Linear independence in linear systems on elliptic curves
| Autor: | Bradley W. Brock, A. J. Scholl, Bjorn Poonen, Joseph L. Wetherell, Bruce W. Jordan |
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| Přispěvatelé: | Apollo - University of Cambridge Repository |
| Rok vydání: | 2021 |
| Předmět: |
Mathematics - Number Theory
Degree (graph theory) Divisor General Mathematics torsion point Order (ring theory) Rational function Combinatorics Mathematics - Algebraic Geometry Elliptic curve Line bundle 14H52 (Primary) 14G35 (Secondary) FOS: Mathematics linear system Linear independence Number Theory (math.NT) Algebraically closed field Algebraic Geometry (math.AG) Mathematics |
| Popis: | 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 $E$. Comment: 10 pages |
| Databáze: | OpenAIRE |
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