### A characterization of integral elliptic automorphic forms

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Let $f$ be a weight $k$ holomorphic automorphic form with respect to ${\mathrm{\Gamma}}_{0}\left(N\right)$. We prove a sufficient condition for the integrality of $f$ over primes dividing $N$. This condition is expressed in terms of the values at particular $CM$ curves of the forms obtained by iterated application of the weight $k$ Maaß operator to $f$ and extends previous results of the Author.

We study the Ekedahl-Oort stratification on moduli spaces of PEL type. The strata are indexed by the classes in a Weyl group modulo a subgroup, and each class has a distinguished representative of minimal length. The main result of this paper is that the dimension of a stratum equals the length of the corresponding Weyl group element. We also discuss some explicit examples.

We give a survey of computational class field theory. We first explain how to compute ray class groups and discriminants of the corresponding ray class fields. We then explain the three main methods in use for computing an equation for the class fields themselves: Kummer theory, Stark units and complex multiplication. Using these techniques we can construct many new number fields, including fields of very small root discriminant.

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In previous articles, we showed that for number fields in a certain large class, there are at most elliptic points on a Shimura curve of Γ₀(p)-type for every sufficiently large prime number p. In this article, we obtain an effective bound for such p.

Let $f$ be an elliptic modular form level of N. We present a criterion for the integrality of $f$ at primes not dividing N. The result is in terms of the values at CM points of the forms obtained applying to $f$ the iterates of the Maaß differential operators.

We study the Iwasawa theory of a CM elliptic curve $E$ in the anticyclotomic ${\mathbf{Z}}_{p}$-extension of the CM field, where $p$ is a prime of good, ordinary reduction for $E$. When the complex $L$-function of $E$ vanishes to even order, Rubin’s proof of the two variable main conjecture of Iwasawa theory implies that the Pontryagin dual of the $p$-power Selmer group over the anticyclotomic extension is a torsion Iwasawa module. When the order of vanishing is odd, work of Greenberg show that it is not a torsion module. In...

One can define class invariants for a quartic primitive CM field $K$ as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to $K$. Such constructions were given by de Shalit-Goren and Lauter. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct $S$-units in certain abelian extensions of a reflex field of $K$, where $S$ is effectively determined by $K$, and to bound the primes appearing...