Using the theory of elliptic curves, we show that the class number hðÀpÞ of the field Qð ffiffiffiffiffiffi ffi Àp p Þ appears in the count of certain factors of the Legendre polynomials P m ðxÞ ðmod pÞ; where p is a prime 43 and m has the form ðp À eÞ=k; with k ¼ 2; 3 or 4 and p e ðmod kÞ: As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y 2 þ axy þ y ¼ x 3 and find an elementary expression for the supersingular polynomial ss p ðxÞ whose roots are the supersingular j-invariants of elliptic curves in characteristic p: As a corollary we show that the class number hðÀpÞ also shows up in the factorization ðmod pÞ of certain Jacobi polynomials.
Perturbations in the concentration of a specific protein are often used to study and control biological networks. The ability to "dial-in" and programmatically control the concentration of a desired protein in cultures of cells would be transformative for applications in research and biotechnology. We developed a culturing apparatus and feedback control scheme which, in combination with an optogenetic system, allows us to generate defined perturbations in the intracellular concentration of a specific protein in microbial cell culture. As light can be easily added and removed, we can control protein concentration in culture more dynamically than would be possible with long-lived chemical inducers. Control of protein concentration is achieved by sampling individual cells from the culture apparatus, imaging and quantifying protein concentration, and adjusting the inducing light appropriately. The culturing apparatus can be operated as a chemostat, allowing us to precisely control microbial growth and providing cell material for downstream assays. We illustrate the potential for this technology by generating fixed and time-varying concentrations of a specific protein in continuous steady-state cultures of the model organism Saccharomyces cerevisiae. We anticipate that this technology will allow for quantitative studies of biological networks as well as external tuning of synthetic gene circuits and bioprocesses.
Wellesley, Mass.) 1. Introduction.Iterating polynomial maps gives a convenient way of finding extensions of Q whose Galois groups are subgroups of special imprimitive groups known as wreath products, as has been shown by Odoni ([o1], [o2]). Subgroups of wreath products occur as Galois groups not only for the iterates studied by Odoni, but also for the algebraic number fields generated by the periodic points of a polynomial map (see [m1] and [vh]). In particular, studying periodic points of iterated maps over a number field leads naturally to some parametrized families of polynomials with special Galois groups. One of the purposes of this paper is to illustrate this by investigating the algebraic and number-theoretic properties of periodic points of order 3 of a quadratic map over an arbitrary field κ whose characteristic is different from 2. The investigation shows that the arithmetic properties of these periodic points are related to an interesting curve of genus 4. This is part of a larger project to study the Galois groups of periodic points of arbitrary polynomial maps. (See [m1] and [pa].) Thus, let σ(x) be a polynomial over a field κ, and denote by σ n (x) the n-fold iteration of σ with itself:
New explicit formulas are given for the supersingular polynomial ss p (t) and the Hasse invariantĤ p (E) of an elliptic curve E in characteristic p. These formulas are used to derive identities for the Hasse invariants of elliptic curves E n in Tate normal form with distinguished points of order n. This yields a proof that H (E 4 ) andĤ (E 5 ) are projective invariants (mod p) for the octahedral group and the icosahedral group, respectively; and that the set of fourth roots λ 1/4 of supersingular parameters of the Legendre normal form Y 2 = X(X −1)(X −λ) in characteristic p has octahedral symmetry. For general n 4, the field of definition of a supersingular E n is determined, along with the field of definition of the points of order n on E n .
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