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Showing 1 to 5 of 5 for “"Quadratic twists"”.
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Distribution of Selmer Groups of Quadratic Twists of a Family of Elliptic Curves
As an most interesting example for the elliptic curve E n : y2 = x 3 - n2x, which is closely related with the congruent number problem, we study the distribution of the size of the six Selmer groups arising from the three 2-isogenies and their dual 2-isogenies. We also describe explicit formulas on …
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Explicit moduli spaces for curves of genus 1 and 2
… or $24$. We then study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all, cases the modular curves in question correspond to the …
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L-functions of twisted elliptic curves over function fields
… the L-function of an elliptic curve and its twists over the function field of the projective line over a finite field. This method requires computing the number of points on an elliptic curve over a finite field, for which we present a novel algorithm. If the j-invariant of an elliptic curve …
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On congruence function fields with many rational places
… arising from a $2$-isogeny for a family of quadratic twists of the elliptic curves with full $2$-torsions over the rational function field $\mathbb{F}_q(x)$ for odd $q$. In particular, we show that the sizes of these Selmer groups are almost always bounded. The calculation relies heavily on …
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On the main conjectures of Iwasawa theory for certain elliptic curves with complex multiplication
… be an elliptic curve defined over an imaginary quadratic field $K$ contained in $\mathbb{C}$, and suppose that $E$ has complex multiplication by the ring of integers of $K$. Let us assume the complex $L$-series $L(E/K,s)$ of $E$ over $K$ does not vanish at $s=1$. K. Rubin showed, using Iwasawa …