Alexandru Gica - A conjecture which implies the theorem of Gauss-Heegner-Stark-Baker

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Alexandru Gica - A conjecture which implies the theorem of Gauss-Heegner-Stark-Baker

ALEXANDRU GICA, Faculty of Mathematics, University of Bucharest, RO-70109 Bucharest 1, Romania

A conjecture which implies the theorem of Gauss-Heegner-Stark-Baker

In 1801 Gauss conjectured that the ring of integers for a quadratic imaginary field $K= {\bf Q} (\sqrt d)$ is principal only for a finite number of d. Heegner, Stark and Baker proved that only for

-d=1,2,3,7,11,19,43,67,163

the ring of integers for ${\bf Q} (\sqrt d)$ (where $d\in {\bf Z}$ , d<0, d squarefree) is principal. We prove two result which are equivalent with this theorem and we pose a conjecture which implies the Gauss-Heegner-Stark-Baker theorem (we will abbreviate: the G.H.S.B. theorem). Finally we pose a ``weaker'' conjecture which seems to be more approachable than the other one.