Arithmetic Schemes Research Articles

A conjecture of Flach and Morin

A conjecture recently stated by Flach and Morin relates the action of the monodromy on the Galois invariant part of the p-adic Beilinson–Hyodo–Kato cohomology of the generic fiber of a scheme defined over a DVR of mixed characteristic to (the cohomology of) its special fiber. We prove the conjecture in the case that the special fiber of the given arithmetic scheme is also a fiber of a geometric family over a curve in positive characteristic.

Dynamical systems for arithmetic schemes

Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vector rings to attach a new ringed space Wrat(X) to every scheme X. We also define R-valued points Wrat(X)(R) of Wrat(X) for every commutative ring R. For normal schemes X of finite type over specZ, using Wrat(X)(ℂ) we construct infinite dimensional R-dynamical systems whose periodic orbits are related to the closed points of X. Various aspects of these topological dynamical systems are studied. We also explain how certain p-adic points of Wrat(X) for X the spectrum of a p-adic local number ring are related to the points of the Fargues–Fontaine curve.

A Potential Enabler for High-Performance In-Memory Multi-Bit Arithmetic Schemes With Unipolar Switching SOT-MRAM

A Potential Enabler for High-Performance In-Memory Multi-Bit Arithmetic Schemes With Unipolar Switching SOT-MRAM

Design of a Scan Chain for Side Channel Attacks on AES Cryptosystem for Improved Security

Scan chain-based attacks are side-channel attacks focusing on one of the most significant features of hardware test circuitry. A technique called Design for Testability (DfT) involves integrating certain testability components into a hardware design. However, this creates a side channel for cryptanalysis, providing crypto devices vulnerable to scan-based attacks. Advanced Encryption Standard (AES) has been proven as the most powerful and secure symmetric encryption algorithm announced by USA Government and it outperforms all other existing cryptographic algorithms. Furthermore, the on-chip implementation of private key algorithms like AES has faced scan-based side-channel attacks. With the aim of protecting the data for secure communication, a new hybrid pipelined AES algorithm with enhanced security features is implemented. This paper proposes testing an AES core with unpredictable response compaction and bit level-masking throughout the scan chain process. A bit-level scan flipflop focused on masking as a scan protection solution for secure testing. The experimental results show that the best security is provided by the randomized addition of masked scan flipflop through the scan chain and also provides minimal design difficulty and power expansion overhead with some negligible delay measures. Thus, the proposed technique outperforms the state-of-the-art LUT-based S-box and the composite sub-byte transformation model regarding throughput rate 2 times and 15 times respectively. And security measured in the avalanche effect for the sub-pipelined model has been increased up to 95 per cent with reduced computational complexity. Also, the proposed sub-pipelined S-box utilizing a composite field arithmetic scheme achieves 7 per cent area effectiveness and 2.5 times the hardware complexity compared to the LUT-based model.

Tori over number fields and special values at $s=1$

We define a Weil-étale complex with compact support for duals (in the sense of the Bloch dualizing cycles complex \mathbb{Z}^c ) of a large class of \mathbb{Z} -constructible sheaves on an integral1-dimensional proper arithmetic scheme flat over Spec( \mathbb{Z} ). This complex can be thought of as computing Weil-étale homology. For those Z-constructible sheaves that are moreover tamely ramified, we define an “additive” complex which we think of as the Lie algebra of the dual of the \mathbb{Z} -constructible sheaf. The product of the determinants of the additive and Weil-étale complex is called the fundamental line. We prove a duality theorem which implies that the fundamental line has a natural trivialization, giving a multiplicative Euler characteristic. We attach a natural L -function to the dual of a \mathbb{Z} -constructible sheaf; up to a finite number of factors, this L -function is an Artin L -function at s = 1 . Our main theorem contains a vanishing order formula at s = 0 for the L -function and states that, in the tamely ramified case, the special value at s = 0 is given up to sign by the Euler characteristic. This generalizes the analytic class number formula for the special value at s = 1 of the Dedekind zeta function. In the function field case, this is a theorem of Geisser–Suzuki.

