Galois representations on the cohomology of hyper-Kähler varieties

We show that the André motive of a hyper-Kähler variety X over a field $$K \subset {\mathbb {C}}$$ K ⊂ C with $$b_2(X)>6$$ b 2 ( X ) > 6 is governed by its component in degree 2. More precisely, we prove that if $$X_1$$ X 1 and $$X_2$$ X 2 are deformation equivalent hyper-Kähler varieties with $$b_2(X_i)>6$$ b 2 ( X i ) > 6 and if there exists a Hodge isometry $$f:H^2(X_1,{\mathbb {Q}})\rightarrow H^2(X_2,{\mathbb {Q}})$$ f : H 2 ( X 1 , Q ) → H 2 ( X 2 , Q ) , then the André motives of $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic after a finite extension of K, up to an additional technical assumption in presence of non-trivial odd cohomology. As a consequence, the Galois representations on the étale cohomology of $$X_1$$ X 1 and $$X_2$$ X 2 are isomorphic as well. We prove a similar result for varieties over a finite field which can be lifted to hyper-Kähler varieties for which the Mumford–Tate conjecture is true.

Mordell–Weil ranks and Tate–Shafarevich groups of elliptic curves with mixed-reduction type over cyclotomic extensions

Let [Formula: see text] be an elliptic curve defined over a number field [Formula: see text] where [Formula: see text] splits completely. Suppose that [Formula: see text] has good reduction at all primes above [Formula: see text]. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic [Formula: see text]-extension of a finite extension [Formula: see text] of [Formula: see text] where [Formula: see text] is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell–Weil ranks of [Formula: see text] over a subextension of the cyclotomic [Formula: see text]-extension are bounded. Furthermore, we derive an aysmptotic formula of the growth of the [Formula: see text]-parts of the Tate–Shafarevich groups of [Formula: see text] over these extensions.

The Third Attempt. A New Chance for an Idealist Philosophy

The message of total non-universality (both sensible content and concept as such are evalued as non-universal) is incompatible with human nature. From this, the exceptional character of my philosophical venture is derived. Essentially, it is a problem of communication. Hence, it is a didactical construction which is central to my paper. The crucial importance of two simple assertions must be emphasized:1) A finite extension is not divisible infinitely (proposed by Berkeley and Hume)2) The line does not consist of points (proposed by Hegel)The above two assertions may be called the hard core of idealist philosophy. Both are instrumental in the conception of original environment. Being tied to the original environment is the outstanding fact that explains why the sensible content should be qualified as non-universal.

On the Equationally Artinian Groups

In this article, we study the property of being equationally Artinian in groups. We define the radical topology corresponding to such groups and investigate the structure of irreducible closed sets of these topologies. We prove that a finite extension of an equationally Artinian group is again equationally Artinian. We also show that a quotient of an equationally Artinian group of the form G[t] by a normal subgroup which is a finite union of radicals, is again equationally Artnian. A necessary and sufficient condition for an Abelian group to be equationally Artinian will be given as the last result. This will provide a large class of examples of equationally Artinian groups

ON THE PRODUCT OF ELEMENTS WITH PRESCRIBED TRACE

This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\text{Tr}$ , for which elements $z$ in $\mathbb{L}$ , and $a$ , $b$ in $\mathbb{K}$ , is it possible to write $z$ as a product $xy$ , where $x,y\in \mathbb{L}$ with $\text{Tr}(x)=a,\text{Tr}(y)=b$ ? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least 5. We also have results for arbitrary fields and extensions of degrees 2, 3 or 4. We then apply our results to the study of perfect nonlinear functions, semifields, irreducible polynomials with prescribed coefficients, and a problem from finite geometry concerning the existence of certain disjoint linear sets.

