Structure of Tate–Shafarevich groups of elliptic curves over global function fields [PDF]
Kyoto Journal of Mathematics, 2015 The structure of the Tate-Shafarevich groups of a class of elliptic curves over global function fields is determined. These are known to be finite abelian groups from the monograph [1] and hence they are direct sums of finite cyclic groups where the ...
Martin L. Brown
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Parity conjectures for elliptic curves over global fields of positive characteristic [PDF]
Compositio Mathematica, 2010 We prove the p-parity conjecture for elliptic curves over global fields of characteristic p>3. We also present partial results on the ℓ-parity conjecture for primes ℓ≠p.
Fabien Trihan, C. Wuthrich
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On ring class eigenspaces of Mordell–Weil groups of elliptic curves over global function fields [PDF]
Journal of Number Theory, 2008 If E is a non-isotrivial elliptic curve over a global function field F of odd characteristic we show that certain Mordell–Weil groups of E have 1-dimensional χ-eigenspace (with χ a complex ring class character) provided that the projection onto this ...
S. Vigni
exaly +7 more sources
Most elliptic curves over global function fields are torsion free [PDF]
Acta Arithmetica, 2021 Given an elliptic curve $E$ over a global function field $K$, the Galois action on the $n$-torsion points of $E$ gives rise to a mod-n Galois representation $\rho_{E,n}$.
T. Phillips
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On the prime Selmer ranks of cyclic prime twist families of elliptic curves over global function fields [PDF]
Compositio Mathematica, 2022 Fix a prime number p. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2 and 3, and containing the primitive pth root of unity $\mu_p$ .
S. Park
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Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields [PDF]
Canadian Journal of Mathematics, 2006 Let $k$ be a global field, $\bar{k}$ a separable closure of $k$ , and ${{G}_{k}}$ the absolute Galois group Gal $(\bar{k}/k)$ of $\bar{k}$ over $k$ .
Florian Breuer, Bo-Hae Im
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International Mathematics Research Notices, 2023 Let $\mathbb{F}_{q}$ be a finite field whose characteristic is relatively prime to $2$ and $3$. Let $p$ be a prime number that is coprime to $q$. Let $E$ be an elliptic curve over the global function field $K = \mathbb{F}_{q}(t)$ such that $\textrm{Gal}
S. Park, Niudun Wang
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Local-global aspects of (hyper)elliptic curves over (in)finite fields
Advances in Mathematics of Communications, 2010 We survey the interaction between local and global theory for studying the arithmetic properties of curves, jacobians, and abelian varieties.
J. Silverman
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Some remarks concerning points of finite order on elliptic curves over global fields
Arkiv för Matematik, 1977 Using the reduction theory of Nrron we give necessary conditions for the existence of points of order q on elliptic curves E rational over global fields. An application is the determination of all elliptic cu rves /Q with integer j and torsion points, generalizing Olson [8]. Another application is a theorem about semistable reduction whose consequences
G. Frey
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The local-global principle for divisibility in CM elliptic curves [PDF]
Journal of Number Theory, 2022 We consider the local-global principle for divisibility in the Mordell-Weil group of a CM elliptic curve defined over a number field. For each prime $p$ we give sharp lower bounds on the degree $d$ of a number field over which there exists a CM elliptic ...
Brendan Creutz, Lu Sheng
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