Uniform algebras on curves

Uniform Algebras on Curves

The proofs use the notion of analytic structure in a maximal ideal space. J. Wermer first obtained results along these lines and further contributions were made by E. Bishop and H. Royden and then by G. Stolzenberg [5] who proved STOLZENBERG'S THEOREM. Let XQC be a polynomially convex set. Let KQC be a finite union of Q-curves. Then (XKJK)*—X\JK is a {possibly empty) pure 1-dimensional analytic...

M-IDEAL STRUCTURE IN UNIFORM ALGEBRAS

It is proved that if A is aregular uniform algebra on a compact Hausdorff space X in which every closed ideal is an M-ideal, then A = C(X).

On Uniform Bounds for Rational Points on Non-rational Curves

We show that the number of rational points of height ≤ H on a non-rational plane curve of degree d is Od(H 2/d−δ), for some δ > 0 depending only on d. The implicit constant depends only on d. This improves a result of Heath-Brown, who proved the bound O (H2/d+ ). We also show that one can take δ = 1/450 in the case d = 3.

Elliptic curves and C*-algebras

We use C-algebras in the context of modular curves and holomorphic 1-forms. We focus on the “leaf space” X and C-algebra C(X) of the Weil 1-form ωN on the modular curve X0(N). It is proved that K0(C (X)) is isomorphic to a stationary dimension group of rank 4g − 3. An application of this result to Hilbert’s 12th problem is discussed.

Vertex Algebras and Algebraic Curves