arXiv subjects

Joseph L. Wetherell

Publications and source records attributed to Joseph L. Wetherell.

5 recordsLinked to original sources

arXiv2020

Linear independence in linear systems on elliptic curves

Let $E$ be an elliptic curve, with identity $O$, and let $C$ be a cyclic subgroup of odd order $N$, over an algebraically closed field $k$ with $\operatorname{char} k \nmid N$. For $P \in C$, let $s_P$ be a rational function with divisor $N \cdot P - N \cdot O$. We ask whether the $N$ functions $s_P$ are linearly independent. For generic $(E,C)$, we prove that the answer is yes. We bound the number of exceptional $(E,C)$ when $N$ is a prime by using the geometry of the universal generalized elliptic curve over $X_1(N)$. The problem can be recast in terms of sections of an arbitrary degree $N$ line bundle on $E$.

math.NT↗

arXiv2008

Polynomials with a common composite

Let f and g be nonconstant polynomials over an arbitrary field K. In this paper we study the intersection of the polynomial rings K[f] and K[g], and in particular we ask whether this intersection is larger than K. We completely resolve this question when K has characteristic zero, and in positive characteristic we present various results, examples, and algorithms.

math.AG↗

arXiv2002

Curves of every genus with many points, II: Asymptotically good families

We resolve a 1983 question of Serre by constructing curves with many points of every genus over every finite field. More precisely, we show that for every prime power q there is a positive constant c_q with the following property: for every non-negative integer g, there is a genus-g curve over F_q with at least c_q * g rational points over F_q. Moreover, we show that there exists a positive constant d such that for every q we can choose c_q = d * (log q). We show also that there is a constant c > 0 such that for every q and every n > 0, and for every sufficiently large g, there is a genus-g curve over F_q that has at least c*g/n rational points and whose Jacobian contains a subgroup of rational points isomorphic to (Z/nZ)^r for some r > c*g/n.

math.AG↗

arXiv2001

Curves of every genus with many points, I: Abelian and toric families

Let N_q(g) denote the maximal number of F_q-rational points on any curve of genus g over the finite field F_q. Ihara (for square q) and Serre (for general q) proved that limsup_{g-->infinity} N_q(g)/g > 0 for any fixed q. In their proofs they constructed curves with many points in infinitely many genera; however, their sequences of genera are somewhat sparse. In this paper, we prove that lim_{g-->infinity} N_q(g) = infinity. More precisely, we use abelian covers of P^1 to prove that liminf_{g-->infinity} N_q(g)/(g/log g) > 0, and we use curves on toric surfaces to prove that liminf_{g-->infty} N_q(g)/g^{1/3} > 0; we also show that these results are the best possible that can be proved with these families of curves.

math.AG↗

arXiv2000

On the genera of X_0(N)

Let g_0(N) be the genus of the modular curve X_0(N). We record several properties of the sequence {g_0(N)}. Even though the average size of g_0(N) is 1.25N/pi^2, a random positive integer has probability zero of being a value of g_0(N). Also, if N is a random positive integer then g_0(N) is odd with probability one. The smallest non-negative integer not occuring as g_0(N) for any N is 150; the smallest such odd integer is 49267.

math.NT↗