Euler characteristics and elliptic curves

Let E be a modular elliptic curve over ℚ, without complex multiplication; let p be a prime number where E has good ordinary reduction; and let F∞ be the field obtained by adjoining to ℚ all p-power division points on E. Write G∞ for the Galois group of F∞ over ℚ. Assume that the complex L-series of E over ℚ does not vanish at s = 1. If p ⩾ 5, we make a precise conjecture about the value of the G∞-Euler characteristic of the Selmer group of E over F∞. If one makes a standard conjecture about the behavior of this Selmer group as a module over the Iwasawa algebra, we are able to prove our conjecture. The crucial local calculations in the proof depend on recent joint work of the first author with R. Greenberg.

Zeta functions and Eisenstein series on classical groups

We construct an Euler product from the Hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a CM field and the eigenform is holomorphic. We also prove analytic continuation of an Eisenstein series on another unitary group, containing the group just mentioned defined with such an eigenform. As an application of our methods, we prove an explicit class number formula for a totally definite hermitian form over a CM field.

Congruences between modular forms: Raising the level and dropping Euler factors

We discuss the relationship among certain generalizations of results of Hida, Ribet, and Wiles on congruences between modular forms. Hida’s result accounts for congruences in terms of the value of an L-function, and Ribet’s result is related to the behavior of the period that appears there. Wiles’ theory leads to a class number formula relating the value of the L-function to the size of a Galois cohomology group. The behavior of the period is used to deduce that a formula at “nonminimal level” is obtained from one at “minimal level” by dropping Euler factors from the L-function.

Regulation of number and size of digits by posterior Hox genes: A dose-dependent mechanism with potential evolutionary implications

The proper development of digits, in tetrapods, requires the activity of several genes of the HoxA and HoxD homeobox gene complexes. By using a variety of loss-of-function alleles involving the five Hox genes that have been described to affect digit patterning, we report here that the group 11, 12, and 13 genes control both the size and number of murine digits in a dose-dependent fashion, rather than through a Hox code involving differential qualitative functions. A similar dose–response is observed in the morphogenesis of the penian bone, the baculum, which further suggests that digits and external genitalia share this genetic control mechanism. A progressive reduction in the dose of Hox gene products led first to ectrodactyly, then to olygodactyly and adactyly. Interestingly, this transition between the pentadactyl to the adactyl formula went through a step of polydactyly. We propose that in the distal appendage of polydactylous short-digited ancestral tetrapods, such as Acanthostega, the HoxA complex was predominantly active. Subsequent recruitment of the HoxD complex contributed to both reductions in digit number and increase in digit length. Thus, transition through a polydactylous limb before reaching and stabilizing the pentadactyl pattern may have relied, at least in part, on asynchronous and independent changes in the regulation of HoxA and HoxD gene complexes.