Special values of L-functions on regular arithmetic schemes of dimension 1

We construct a well-behaved Weil-étale complex for a large class of Z-constructible sheaves on a regular irreducible scheme U of finite type over Z and of dimension 1. We then give a formula for the special value at s=0 of the L-function associated to any Z-constructible sheaf on U in terms of Euler characteristics of Weil-étale cohomology; for smooth proper curves, we obtain the formula of [GS20]. We deduce a special value formula for Artin L-functions of integral representations twisted by a singular irreducible scheme X of finite type over Z and of dimension 1. This generalizes and improves all results in [Tra16]; as a special case, we obtain a special value formula for the arithmetic zeta function of X.

Fit the Joint Moments: How to Attack Any Masking Scheme

Side-Channel Analysis (SCA) allows extracting secret keys manipulated by cryptographic primitives through leakages of their physical implementations. Supervised attacks, known to be optimal, can theoretically defeat any countermeasure, including masking, by learning the dependency between the leakage and the secret through the profiling phase. However, defeating masking is less trivial when it comes to unsupervised attacks. While classical strategies such as correlation power analysis or linear regression analysis have been extended to masked implementations, we show that these extensions only hold for Boolean and arithmetic schemes. Therefore, we propose a new unsupervised strategy, the Joint Moments Regression (JMR), able to defeat any masking schemes (multiplicative, affine, polynomial, inner product...), which are gaining popularity in real implementations. The main idea behind JMR is to directly regress the leakage model of the shares by fitting a system based on higher-order joint moments conditions. We show that this idea can be seen as part of a more general framework known as the Generalized Method of Moments (GMM). This offers mathematical foundations on which we rely to derive optimizations of JMR. Simulations results confirm the interest of JMR over state-of-the-art attacks, even in the case of Boolean and arithmetic masking. Eventually, we apply this strategy to real traces and provide, to the best of our knowledge, the first unsupervised attack on the protected AES implementation proposed by the ANSSI for SCA research, which embeds an affine masking and shuffling counter-measures.

The Hilbert property for arithmetic schemes

We extend the usual Hilbert property for varieties over fields to arithmetic schemes over integral domains by demanding the set of near-integral points (as defined by Vojta) to be non-thin. We then generalize results of Bary-Soroker–Fehm–Petersen and Corv

Aviv barley and calendar diversity among Jews in eleventh-century Palestine

One of the most salient medieval Qaraite practices was setting the calendar by observation of natural phenomena. While the Rabbanites followed arithmetical schemes, Qaraites set months by sighting the new moon and intercalated years on the basis of the state of ripeness of barley crops ( aviv ). Multiple Qaraite treatises on the aviv are preserved, but documentary evidence of empirical intercalation is scarce, making it difficult to learn how it was performed in practice. This article examines two Qaraite calendar chronicles that document barley observations and decisions regarding intercalation in a range of years in the eleventh century. They shed important light on how the Qaraite calendar operated over periods of time and attest to frequent calendar difference within the Qaraite movement and between Qaraites and Rabbanites. The chronicles make it clear that the Qaraite calendar of the period was not a monolithic system counterposed to that of the Rabbanites.

Étale cohomology of arithmetic schemes and zeta values of arithmetic surfaces

In this paper, we deal with the étale cohomology of a proper regular arithmetic scheme X with Zp(r) and Qp(r)-coefficients, where the coefficients are complexes of étale sheaves that the author introduced in [SH]. We will prove that the étale cohomology of X with Qp(r)-coefficients agrees with the Selmer group of Bloch-Kato for any r≧dim(X). Using this fundamental result, we further discuss an approach to the study of zeta values (or residue) at s=r, via the étale cohomology with Zp(r)-coefficients, relating Tamagawa number conjecture of Bloch-Kato with a zeta value formula. As a consequence, we will obtain an unconditional example of an arithmetic surface for which the residue of its zeta function at s=2 is computed modulo rational numbers prime to p, for infinitely many p's.

