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Quantum key distribution is emerging as a cutting-edge application of quantum technology, gradually integrating into the industrial landscape. Many protocols employing discrete or continuous variables have been developed over time. Whereas the firsts usually excel in covering longer distances, the seconds are typically superior in producing higher secret key rates at short distances. Present efforts aim to create systems that can exploit both these strengths, foreseeing the future challenge regarding the realization of a quantum network consisting of multiple and heterogeneous interconnected nodes. Within such a context, a possible solution is systems able to efficiently toggle between discrete and continuous variable working modes with hybrid quantum state encoders. Therefore, this study presents a new hybrid encoder based on an iPOGNAC modulator, ensuring compatibility with DV and CV QKD systems that can be assembled entirely with commercial-off-the-shelf components. The proposed scheme is the first supporting DV polarization protocols, thus making it an appealing candidate for space nodes of a future quantum network, given that polarization-based protocols are well suited for space links., Comment: 8 pages, 5 figures

A wide range of applications require, by hypothesis, to have access to a high-speed, private, and genuine random source. Quantum Random Number Generators (QRNGs) are currently the sole technology capable of producing true randomness. However, the bulkiness of current implementations significantly limits their adoption. In this work, we present a high-performance source-device independent QRNG leveraging a custom made integrated photonic chip. The proposed scheme exploits the properties of a heterodyne receiver to enhance security and integration to promote spatial footprint reduction while simplifying its implementation. This characteristics could represents a significant advancement toward the development of generators better suited to meet the demands of portable and space applications. The system can deliver secure random numbers at a rate greater than 20 Gbps with a reduced spatial and power footprint., Comment: 9 pages, 7 figures

Let $\mathscr{O}_K$ be a henselian DVR with field of fractions $K$ and residue field of characteristic $p>0$. Let $S$ denote $\mathop{\mathrm{Spec}} \mathscr{O}_K$ endowed with the canonical log structure. We show that the generic fiber functor $\mathbf{BT}_{S, {\mathrm{d}}}^{\log}\to \mathbf{BT}^{\mathrm{st}}_K$ between the category of dual representable log $p$-divisible groups over $S$ and the category of $p$-divisible groups with semistable reduction over $K$ is an equivalence. If $\mathscr{O}_K$ is further complete with perfect residue field and of mixed characteristic, we show that $\mathbf{BT}_{S, {\mathrm{d}}}^{\log}$ is also equivalent to the category of semistable Galois $\mathbb{Z}_p$-representations with Hodge-Tate weights in $\{0,1\}$. Finally, we show that the above equivalences respect monodromies., Comment: 41 pages. We have rewritten part of the paper, in particular the introduction in order to make more evident the novelty of our results and added a corollary on the antiequivalence between the category of log $p$-divisible groups and the category of strongly divisible modules of weight less or equal to 1

We study the $n$-th arithmetic jet space of the $p$-torsion subgroup attached to a smooth commutative formal group scheme. We show that the $n$-th jet space above fits in the middle of a canonical short exact sequence between a power of the formal scheme of Witt vectors of length $n$ and the $p$-torsion subgroup we started with. This result generalizes a result of Buium on roots of unity.

Current technological progress is driving Quantum Key Distribution towards a commercial and world widescale expansion. Its capability to deliver unconditionally secure communication will be a fundamental feature in the next generations of telecommunication networks. Nevertheless, demonstrations of QKD implementation in a real operating scenario and their coexistence with the classical telecom infrastructure are of fundamental importance for reliable exploitation. Here we present a Quantum Key Distribution application implemented overa classical fiber-based infrastructure. By exploiting just a single fiber cable for both the quantum and the classical channel and by using a simplified receiver scheme with just one single-photon detector, we demonstrate the feasibility of low-cost and ready-to-use Quantum Key Distribution systems compatible with standard classical infrastructure., Comment: 5 pages, 5 figures

