ON STARK’S CLASS NUMBER CONJECTURE AND THE GENERALISED BRAUER–SIEGEL CONJECTURE

Stark conjectured that for any $h\in \Bbb {N}$ , there are only finitely many CM-fields with class number h. Let $\mathcal {C}$ be the class of number fields L for which L has an almost normal subfield K such that $L/K$ has solvable Galois closure. We prove Stark’s conjecture for $L\in \mathcal {C}$ of degree greater than or equal to 6. Moreover, we show that the generalised Brauer–Siegel conjecture is true for asymptotically good towers of number fields $L\in \mathcal {C}$ and asymptotically bad families of $L\in \mathcal {C}$ .

Fermat curves and a refinement of the reciprocity law on cyclotomic units

We define a “period-ring-valued beta function” and give a reciprocity law on its special values. The proof is based on some results of Rohrlich and Coleman concerning Fermat curves. We also have the following application. Stark’s conjecture implies that the exponentials of the derivatives at s=0 of partial zeta functions are algebraic numbers which satisfy a reciprocity law under certain conditions. It follows from Euler’s formulas and properties of cyclotomic units when the base field is the rational number field. In this paper, we provide an alternative proof of a weaker result by using the reciprocity law on the period-ring-valued beta function. In other words, the reciprocity law given in this paper is a refinement of the reciprocity law on cyclotomic units.

Real-dihedral harmonic Maass forms and CM-values of Hilbert modular functions

In this paper, we study real-dihedral harmonic Maass forms and their Fourier coefficients. The main result expresses the values of Hilbert modular forms at twisted CM 0-cycles in terms of these Fourier coefficients. This is a twisted version of the main theorem in Bruinier and Yang [CM-values of Hilbert modular functions, Invent. Math. 163 (2006), 229–288] and provides evidence that the individual Fourier coefficients are logarithms of algebraic numbers in the appropriate real-quadratic field. From this result and numerical calculations, we formulate an algebraicity conjecture, which is an analogue of Stark’s conjecture in the setting of harmonic Maass forms. Also, we give a conjectural description of the primes appearing in CM-values of Hilbert modular functions.

ON HIGHER-ORDER STICKELBERGER-TYPE THEOREMS FOR MULTI-QUADRATIC EXTENSIONS

We prove, for all quadratic and a wide range of multi-quadratic extensions of global fields, a result concerning the annihilation as Galois modules of ideal class groups by explicit elements constructed from the values of higher-order derivatives of Dirichlet L-functions. This result simultaneously refines Rubin's integral version of Stark's Conjecture and provides evidence for the relevant case of the Equivariant Tamagawa Number Conjecture of Burns and Flach.

Stark's Conjecture and New Stickelberger Phenomena

We introduce a new conjecture concerning the construction of elements in the annihilator ideal associated to a Galois action on the higher-dimensional algebraicK–groups of rings of integers in number fields. Our conjecture ismotivic in the sense that it involves the (transcendental) Borel regulator as well as being related tol–adic étale cohomology. In addition, the conjecture generalises the wellknown Coates–Sinnott conjecture. For example, for a totally real extension whenr= –2,–4,–6, … the Coates–Sinnott conjecturemerely predicts that zero annihilatesK–2rof the ring ofS–integers while our conjecture predicts a non-trivial annihilator. By way of supporting evidence, we prove the corresponding (conjecturally equivalent) conjecture for the Galois action on the étale cohomology of the cyclotomic extensions of the rationals.