Bivergent extension in the overriding plate above a slab tear (Dodecanese, Greece)

<p><span lang="EN-US">Tearing in the Hellenic slab below the transition between the Aegean and Anatolian plate is considered to have significantly affected Miocene tectonic and magmatic evolution of the eastern Mediterranean by causing a toroidal flow of asthenosphere and a lateral gradient of extension in the upper plate. Some studies suggest that this lateral gradient is accommodated by a distributed sinistral lithospheric-scale shear zone whereas other studies favor a localized NE-SW striking transfer zone. Recent studies in the northern Dodecanese demonstrate that the transition zone between the Aegean and Anatolian plate is characterized by Miocene extension with a constant NNE-SSW sense of shear accommodating the difference in finite extension rates in the middle-lower crust. Neither localized or distributed strike-slip faults nor rotation of blocks about a vertical axis have been observed.</span></p> <p><span lang="EN-US">In this work we focus on the geology Kalymnos located in the central Dodecanese. Based on our new geological map, three major tectonic units can be distinguished: (i) Low-grade, fossil-rich late Paleozoic marbles, which have been deformed into S-vergent folds and out-of-sequence thrusts. This fold-and-thrust belt is sealed by an up to 200 m thick wildflysch-type deposit consisting of low-grade metamorphic radiolarites and conglomerates with tens of meters-scale marbles and ultramafics blocks. (ii) Above this unit, amphibolite facies schists, quartzites and amphibolites are tectonically juxtaposed along a several meter-thick thrust fault with low-grade ultramylonites and cohesive ultracataclasites/pseudotachylites with top-to-N kinematics. (iii) At highest structural levels, a major cataclastic low-angle normal fault zone localized in Verrucano-type violet slates separates Mesozoic unmetamorphosed limestones in the hanging wall. The sense of shear of the normal fault is top-to-SSW. All units are cut by brittle high-angle normal faults shaping the geomorphology of Kalymnos, which is characterized by three major NNW-SSE trending graben systems.</span></p> <p><span lang="EN-US">New white mica Ar-Ar ages suggests that the middle units represent relics of a Variscan basement, which was thrusted on top of a fold-and-thrust belt during an Eo-Cimmerian event. Zircon (U-Th)/He ages from the Variscan basement are c. 28 Ma, indicating that the lower units were exhumed below the Mesozoic carbonates during the Oligocene-Miocene. Since Miocene extension in the northern Dodecanese records top-to-NNE kinematics, we suggest that back-arc extension in the whole Aegean realm and transition to the Anatolian plate is bivergent, and tearing in the Hellenic slab did not significantly affected the extension pattern in the upper crust.</span></p>

Deformation Theory of the Trivial mod p Galois Representation for GLn

We study the rigid generic fiber $\mathcal{X}^\square _{\overline \rho }$ of the framed deformation space of the trivial representation $\overline \rho : G_K \to \textrm{GL}_n(k)$ where $k$ is a finite field of characteristic $p>0$ and $G_K$ is the absolute Galois group of a finite extension $K/\textbf{Q}_p$. Under some mild conditions on $K$ we prove that $\mathcal{X}^\square _{\overline \rho }$ is normal. When $p> n$ we describe its irreducible components and show Zariski density of its crystalline points.

ON THE EXISTENCE OF ADMISSIBLE SUPERSINGULAR REPRESENTATIONS OF -ADIC REDUCTIVE GROUPS

Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_{p}$ and that $C$ is a field of characteristic $p$ . We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $C$ .

ON THE IRREDUCIBLE COMPONENTS OF SOME CRYSTALLINE DEFORMATION RINGS

We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_{K}$ for a finite extension $K/\mathbb{Q}_{p}$ . This is done by considering a moduli space of Breuil–Kisin modules, satisfying an additional Galois condition, over the unrestricted deformation ring. For $K$ unramified over $\mathbb{Q}_{p}$ and Hodge–Tate weights in $[0,p]$ , we study the geometry of this space. As a consequence, we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of $\mathbb{Q}_{p}$ , with Hodge–Tate weights in $[0,p]$ , are potentially diagonalizable.