We determine the universal deformation ring, in the sense of Mazur, of a residual representation $\bar \rho :G_K\to {\rm GL}_2(k)$, where k is a finite field of characteristic p and K is a local field of residue characteristic p. As one might hope for, but is not proven in the global case, the deformation ring is a complete intersection, flat over W(k), with the exact number of equations given by the dimension of $H^2(G_K,{\rm ad}_{\bar \rho})$. We then go on to determine the ordinary locus inside the deformation space and, using ideas of Mazur, apply this to compare the universal and the universal ordinary deformation spaces. Provided that the universal ring for ordinary deformations with fixed determinant is finite flat over W(k), as was shown in many cases by Diamond, Fujiwara, Taylor–Wiles and Wiles, we show that the corresponding universal deformation ring – with no restriction of ordinariness or fixed determinant – is a complete intersection, finite flat over W(k) of the dimension conjectured by Mazur, provided that the restriction of $\det (\bar \rho)$ to the inertia subgroup is different from the inverse cyclotomic character.

For a domain Ω of ${\mathbb C}^N$ we introduce a fairly general and intrinsic condition of weak q-pseudoconvexity, and prove, in Theorem 4, solvability of the $\bar \partial$-complex for forms with $C^\infty (\bar \Omega)$-coefficients in degree $\geq q+1$. All domains whose boundary have a constant number of negative Levi eigenvalues are easily recognized to fulfill our condition of q-pseudoconvexity; thus we regain the result of Michel (with a simplified proof). Our method deeply relies on the $L^2$-estimates by Hörmander (with some variants). The main point of our proof is that our estimates (both in weightened-$L^2$ and in Sobolev norms) are sufficiently accurate to permit us to exploit the technique by Dufresnoy for regularity up to the boundary.

The number N of rational points on an algebraic curve of genus g over a finite field ${\mathbb F}_q$ satisfies the Hasse–Weil bound $N \leq q + 1 +2g\sqrt {q}$. A curve that attains this bound is called maximal. With $g_0 =\textstyle {1\over 2}(q -\sqrt {q})$ and $g_1= \textstyle {1\over 4}(\sqrt {q}- 1)^2$, it is known that maximalcurves have $g= g_0$ or $g \leq g_1$. Maximal curves with $g=g_0$ or $g_1$ have been characterized up to isomorphism. A natural genus to be studied is $g_2 =\textstyle {1 \over 8}(\sqrt {q}- 1)(\sqrt {q}- 3),$ and for this genus there are two non-isomorphic maximal curves known when $\sqrt {q}\equiv 3\;({\rm mod}\; {4})$. Here, a maximal curve with genus $g_2$ and a non-singular plane model is characterized as a Fermat curve of degree $ {1\over 2}(\sqrt {q} + 1)$.

In this paper, we study polynomial structures by starting on the Lie algebra level, then passing to Lie groups to finally arrive at the polycyclic-by-finite group level. To be more precise, we first show how a general solvable Lie algebra can be decomposed into a sum of two nilpotent subalgebras. Using this result, we construct, for any simply connected, connected solvable Lie group G of dim n, a simply transitive action on Rn which is polynomial and of degree ≤ n3. Finally, we show the existence of a polynomial structure on any polycyclic-by-finite group Γ, which is of degree ≤ h(Γ)3 on almost the entire group (h (Γ) being the Hirsch length of Γ).

Let A be a finite-dimensional k-algebra over algebraically closed field k and $m{\rm mod}\, A$ be the category of finite-dimensional left A-modules. We show that a module M in ${\rm mod}\, A$ degenerates to another module N in ${\rm mod}\, A$ if and only if there is an exact sequence $0\to N\to M\oplus Z\to Z\to 0$ in ${\rm mod}\, A$ for some A-module Z. Moreover, we give a description of minimal degenerations of finite-dimensional A-modules.