Pretentiously detecting power cancellation

Granville and Soundararajan have recently introduced the notion of pretentiousness in the study of multiplicative functions of modulus bounded by 1, essentially the idea that two functions which are similar in a precise sense should exhibit similar behavior. It turns out, somewhat surprisingly, that this does not directly extend to detecting power cancellation - there are multiplicative functions which exhibit as much cancellation as possible in their partial sums that, modified slightly, give rise to functions which exhibit almost as little as possible. We develop two new notions of pretentiousness under which power cancellation can be detected, one of which applies to a much broader class of multiplicative functions

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 $2$-Selmer group of an elliptic curve over $\mathbb{Q}$ is exactly $3$, and Bhargava and Ho have shown that the average size of the $2$-Selmer group in the family of elliptic curves with a marked point is exactly $6$. In contrast to these results, we show that the average size of the $2$-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: 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.