4,441 research outputs found
The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point
In recent work, Bhargava and Shankar have shown that the average size of the
-Selmer group of an elliptic curve over is exactly , and
Bhargava and Ho have shown that the average size of the -Selmer group in the
family of elliptic curves with a marked point is exactly . In contrast to
these results, we show that the average size of the -Selmer group in the
family of elliptic curves with a two-torsion point is unbounded. In particular,
the existence of a two-torsion point implies the existence of rational isogeny.
A fundamental quantity attached to a pair of isogenous curves is the Tamagawa
ratio, which measures the relative sizes of the Selmer groups associated to the
isogeny and its dual. Building on previous work in which we considered the
Tamagawa ratio in quadratic twist families, we show that, in the family of all
elliptic curves with a two-torsion point, the Tamagawa ratio is essentially
governed by a normal distribution with mean zero and growing variance
When Winning is the Only Thing:Â Pure Strategy Nash Equilibria in a Three-Candidate Spatial Voting Model
It is well-known that there are no pure strategy Nash equilibria (PSNE) in the standard three-candidate spatial voting model when candidates maximize their share of the vote. When all that matters to the candidates is winning the election, however, we show that PSNE do exist. We provide a complete characterization of such equilibria and then extend our results to elections with an arbitrary number of candidates. Finally, when two candidates face the potential entrant of a third, we show that PSNE no longer exist, however, they do exist when the number of existing candidates is at least three.Voting, spatial equilibrium, location models, entry.
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