A Linear “Microscope” for Interventions and Counterfactuals

This note illustrates, using simple examples, how causal questions of non-trivial character can be represented, analyzed and solved using linear analysis and path diagrams. By producing closed form solutions, linear analysis allows for swift assessment of how various features of the model impact the questions under investigation. We discuss conditions for identifying total and direct effects, representation and identification of counterfactual expressions, robustness to model misspecification, and generalization across populations.

THE HOLOMORPHY CONJECTURE FOR NONDEGENERATE SURFACE SINGULARITIES

The holomorphy conjecture roughly states that Igusa’s zeta function associated to a hypersurface and a character is holomorphic on$\mathbb{C}$whenever the order of the character does not divide the order of any eigenvalue of the local monodromy of the hypersurface. In this article, we prove the holomorphy conjecture for surface singularities that are nondegenerate over$\mathbb{C}$with respect to their Newton polyhedron. In order to provide relevant eigenvalues of monodromy, we first show a relation between the normalized volumes (which appear in the formula of Varchenko for the zeta function of monodromy) of the faces in a simplex in arbitrary dimension. We then study some specific character sums that show up when dealing with false poles. In contrast to the context of the trivial character, we here need to show fakeness of certain candidate poles other than those contributed by$B_{1}$-facets.

THE IWASAWA MAIN CONJECTURE FOR HILBERT MODULAR FORMS

Following the ideas and methods of a recent work of Skinner and Urban, we prove the one divisibility of the Iwasawa main conjecture for nearly ordinary Hilbert modular forms under certain local hypotheses. As a consequence, we prove that for a Hilbert modular form of parallel weight, trivial character, and good ordinary reduction at all primes dividing$p$, if the central critical$L$-value is zero then the$p$-adic Selmer group of it has rank at least one. We also prove that one of the local assumptions in the main result of Skinner and Urban can be removed by a base-change trick.

Linear Models: A Useful “Microscope” for Causal Analysis

This note reviews basic techniques of linear path analysis and demonstrates, using simple examples, how causal phenomena of non-trivial character can be understood, exemplified and analyzed using diagrams and a few algebraic steps. The techniques allow for swift assessment of how various features of the model impact the phenomenon under investigation. This includes: Simpson’s paradox, case–control bias, selection bias, missing data, collider bias, reverse regression, bias amplification, near instruments, and measurement errors.

The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey–Jarden Conjecture

Frey and Jarden asked if any abelian variety over a number field K has the infinite Mordell–Weil rank over the maximal abelian extension Kab. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve C over K such that #C(Kab) = ∞ and for any abelian variety of GL2-type with trivial character.

CRISES AS A DISEASE OF THE BODY POLITICK. A METAPHOR IN THE HISTORY OF NINETEENTH-CENTURY ECONOMICS

This paper examines the use of the medical metaphor in the early theories of crises. It first considers the borrowing of medical terminology and generic references to disease, which, notwithstanding their relatively trivial character, illustrate how crises were originally conceived as disturbances (often of a political nature) to a naturally healthy system. Then it shows how a more specific metaphor, the fever of speculation, shifted the emphasis by treating prosperity as the diseased phase, to which crises are a remedy. The metaphor of the epidemic spreading of the disease introduced the theme of the cumulative character of both upswing and downswing, while the similitude with intermittent fevers accounted for the recurring nature of crises. Finally, the paper examines how the medical reflections on the causality of diseases contributed to the epistemology of crises theory, and reflects on the metaphysical shift accompanying the transition from the theories of crises to the theories of cycles.

CONSTRUCTING SIMULTANEOUS HECKE EIGENFORMS

It is well known that newforms of integral weight are simultaneous eigenforms for all the Hecke operators, and that the converse is not true. In this paper, we give a characterization of all simultaneous Hecke eigenforms associated to a given newform, and provide several applications. These include determining the number of linearly independent simultaneous eigenforms in a fixed space which correspond to a given newform, and characterizing several situations in which the full space of cusp forms is spanned by a basis consisting of such eigenforms. Part of our results can be seen as a generalization of results of Choie–Kohnen who considered diagonalization of "bad" Hecke operators on spaces with square-free level and trivial character. Of independent interest, but used herein, is a lower bound for the dimension of the space of newforms with arbitrary character.