Volume 2019, Issue 749

We study the structure of parabolic inductions of a pro- p -Iwahori Hecke algebra. In particular, we give a classification of irreducible modulo p representations of pro- p -Iwahori Hecke algebras in terms of supersingular representations. Since supersingular representations are classified by Ollivier and Vignéras, it completes the classification of irreducible modulo p representations.

We show that a nontrivial abelian variety over a Hilbertian field in which the weak Mordell–Weil theorem holds admits infinitely many torsors with period any given n>1n>1 that is not divisible by the characteristic. The corresponding statement with “period” replaced by “index” is plausible but open, and it seems much more challenging. We show that for every infinite, finitely generated field K , there is an elliptic curve E/KE_{/K} which admits infinitely many torsors with index any given n>1n>1.

Kuranishi’s fundamental result (1962) associates to any compact complex manifold X0{X_{0}} a finite-dimensional analytic space which has to be thought of as a local moduli space of complex structures close to X0{X_{0}}. In this paper, we give an analogous statement for Levi-flat CR-manifolds fibering properly over the circle by associating to any such 𝒳0{\mathcal{X}_{0}} the loop space of a finite-dimensional analytic space which serves as a local moduli space of CR-structures close to 𝒳0{\mathcal{X}_{0}}. We then develop in this context a Kodaira–Spencer deformation theory making clear the likenesses as well as the differences with the classical case. The article ends with applications and examples.

In this paper we study the regularized Petersson product between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight-one modular form with integral Fourier coefficients. In [18], we proved that these Petersson products posses remarkable arithmetic properties. Namely, such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. A similar result was obtained independently using a different method by W. Duke and Y. Li [5]. The main result of this paper is an explicit factorization formula for the algebraic number obtained by exponentiating a Petersson product.

Severi varieties and Brill–Noether theory of curves on K3 surfaces are well understood. Yet, quite little is known for curves on abelian surfaces. Given a general abelian surface S with polarization L of type (1,n){(1,n)}, we prove nonemptiness and regularity of the Severi variety parametrizing δ-nodal curves in the linear system |L|{|L|} for 0≤δ≤n-1=p-2{0\leq\delta\leq n-1=p-2} (here p is the arithmetic genus of any curve in |L|{|L|}). We also show that a general genus g curve having as nodal model a hyperplane section of some (1,n){(1,n)}-polarized abelian surface admits only finitely many such models up to translation; moreover, any such model lies on finitely many (1,n){(1,n)}-polarized abelian surfaces. Under certain assumptions, a conjecture of Dedieu and Sernesi is proved concerning the possibility of deforming a genus g curve in S equigenerically to a nodal curve. The rest of the paper deals with the Brill–Noether theory of curves in |L|{|L|}. It turns out that a general curve in |L|{|L|} is Brill–Noether general. However, as soon as the Brill–Noether number is negative and some other inequalities are satisfied, the locus |L|dr{|L|^{r}_{d}} of smooth curves in |L|{|L|} possessing a gdr{g^{r}_{d}} is nonempty and has a component of the expected dimension. As an application, we obtain the existence of a component of the Brill–Noether locus ℳp,dr{{\mathcal{M}}^{r}_{p,d}} having the expected codimension in the moduli space of curves ℳp{{\mathcal{M}}_{p}}. For r=1{r=1}, the results are generalized to nodal curves.

We construct a complete convergent normal form for a real hypersurface in ℂN{\mathbb{C}^{N}}, N≥2{N\geq 2}, at a generic Levi-degeneracy. This seems to be the first convergent normal form for a Levi-degenerate hypersurface. As an application of the convergence result, we obtain an explicit description of the moduli space of germs of real-analytic hypersurfaces with a generic Levi-degeneracy. As another application, we obtain, in the spirit of the work of Chern and Moser [6], distinguished curves inside the Levi-degeneracy set that we call degenerate chains.

We investigate a semi-continuity property for stability conditions for sheaves that is important for the problem of variation of the moduli spaces as the stability condition changes. We place this in the context of a notion of stability previously considered by the authors, called multi-Gieseker-stability, that generalises the classical notion of Gieseker-stability to allow for several polarisations. As such we are able to prove that on smooth threefolds certain moduli spaces of Gieseker-stable sheaves are related by a finite number of Thaddeus-flips (that is flips arising for Variation of Geometric Invariant Theory) whose intermediate spaces are themselves moduli spaces of sheaves.

We sharpen the constant in the KS2{\mathrm{KS}_{2}} Conjecture of Weaver [31] that was given by Marcus, Spielman and Srivastava [28] in their solution of the Kadison–Singer problem. We then apply this result to prove optimal asymptotic bounds on the size of partitions in the Feichtinger Conjecture.

Let K be a complete discrete valuation field whose residue field is perfect and of positive characteristic, let X be a connected, proper scheme over 𝒪K\mathcal{O}_{K}, and let U be the complement in X of a divisor with simple normal crossings. Assume that the pair (X,U)(X,U) is strictly semi-stable over 𝒪K\mathcal{O}_{K} of relative dimension one and K is of equal characteristic. We prove that, for any smooth ℓ\ell-adic sheaf 𝒢\mathcal{G} on U of rank one, at most tamely ramified on the generic fiber, if the ramification of 𝒢\mathcal{G} is bounded by t+t+ for the logarithmic upper ramification groups of Abbes–Saito at points of codimension one of X , then the ramification of the étale cohomology groups with compact support of 𝒢\mathcal{G} is bounded by t+t+ in the same sense.