Chern numbers of Hilbert modular varieties.
Chowla's conjecture
Circular units of real abelian fields with four ramified primes
In this paper we study the groups of circular numbers and circular units in Sinnott’s sense in real abelian fields with exactly four ramified primes under certain conditions. More specifically, we construct -bases for them in five special infinite families of cases. We also derive some results about the corresponding module of relations (in one family of cases, we show that the module of Ennola relations is cyclic). The paper is based upon the thesis [6], which builds upon the results of the paper...
Class 2 Galois representations of Kummer type.
Class fields of abelian extensions of Q.
Class fields towers of imaginary quadratic fields.
Class group of a cyclotomic -extension
Class Group Rank Relations in Zl-Extensions.
Class groups and general linear group cohomology for a ring of algebraic integers.
Class groups of abelian fields, and the main conjecture
This first part of this paper gives a proof of the main conjecture of Iwasawa theory for abelian base fields, including the case , by Kolyvagin’s method of Euler systems. On the way, one obtains a general result on local units modulo circular units. This is then used to deduce theorems on the order of -parts of -class groups of abelian number fields: first for relative class groups of real fields (again including the case ). As a consequence, a generalization of the Gras conjecture is stated...
Class groups of global fields.
Class groups under relative quadratic extensions
Class invariants and cyclotomic unit groups from special values of modular units
In this article we obtain class invariants and cyclotomic unit groups by considering specializations of modular units. We construct these modular units from functional solutions to higher order -recurrence equations given by Selberg in his work generalizing the Rogers-Ramanujan identities. As a corollary, we provide a new proof of a result of Zagier and Gupta, originally considered by Gauss, regarding the Gauss periods. These results comprise part of the author’s 2006 Ph.D. thesis [6] in which...
Class invariants by Shimura's reciprocity law
We apply the Shimura reciprocity law to determine when values of modular functions of higher level can be used to generate the Hilbert class field of an imaginary quadratic field. In addition, we show how to find the corresponding polynomial in these cases. This yields a proof for conjectural formulas of Morain and Zagier concerning such polynomials.
Class Invariants for Quartic CM Fields
One can define class invariants for a quartic primitive CM field as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to . 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 -units in certain abelian extensions of a reflex field of , where is effectively determined by , and to bound the primes appearing...
Class invariants of Mordell-Weil groups.
Class number and factorization in quadratic number fields
Class number calculation of a cubic field from the elliptic unit.
Class number calculation of a quartic field from the elliptic unit
Class number calculation of a sextic field from the elliptic unit