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“Numbers and functions” – Memorial conference for 80th birthday of Alexey Nikolaevich Parshin
November 29, 2022 17:00–17:50, Moscow, Steklov Mathematical Institute of RAS, 8, Gubkina str., room 104
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Non-isogenous elliptic curves (Zoom)
Yu. G. Zarhin |
Number of views: |
This page: | 145 | Materials: | 14 |
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Abstract:
Let $E_f: y^2=f(x)$ and $E_h: y^2=h(x)$ be elliptic curves over a field $K$ of characteristic zero that are defined by cubic polynomials
$f(x)$ and $h(x)$ with coefficients in $K$.
Suppose that one of the polynomials is irreducible and the other reducible.
We prove that if $E_f$ and $E_h$ are isogenous over an algebraic closure $\bar{K}$ of $K$ then they both are isogenous over $\bar{K}$
to the elliptic curve
$$y^2=x^3-1.$$
References.
[1] Yu. G. Zarhin, Homomorphisms of hyperelliptic jacobians. Trudy Math. Inst.
Steklova 241 (2003), 79–92; Proc. Steklov Institute of Mathematics 241
(2003), 90–104.
[2] Yu. G. Zarhin, Non-isogenous superelliptic jacobians. Math. Z. 253 (2006),
537–554.
[3] Yu. G. Zarhin, Non-isogenous elliptic curves and
hyperelliptic jacobians. Math Research Letters, to appear; arXiv:2105.03783 [math.NT].
[4] Yu. G. Zarhin, Non-isogenous elliptic curves and
hyperelliptic jacobians. II. MPIM preprint series 29 (2022); arXiv:2204.10567 [math.NT].
Supplementary materials:
Zarhin_slides.pdf (1.6 Mb)
Language: English
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