Investigating the transformation of a secondary teacher’s knowledge of trigonometric functions

Shulman (1987) defined pedagogical content knowledge as the knowledge required to transform subject-matter knowledge into curricular material and pedagogical representations. This paper presents the results of an exploratory case study that examined a secondary teacher’s knowledge of sine and cosine values in both clinical and professional settings to discern the characteristics of mathematical schemes that facilitate their transformation into learning artifacts and experiences for students. My analysis revealed that the teacher’s knowledge of sine and cosine values consisted of uncoordinated quantitative and arithmetic schemes and that he was cognizant only of the behavioral proficiencies these schemes enable, not the mental actions and conceptual operations they entail. Based on these findings, I hypothesize that the extent to which a teacher is consciously aware of the mental activity that comprises their mathematical conceptions influences their capacity to transform their mathematical knowledge into curricular material and pedagogical representations to effectively support students’ conceptual learning.

SPECIAL VALUES OF THE ZETA FUNCTION OF AN ARITHMETIC SURFACE

AbstractWe prove that the special-value conjecture for the zeta function of a proper, regular, flat arithmetic surface formulated in [6] at $s=1$ is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian of the generic fibre. There are two key results in the proof. The first is the triviality of the correction factor of [6, Conjecture 5.12], which we show for arbitrary regular proper arithmetic schemes. In the proof we need to develop some results for the eh-topology on schemes over finite fields which might be of independent interest. The second result is a different proof of a formula due to Geisser, relating the cardinalities of the Brauer and the Tate–Shafarevich group, which applies to arbitrary rather than only totally imaginary base fields.

Fibres non r\xe9duites d\u2019un sch\xe9ma arithm\xe9tique

(Non-reduced fibers of an arithmetic scheme) For a reduced projective scheme over the ring of integers of a number field, the set of places over which the fibres of the scheme are not reduced is a finite set. We give an explicit upper bound for the product of the norms of places in this set. For this purpose, we introduce a generalization of the notion of height over the adelic ring. We reduce the general case of a scheme of pure dimension to the case of a hypersurface by using the theory of Chow varieties. The case of a hypersurface is then treated with the help of the resultant of the equation of the hypersurface with some partial derivatives of the equation.

Compatibility of special value conjectures with the functional equation of zeta functions

We prove that the special value conjecture for the Zeta function \zeta(\mathcal{X},s) of a proper, regular arithmetic scheme \mathcal{X} that we formulated in [ M. Flach and B. Morin , Doc. Math. 23, 1425–1560 (2018; Zbl 1404.14024)] is compatible with the functional equation of \zeta(\mathcal{X},s) provided that the rational factor C(\mathcal{X},n) we were not able to compute previously has the simple explicit form given in the introduction below.

A local to global principle for higher zero-cycles

We study a local to global principle for certain higher zero-cycles over global fields. We thereby verify a conjecture of Colliot-Thélène for these cycles. Our main tool are the Kato conjectures proved by Jannsen, Kerz and Saito. Our approach also allows to reprove the ramified global class field theory of Kato and Saito. Finally, we apply the Kato conjectures to study the p-adic cycle class map over henselian discrete valuation rings of mixed characteristic and to deduce finiteness theorems for arithmetic schemes in low degree.

An arithmetic Lefschetz–Riemann–Roch theorem

In this article, we consider regular projective arithmetic schemes in the context of Arakelov geometry, any of which is endowed with an action of the diagonalisable group scheme associated to a finite cyclic group and with an equivariant very ample invertible sheaf. For any equivariant morphism between such arithmetic schemes, which is smooth over the generic fibre, we define a direct image map between corresponding higher equivariant arithmetic K-groups and we discuss its transitivity property. Then we use the localization sequence of higher arithmetic K-groups and the higher arithmetic concentration theorem developed in \cite{T3} to prove an arithmetic Lefschetz-Riemann-Roch theorem. This theorem can be viewed as a generalization, to the higher equivariant arithmetic K-theory, of the fixed point formula of Lefschetz type proved by K. K\"{o}hler and D. Roessler in \cite{KR1}.

