Rational Equivalences on Products of Elliptic Curves in a Family

Given a pair of elliptic curves $E_1,E_2$ over a field $k$, we have a natural map $\text{CH}^1(E_1)_0\otimes\text{CH}^1(E_2)_0\to\text{CH}^2(E_1\times E_2)$, and a conjecture due to Beilinson predicts that the image of this map is finite when $k$ is a number field. We construct a $2$-parameter family of elliptic curves that can be used to produce examples of pairs $E_1,E_2$ where this image is finite. The family is constructed to guarantee the existence of a rational curve passing through a specified point in the Kummer surface of $E_1\times E_2$.Comment: 12 pages, 1 figure, 2 tables. Organizational changes, expanded some comments into lemmas. Content is identical to the published version, aside from the organization of information in the Appendi

Local and local-to-global Principles for zero-cycles on geometrically Kummer K3K3 surfaces

Let $X$ be a $K3$ surface over a $p$-adic field $k$ such that for some abelian surface $A$ isogenous to a product of two elliptic curves, there is an isomorphism over the algebraic closure of $k$ between $X$ and the Kummer surface associated to $A$. Under some assumptions on the reduction types of the elliptic curve factors of $A$, we prove that the Chow group $A_0(X)$ of zero-cycles of degree $0$ on $X$ is the direct sum of a divisible group and a finite group. This proves a conjecture of Raskind and Spiess and of Colliot-Th\'{e}l\`{e}ne and it is the first instance for $K3$ surfaces when this conjecture is proved in full. This class of $K3$'s includes, among others, the diagonal quartic surfaces. In the case of good ordinary reduction we describe many cases when the finite summand of $A_0(X)$ can be completely determined. Using these results, we explore a local-to-global conjecture of Colliot-Th\'{e}lene, Sansuc, Kato and Saito which, roughly speaking, predicts that the Brauer-Manin obstruction is the only obstruction to Weak Approximation for zero-cycles. We give examples of Kummer surfaces over a number field $F$ where the ramified places of good ordinary reduction contribute nontrivially to the Brauer set for zero-cycles of degree $0$ and we describe cases when an unconditional local-to-global principle can be proved, giving the first unconditional evidence for this conjecture in the case of $K3$ surfaces.Comment: 27 page

The effect of photobleaching on bee (Hymenoptera: Apoidea) setae color and its implications for studying aging and behavior

Historically, bee age has been estimated using measurements of wing wear and integument color change.  These measurements have been useful in studies of foraging ecology and plant-pollinator interactions.  Wing wear is speculated to be affected by the behaviors associated with foraging, nesting, and mating activities.  Setal color change may be an additional parameter used to measure bee age if it is affected by sun exposure during these same activities.  The objectives of this study were to experimentally assess the effect of direct sun exposure on setal color, unicellular hair-like processes of the integument, and determine whether wing wear and integument photobleaching are correlated.  To quantify photobleaching of setae, we measured changes in hue of lab-reared Bombus huntii Greene (Apidae) exposed to natural sunlight.  We found that sun exposure was a significant variable in determining setal bleaching.  To assess the relationship between wing wear and setal photobleaching, we scored wing wear and measured setal hue of B. huntii, Melecta pacifica fulvida Cresson (Apidae), and Osmia integra Cresson (Megachilidae) from museum specimens.  Wing wear and setal hue values were positively correlated for all three species; however, the strength of the relationship varies across bee species as indicated by correlation coefficient estimates.  Our results suggest that setal color change is affected by sun exposure, and is likely an accurate estimate of bee age.  We suggest that future investigations of bee aging consider a suite of morphometric characteristics due to differences in natural history and sociobiology that may be confounded by the use of a single characteristic