Joint distribution for the Selmer ranks of the congruent number curves

summary:We determine the distribution over square-free integers $n$ of the pair $(\dim _{\mathbb {F}_2}{\rm Sel}^\Phi (E_n/\mathbb {Q}),\dim _{\mathbb {F}_2} {\rm Sel}^{\widehat {\Phi }}(E_n'/\mathbb {Q}))$, where $E_n$ is a curve in the congruent number curve family, $E_n'\colon y^2=x^3+4n^2x$ is the image of isogeny $\Phi \colon E_n\rightarrow E_n'$, $\Phi (x,y)=(y^2/x^2,y(n^2-x^2)/x^2)$, and $\widehat {\Phi }$ is the isogeny dual to $\Phi$

Projective center point and Tverberg theorems

We present projective versions of the center point theorem and Tverberg's theorem, interpolating between the original and the so-called "dual" center point and Tverberg theorems. Furthermore we give a common generalization of these and many other known (transversal, constraint, dual, and colorful) Tverberg type results in a single theorem, as well as some essentially new results about partitioning measures in projective space.Comment: 10 page

Combinatorics of unavoidable complexes

© 2019 Elsevier Ltd The partition number π(K) of a simplicial complex K⊆2[n] is the minimum integer k such that for each partition A1⊎…⊎Ak=[n] of [n] at least one of the sets Ai is in K. A complex K is r-unavoidable if π(K)≤r. Simplicial complexes with small π(K) are important for applications of the “constraint method” (Blagojević et al., 2014) and serve as an input for the “index inequalities” (Jojić et al., 2018), such as (1.1). We introduce a “threshold characteristic” ρ(K) of K (Section 3) and define a fractional (linear programming) relaxation of π(K) (Section 4), which allows us to systematically generate interesting examples of r-unavoidable complexes and pave the way for new results of Van Kampen–Flores–Tverberg type