$p$-Adic limits of renormalized logarithmic Euler characteristics

Given a countable residually finite group \Gamma , we write \Gamma_n \to e if (\Gamma_n) is a sequence of normal subgroups of finite index such that any infinite intersection of \Gamma_n 's contains only the unit element e of \Gamma . Given a \Gamma -module M we are interested in the multiplicative Euler characteristics \chi (\Gamma_n , M) = \prod_i |H_i (\Gamma_n , M)|^{(-1)^i} and the limit in the field \mathbb Q_p of p -adic numbers h_p := \lim_{n\to\infty} (\Gamma : \Gamma_n)^{-1} \log_p \chi (\Gamma_n , M) \; . Here \log_p : \mathbb Q^{\times}_p \to \mathbb Z_p is the branch of the p -adic logarithm with \log_p (p) = 0 . Of course, neither expression will exist in general. We isolate conditions on M , in particular p -adic expansiveness which guarantee that the Euler characteristics \chi (\Gamma_n,M) are well defined. That notion is a p -adic analogue of expansiveness of the dynamical system given by the \Gamma -action on the compact Pontrjagin dual X=M^∗ of M . Under further conditions on \Gamma we also show that the renormalized p -adic limit in the second formula exists and equals the p -adic R -torsion of M . The latter is a p -adic analogue of the Li–Thom L2 R -torsion of a \Gamma -module M which they related to the entropy h of the \Gamma -action on X . We view the limit h_p as a version of entropy which values in the p -adic numbers and the equality with p -adic R -torsion as an analogue of the Li–Thom formula in the expansive case. We discuss the case \Gamma = \mathbb{Z}^n in more detail where our theory is related to Serre's intersection numbers on arithmetic schemes.

Francisco Guerrero héritier de la tradition franco-flamande: onze pièces basées sur un ostinato

Dans son œuvre religieuse, Francisco Guerrero exploite à plusieurs reprises un type de composition contrapuntique relativement peu répandu à son époque: l'écriture sur cantus firmus ostinato. Pour les musicologues germaniques, ceci représente une des formes ultimes du motet isorythmique médiéval. Le travail architectural, particulièrement soigné, y est ordonné par une véritable arithmétique musicale, issue de l'ars música mathematica. Guerrero honore ainsi la tradition de ses prédécesseurs, illustrée par le Miserere à 5 voix de Josquin des Prés. Il suit également son maître Cristobal de Morales en lui empruntant, semble-t-il, quelques modèles. Mais son écriture sait se mettre au goût du jour, et confère à l'ostinato un sens et une efficacité oratoires dignes d'un grand rhétoricien. Cet aspect de sa personnalité musicale s'unit harmonieusement à son côté novateur -celui des villancicos et des motets polychorauxpour témoigner de son génie universel.

Poitou–Tate duality for arithmetic schemes

We give a generalization of Poitou–Tate duality to schemes of finite type over rings of integers of global fields.

Finiteness of Frobenius Traces of a Sheaf on a Flat Arithmetic Scheme

Abstract For a lisse $\ell $-adic sheaf on a scheme flat and of finite type over $\mathbb{Z}$, we consider the field generated over $ \mathbb{Q}$ by Frobenius traces of the sheaf at closed points. Assuming conjectural properties of geometric Galois representations of number fields and the Generalized Riemann Hypothesis, we prove that the field is finite over $\mathbb{Q}$ when the sheaf is de Rham at $\ell $ pointwise. This is a number field analog of Deligne’s finiteness result about Frobenius traces in the function field case.