This work presents a hardware and software architecture which can be used in those systems that implement practical Quantum Key Distribution (QKD) and Quantum Random Number Generation (QRNG) schemes. This architecture fully exploits the capability of a System-on-a-Chip (SoC) which comprehends both a Field Programmable Gate Array (FPGA) and a dual core CPU unit. By assigning the time-related tasks to the FPGA and the management to the CPU, we built a flexible system with optimized resource sharing on a commercial off-the-shelf (COTS) evaluation board which includes a SoC. Furthermore, by changing the dataflow direction, the versatile system architecture can be exploited as a QKD transmitter, QKD receiver and QRNG control-acquiring unit. Finally, we exploited the dual core functionality and realized a concurrent stream device to implement a practical QKD transmitter where one core continuously receives fresh data at a sustained rate from an external QRNG source while the other operates with the FPGA to drive the qubits transmission to the QKD receiver. The system was successfully tested on a long-term run proving its stability and security. This demonstration paves the way towards a more secure QKD implementation, with fully unconditional security as the QKD states are entirely generated by a true random process and not by deterministic expansion algorithms. Eventually, this enables the realization of a standalone quantum transmitter, including both the random numbers and the qubits generation., Comment: 7 pages, 4 figures

Given a scheme over a complete discrete valuation ring of mixed characteristic with perfect residue field, the Greenberg transform produces a new scheme over the residue field thicker than the special fiber. In this paper, we will generalize this transform to the case of imperfect residue field. We will then construct a certain kind of cycle class map defined on this generalized Greenberg transform applied to the N\'eron model of a semi-abelian variety, which takes values in the relatively perfect nearby cycle functor defined by Kato and the second author., Comment: Accepted for publication in manuscripta mathematica. No changes in the text from v2. 43 pages

We extend Borger's construction of algebraic jet spaces to allow for an arbitrary prolongation sequence, clarify the relation between Borger's and Buium's jet spaces and compare them in the extended sense. As a result, we strengthen a result of Buium on the relation between Greenberg's transform and the special fiber of jet spaces, including the ramified case., Comment: 25 pages

Let O be a complete discrete valuation ring of mixed characteristic and with finite residue field k. We study a natural morphism between the Greenberg algebra of O and the special fiber of the scheme of ramified Witt vectors over O. It is a universal homeomorphism with pro-infinitesimal kernel that can be explicitly described in some cases., Comment: We improved some results. 19 pages

We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. These divisibility properties are treated in an appendix by B. Kahn (extending previous work of Bloch and Colliot-Th\'el\`ene--Raskind)., Comment: Appendix by B. Kahn added. Revised version to appear on Adv. Math

According to a well-known theorem of Serre and Tate, the infinitesimal deformation theory of an abelian variety in positive characteristic is equivalent to the infinitesimal deformation theory of its Barsotti-Tate group. We extend this result to $1$-motives., Comment: Minor changes

This is a chiefly expository paper on the subject of the title which, in our view, has not received a detailed treatment in the literature which is commensurate with its importance. We expect the results presented here to be useful in a number of contexts. For example, several of them will be applied in a forthcoming paper by the authors., Comment: 25 pages

After introducing the Ogus realization of 1-motives we prove that it is a fully faithful functor. More precisely, following a framework introduced by Ogus, considering an enriched structure on the de Rham realization of 1-motives over a number field, we show that it yields a full functor by making use of an algebraicity theorem of Bost.

Let $k$ be a field, $A$ a finite $k$-algebra and $X$ a smooth $A$-scheme. We describe the Galois set of connected components of the Weil restriction $\Re_{A/k}(X)$ in terms of the sets of connected components of the geometric fibers of $X$., Comment: 6 pages. Accepted for publication in Mathematische Zeitschrift

We determine the behavior under Weil restriction of the group of connected components of the special fiber of an arbitrary smooth group scheme (whose Weil restriction exists) over an arbitrary (commutative and unital) local ring. Applications to N\'eron models are given., Comment: 7 pages. We decided not to publish it for the moment

If T is an algebraic torus defined over a discretely valued field K with perfect residue field k, we relate the K-cohomology of T to the k-cohomology of certain objects associated to T. When k has cohomological dimension <= 1, our results have a particularly simple form and yield, more generally, isomorphisms between the abelian K-cohomology of a reductive group G over K and the k-cohomology of a certain quotient of the algebraic fundamental group of G., Comment: Version submitted for publication

The proof, but not the statement, of Proposition 18.2 contained an error which is repaired in this version. See Remark 18.3 in this version. No other changes. We extend Greenberg's original construction to arbitrary (in particular, non-reduced) schemes over (certain types of) local artinian rings. We then establish a number of basic properties of the extended functor and determine, for example, its behavior under Weil restriction. We also discuss a formal analog of the functor., Comment: 120 pages. The proof, but not the statement, of Proposition 18.2 contained an error which is repaired in this version. See the new Remark 18.3. No other changes

