21 research outputs found
Abelian varieties over Q with bad reduction in one prime only
Let l be a prime. We show that there do not exist any non-zero semi-stable
abelian varieties over Q with good reduction outside l if and only if l=2, 3,
5, 7 or 13. We show that any semi-stable abelian variety over Q with good
reduction outside l=11 is isogenous to a power of the Jacobian variety of the
modular curve X_0(11). In addition we show that there do not exist any non-zero
abelian varieties over Q with good reduction outside l acquiring semi-stable
reduction at l over a tamely ramified extension if and only if l \le 5
Four primality testing algorithms
In this expository paper we describe four primality tests. The first test is
very efficient, but is only capable of proving that a given number is either
composite or 'very probably' prime. The second test is a deterministic
polynomial time algorithm to prove that a given numer is either prime or
composite. The third and fourth primality tests are at present most widely used
in practice. Both tests are capable of proving that a given number is prime or
composite, but neither algorithm is deterministic. The third algorithm exploits
the arithmetic of cyclotomic fields. Its running time is almost, but not quite
polynomial time. The fourth algorithm exploits elliptic curves. Its running
time is difficult to estimate, but it behaves well in practice.Comment: 21 page
Class groups of real cyclotomic fields
We prove that every finite abelian group G occurs as a subgroup of the class
group of infinitely many real cyclotomic fields.Comment: References have been updated and some typos have been corrected. To
appear in Monatshefte f\"ur Mathemati
miR-126 Regulates Distinct Self-Renewal Outcomes in Normal and Malignant Hematopoietic Stem Cells
SummaryTo investigate miRNA function in human acute myeloid leukemia (AML) stem cells (LSC), we generated a prognostic LSC-associated miRNA signature derived from functionally validated subpopulations of AML samples. For one signature miRNA, miR-126, high bioactivity aggregated all in vivo patient sample LSC activity into a single sorted population, tightly coupling miR-126 expression to LSC function. Through functional studies, miR-126 was found to restrain cell cycle progression, prevent differentiation, and increase self-renewal of primary LSC in vivo. Compared with prior results showing miR-126 regulation of normal hematopoietic stem cell (HSC) cycling, these functional stem effects are opposite between LSC and HSC. Combined transcriptome and proteome analysis demonstrates that miR-126 targets the PI3K/AKT/MTOR signaling pathway, preserving LSC quiescence and promoting chemotherapy resistance
Computing Arakelov class groups
Shanks’s infrastructure algorithm and Buchmann’s algorithm for computing class groups and unit groups of rings of integers of algebraic number fields are most naturally viewed as computations inside Arakelov class groups. In this paper we discuss the basic properties of Arakelov class groups and of the set of reduced Arakelov divisors. As an application we describe Buchmann’s algorithm in this context