TITLE: Linearly Irregular Primes- A first report
SPEAKERS: Alyson Deines and Ben Lundell
LOCATION: 3:35 in Padelford C401 on March 8, 2012
ABSTRACT: A prime number nt(2f)20120308/latex_07e287db17d9adad69e31ec444cbc74de07b09ac_p1.png "$p$")
is called irregular if divides the class number of the -th cyclotomic field. Through work of Kummer, Kubota-Leopoldt, and Iwasawa, this irregularity condition is related to the -divisibility of certain Bernoulli numbers, which themselves arise as the constant terms of -adic nt(2f)20120308/latex_ed45432016aa8f461445b268b596ea92edbcc79e_p1.png "$L$")
-functions. In this talk, we'll explain these ideas and how to extend them to relate congruences of Bernoulli numbers modulo nt(2f)20120308/latex_0848f6c83dd3425ed0a31c4198df5c33ae285784_p1.png "$p^2$")
to the -divisibility of the linear term of the -adic -function. We will also explain how to efficiently compute these congruences and provide some preliminary computational evidence which suggests a definite pattern.