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