We extend the construction of the category of 1-motivic sheaves (introduced by Barbieri-Viale and Kahn) allowing quotients of connected algebraic k-groups by formal k-groups. We show that its bounded derived category is equivalent to the bounded derived category of the category of generalized 1-motives with torsion introduced in a previous paper by Barbieri-Viale and the author., Comment: 10 pages

Let $K$ be a local field with algebraically closed residue field and $X_K$ a torsor under an elliptic curve $J_K$ over $K$. Let $X$ be a proper minimal regular model of $X_K$ over the ring of integers of $K$ and $J$ the identity component of the N\'eron model of $J_K$. We study the canonical morphism $q\colon \mathrm{Pic}^{0}_{X/S}\to J$ which extends the biduality isomorphism on generic fibres. We show that $q$ is pro-algebraic in nature with a construction that recalls Serre's work on local class field theory. Furthermore we interpret our results in relation to Shafarevich's duality theory for torsors under abelian varieties., Comment: This paper arises from the confluence and the comparison of the results contained in the preprints arXiv:1106.1540v2 and arXiv:1005.0462v1 of the two authors. Final version and to appear in Math. Zeit. We thank the referee for the very detail review of our paper

Let A be an abelian variety over a local field K of mixed characteristic and with algebraically closed residue field. We provide a geometric construction (via the relative Picard functor) of the Shafarevich duality between the group of isomorphism classes of torsors under A and the "fundamental group" of the N\'eron model of the dual abelian variety A'. An analogous construction works over fields of positive characteristic p providing a duality on the prime-to-p parts.

We generalize the construction of the category of 1-motives with torsion ${}^tM_1$ (introduced by Barbieri-Viale, Rosenschon and Saito) as well as the construction of the category of 1-motivic sheaves ${\rm Shv}_1$ (defined by Barbieri-Viale and Kahn) to perfect fields $k$ (without inverting the exponential characteristic). For $k$ transcendental over the prime field we extend a result of Barbieri-Viale and Kahn, showing that ${}^tM$ and ${\rm Shv}_1$ have equivalent bounded derived categories., Comment: 14 pages. Shortened, corrected version. Results on Laumon 1-motives will appear in another paper

We introduce the "sharp" (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the "sharp de Rham realization" by passing to the Lie-algebra. Over the complex numbers we then show a (sharp de Rham) comparison theorem in the category of formal Hodge structures. For a free 1-motive along with its Cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations. We thus provide "one-dimensional sharp de Rham cohomology" of algebraic varieties., Comment: 30 pages

We show that the pairing on de Rham realizations of 1-motives in "Theorie di Hodge III", IHES 44, can be defined over any base scheme and we prove that it gives rise to a perfect duality if one is working with a 1-motive and its Cartier dual. Furthermore we study universal extensions of 1-motives and their relation with $\natural$-extensions., Comment: 21 pages. Last section rewritten. New proofs

Let $R$ be a complete discrete valuation ring with perfect residue field $k$ of positive characteristic $p$ and field of fractions $K$ of characteristic 0. In this paper we consider a $K$-1-motive $M_K$ as in [Ra] and its associated Barsotti-Tate group. This last does not in general extend to a Barsotti-Tate group over $R$. However, with some assumptions, it extends to a logarithmic Barsotti-Tate group over $R$. This follows from [Ra] and Kato's results on finite logarithmic group schemes. Once chosen a uniformizing parameter $\pi$ of $R$, any logarithmic Barsotti-Tate group over $R$ is described by two data $(G,N)$ where $G$ is a classical Barsotti-Tate group over $R$ and $N$ is a homomorphism of classical Barsotti-Tate groups. Moreover, if $R=W(k)$, $N$ induces a $W(k)$-homorphism ${\cal N}\colon M(G_k)\to M(G_k)$ on Dieudonn\'e modules such that $F{\cal N}V={\cal N}$ and ${\cal N}^2=0$. In the first part of the paper we recall these constructions and we show how to relate $N$ with the ``geometric monodromy'' introduced by Raynaud. In the second part of the paper we give an explicit description of ${\cal N}$ in terms of additive extensions and integrals. In the last part of the paper we describe how to recover the logarithmic Barsotti-Tate group attached to a 1-motive from a concrete scheme endowed with a suitable logarithmic structure., Comment: 22 pages