For a prime p and a rational elliptic curve $E_{/\mathbb {Q}}$, set $K=\mathbb {Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname {ker}\{E\xrightarrow {p} E\}$. The class group $\operatorname {Cl}_K$ is a module over $\operatorname {Gal}(K/\mathbb {Q})$. Given a fixed odd prime number p, we study the average nonvanishing of certain Galois stable quotients of the mod-p class group $\operatorname {Cl}_K/p\operatorname {Cl}_K$. Here, E varies over all rational elliptic curves, ordered according to height. Our results are conditional, since we assume that the p-primary part of the Tate–Shafarevich group is finite. Furthermore, we assume predictions made by Delaunay for the statistical variation of the p-primary parts of Tate–Shafarevich groups. We also prove results in the case when the elliptic curve $E_{/\mathbb {Q}}$ is fixed and the prime p is allowed to vary.

We prove the μ-part of the main conjecture for modular forms along the anticyclotomic Zp-extension of a quadratic imaginary field. Our proof consists of first giving an explicit formula for the algebraic μ-invariant, and then using results of Ribet and Takahashi showing that our formula agrees with Vatsal’s formula for the analytic μ-invariant.

Measurements of the release of dissolved organic nitrogen (DON) and carbon (DOC) were carried out on an axenic batch culture of the coccolithophorid Emiliania huxleyi. This unicellular marine alga was cultured using a media with nitrate as the sole N source and the changes of DOM concentrations measured over 14 days. Results showed that there was a significant release of DON, i.e.7.6 µM N day−1 during mid-exponential growth phase (days 5–7). The highest release of DOC was also recorded in the same growth phase and accounted for 24.0 μM C day−1.

Let $\rho :\,{{G}_{Q}}\,\to \,\text{G}{{\text{L}}_{n}}\left( {{Q}_{\ell }} \right)$ be a motivic $\ell $ -adic Galois representation. For fixed $m\,>\,1$ we initiate an investigation of the density of the set of primes $p$ such that the trace of the image of an arithmetic Frobenius at $p$ under $\rho $ is an $m$ -th power residue modulo $p$ . Based on numerical investigations with modular forms we conjecture (with Ramakrishna) that this density equals $1/m$ whenever the image of $\rho $ is open. We further conjecture that for such $\rho $ the set of these primes $p$ is independent of any set defined by Cebatorev-style Galois-theoretic conditions (in an appropriate sense). We then compute these densities for certain $m$ in the complementary case of modular forms of $\text{CM}$ -type with rational Fourier coefficients; our proofs are a combination of the Cebatorev density theorem (which does apply in the $\text{CM}$ case) and reciprocity laws applied to Hecke characters. We also discuss a potential application (suggested by Ramakrishna) to computing inertial degrees at $p$ in abelian extensions of imaginary quadratic fields unramified away from $p$ .

In this paper, we construct a theory of geometric Euler systems, complementary to the arithmetic theory of Rubin, Kato and Perrin-Riou. We show that geometric Euler systems can be used to prove the finiteness of certain Galois representations of weight zero and we discuss a conjectural framework for the existence of geometric Euler systems for motivic Galois representations. We give applications to adjoint Selmer groups of certain classical and Drinfeld modular forms.