot A^{n}
Externí odkaz:
http://arxiv.org/abs/1910.01001
Autor:
Cerchia, Michael, Rouse, Jeremy
Let $\ell$ be a prime number and let $F$ be a number field and $E/F$ a non-CM elliptic curve with a point $\alpha \in E(F)$ of infinite order. Attached to the pair $(E,\alpha)$ is the $\ell$-adic arboreal Galois representation $\omega_{E,\alpha,\ell^
Externí odkaz:
http://arxiv.org/abs/1909.07468