Heuristics on Tate-Shafarevitch groups of elliptic curves defined over .
Delaunay, Christophe (2001)
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Delaunay, Christophe (2001)
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We give an asymptotic formula for the number of elliptic curves over ℚ with bounded Faltings height. Silverman (1986) showed that the Faltings height for elliptic curves over number fields can be expressed in terms of modular functions and the minimal discriminant of the elliptic curve. We use this to recast the problem as one of counting lattice points in a particular region in ℝ².
Andrej Dujella, Kálmán Győry, Ákos Pintér (2012)
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Clemens Fuchs, Rafael von Känel, Gisbert Wüstholz (2011)
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Tom Fisher (2015)
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We compute equations for the families of elliptic curves 9-congruent to a given elliptic curve. We use these to find infinitely many non-trivial pairs of 9-congruent elliptic curves over ℚ, i.e. pairs of non-isogenous elliptic curves over ℚ whose 9-torsion subgroups are isomorphic as Galois modules.
Besche, Hans Ulrich, Eick, Bettina, O'Brien, E.A. (2001)
Electronic Research Announcements of the American Mathematical Society [electronic only]
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Delaunay, C., Duquesne, S. (2003)
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