Visibility and the Birch and Swinnerton-Dyer conjecture for analytic rank one

Let $E$ be an optimal elliptic curve over \Q of conductor $N$ having analytic rank one, i.e., such that the $L$-function $L_E(s)$ of $E$ vanishes to order one at $s=1$. Let $K$ be a quadratic imaginary field in which all the primes dividing $N$ split and such that the $L$-function of $E$ over $K$ vanishes to order one at $s=1$. Suppose there is another optimal elliptic curve over \Q of the same conductor $N$ whose Mordell-Weil rank is greater than one and whose associated newform is congruent to the newform associated to $E$ modulo an integer $r$. The theory of visibility then shows that under certain additional hypotheses, $r$ divides the order of the Shafarevich-Tate group of $E$ over $K$. We show that under somewhat similar hypotheses, $r$ divides the order of the Shafarevich-Tate group of $E$ over $K$. We show that under somewhat similar hypotheses, $r$ also divides the Birch and Swinnerton-Dyer {\em conjectural} order of the Shafarevich-Tate group of $E$ over $K$, which provides new theoretical evidence for the second part of the Birch and Swinnerton-Dyer conjecture in the analytic rank one case

A proposed testbed for detector tomography

Measurement is the only part of a general quantum system that has yet to be characterized experimentally in a complete manner. Detector tomography provides a procedure for doing just this; an arbitrary measurement device can be fully characterized, and thus calibrated, in a systematic way without access to its components or its design. The result is a reconstructed POVM containing the measurement operators associated with each measurement outcome. We consider two detectors, a single-photon detector and a photon-number counter, and propose an easily realized experimental apparatus to perform detector tomography on them. We also present a method of visualizing the resulting measurement operators.Comment: 9 pages, 4 figure