Parity conjectures for elliptic curves over global fields of positive characteristic [PDF]
Compositio Mathematica, 2011 Abstract 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
Trihan, Fabien, Wuthrich, Christian
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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 orders of these cyclic components are invariants of the Tate-Shafarevich group.
Brown, Martin L.
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Densities for Elliptic Curves over Global Function Fields
, 2023 Let $K$ be a global function field. We obtain a set of formulas for the densities of the Kodaira types and Tamagawa numbers of elliptic curves over a completion of $K$ that is independent of the field's characteristic. Furthermore, for a finite field $F$ and real numbers $s$ and $ε$ such that $s>1$ and $ε>0$, we prove that there exists a global ...
Yao, Andrew
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Torsion points on elliptic curves over a global field
Manuscripta Mathematica, 1979 Let C be an elliptic curve defined over a global field K and denote by CK the group of rational points of C over K. The classical Nagell-Lutz-Cassels theorem states, in the case of an algebraic number field K as groud field, a necessary condition for a point in CK to be a torsion point, i.e. a point of finite order.
Horst Günter Zimmer
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Most elliptic curves over global function fields are torsion free [PDF]
Acta Arithmetica, 2022 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 $ρ_{E,n}$. For $K$ satisfying some mild conditions, we show that the set of $E$ for which $ρ_{E,n}$ is as large as possible for all $n$, has density $1$.
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Heegner Points and the Rank of Elliptic Curves over Large Extensions of Global Fields [PDF]
Canadian Journal of Mathematics, 2008 AbstractLetkbe a global field,a separable closure ofk, andGkthe absolute Galois groupofoverk. For everyσ ∈ Gk, letbe the fixed subfield ofunderσ. LetE/kbe an elliptic curve overk. It is known that the Mordell–Weil grouphas infinite rank. We present a new proof of this fact in the following two cases.
Breuer, Florian, Im, Bo-Hae
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International Mathematics Research Notices, 2023 Abstract 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 $\textrm{Gal}(K(E[p])/K)$ contains the special linear group $\textrm{SL}_{2}(\mathbb{F}_{p}
Park, Sun Woo, Wang, Niudun
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First and second
Annales de l'Institut Fourier, 2018 In this paper, we show that the maximal divisible subgroup of groups K 1 and K 2
Satoshi Kondo, Seidai Yasuda
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The Frobenius twists of elliptic curves over global function fields
, 2023 25 pages, 0 ...
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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.
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