manuscripta mathematica

Ap-adic subanalytic set shares with a real subanalytic set the fundamental property that its singular locus is itself subanalytic. Furthermore, given ap-adic subanalytic function ƒ with domain contained in ℤ , there is an integerL such that for any pointx 0 ∈ ℤ in a neighborhood of whichf is defined,f has a Taylor approximation up to orderL atx 0 if, and only if, ƒ is analytic aroundx 0. These results extend to thep-adic fields real variables theorems by M. Tamm [21].

We consider the semilinear eigenvalue problem $$ - \Delta u - q(x)\left| u \right|^\sigma u = \lambda \cdot u$$ on ℛN (N ≥ 2) (N≥2) and investigate the question under which conditions on the radially symmetric function q, λ=0 is a bifurcation point for this equation in H1, In H2 and in Lp for 2≤p≤+∞.

Regularity problems are studied for a PDO P(D) with constant coefficients using a new variant of the parametrix-method. A new Characterization of the index of hypoellipticity r and a pointwise uniquness theorem for the Cauchy-problem is given. For semielliptic P(D) the functional dimension of C p ∞ (R n ) is proved to be ¦r¦. The equation P(D)g=f Is solved with distributions with part. bounded support.

We discuss the harmonic properties of a class of sequences which contains the so-called paper folding sequences; meanperiodicity is first studied, then a sufficient condition is given for the existence of a spectral measure.

We prove a rigidity theorem for a space-like graph with parallel mean curvature of arbitrary dimension and codimension in pseudo-Euclidean space via properties of its harmonic Gauss map. We also give an estimate of the squared norm of the second fundamental form in terms of the mean curvature and the image diameter under the Gauss map for space-like submanifolds with parallel mean curvature in pseudo-Euclidean space. The estimate also implies the former theorem.

In a recent paper I generalized the Thue-Siegel-Roth-Schmidt theorem on simultaneous rational approximation of n real algebraic numbers to include also the p-adic case. In the present paper I shall carry the argument further and prove analogous results for systems of real and p-adic linear forms with algebraic coefficients.

Given a newform f, we extend Howard’s results on the variation of Heegner points in the Hida family of f to a general quaternionic setting. More precisely, we build big Heegner points and big Heegner classes in terms of compatible families of Heegner points on towers of Shimura curves. The novelty of our approach, which systematically exploits the theory of optimal embeddings, consists in treating both the case of definite quaternion algebras and the case of indefinite quaternion algebras in a uniform way. We prove results on the size of Nekovář’s extended Selmer groups attached to suitable big Galois representations and we formulate two-variable Iwasawa main conjectures both in the definite case and in the indefinite case. Moreover, in the definite case we propose refined conjectures à la Greenberg on the vanishing at the critical points of (twists of) the L-functions of the modular forms in the Hida family of f living on the same branch as f.