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Baptiste Morin

Publications and source records attributed to Baptiste Morin.

At least 19 recordsLinked to original sources
arXiv2026

The de Rham and the syntomic logarithm

We define and study an integral refinement of the inverse of the Bloch-Kato exponential map which we call the de Rham logarithm. Our main tool to analyze the de Rham logarithm is the syntomic logarithm, a certain limit construction based on the theory of filtered prismatic cohomology initiated by Antieau, Krause and Nikolaus. We use the syntomic logarithm to prove a version of the Beilinson fibre square for all quasicompact, quasiseparated derived formal schemes. We also use our techniques to prove Conjecture $C_{EP}(\bq_p(n))$ of Fontaine and Perrin-Riou for all local fields $K/\bq_p$ and to compute the correction factor $C(X,n)$ introduced by Flach and Morin in their reformulation of the Bloch-Kato Tamagawa number conjecture for the Zeta function of a smooth projective scheme $X$ over a number ring.

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arXiv2022

On Integral Class field theory for varieties over $p$-adic fields

Let $K$ be a finite extension of the $p$-adic numbers $\mathbb Q_p$ with ring of integers $\mathcal O_K$, $\mathcal X$ a regular scheme, proper, flat, and geometrically irreducible over $\mathcal O_K$ of dimension $d$, and $\mathcal X_K$ its generic fiber. We show, under some assumptions on $\mathcal X_K$, that there is a reciprocity isomorphism of locally compact groups $H_{ar}^{2d-1}(\mathcal X_K, \mathbb Z(d)) \simeq π_1^{ab}(\mathcal X_K)_{W}$ from a new cohomology theory to an integral model $π_1^{ab}(\mathcal X_K)_{W}$ of the abelianized geometric fundamental groups $π_1^{ab}(\mathcal X_K)^{geo}$. After removing the contribution from the base field, the map becomes an isomorphism of finitely generated abelian groups.

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arXiv2021

Pontryagin duality for varieties over $p$-adic fields

We define cohomological complexes of locally compact abelian groups associated with varieties over $p$-adic fields and prove a duality theorem under some assumption. Our duality takes the form of Pontryagin duality between locally compact motivic cohomology groups.

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arXiv2021

Topological Hochschild homology and Zeta-values

Using work of Antieau and Bhatt-Morrow-Scholze, we define a filtration on topological Hochschild homology and its variants $TP$ and $TC^-$ of quasi-lci rings with bounded torsion, which recovers the BMS-filtration after $p$-adic completion. Then we compute the graded pieces of this filtration in terms of Hodge completed derived de Rham cohomology relative to the base ring $\mathbb{Z}$. We denote the cofiber of the canonical map from $\mathrm{gr}^{n}TC^-(-)$ to $\mathrm{gr}^{n}TP(-)$ by $LΩ^{<n}_{-/\mathbb{S}}[2n]$. Let $\mathcal{X}$ be a regular connected scheme of dimension $d$ proper over $\mathrm{Spec}(\mathbb{Z})$ and let $n\in\mathbb{Z}$ be an arbitrary integer. Together with Weil-étale cohomology with compact support $RΓ_{W,c}(\mathcal{X},\mathbb{Z}(n))$, the complex $LΩ^{<n}_{\mathcal{X}/\mathbb{S}}$ is expected to give the Zeta-value $\pmζ^*(\mathcal{X},n)$ on the nose. Combining the results proven here with a theorem recently proven in joint work with Flach, we obtain a formula relating $LΩ^{<n}_{\mathcal{X}/\mathbb{S}}$, $LΩ^{<d-n}_{\mathcal{X}/\mathbb{S}}$, Weil-étale cohomology of the archimedean fiber $\mathcal{X}_{\infty}$ with Tate twists $n$ and $d-n$, the Bloch conductor $A(\mathcal{X})$ and the special values of the archimedean Euler factor of the Zeta-function $ζ(\mathcal{X},s)$ at $s=n$ and $s=d-n$. This formula is a shadow of the functional equation of Zeta-functions.

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arXiv2021

On the kernel of the Brauer-Manin pairing

Let $\mathcal X$ be a regular scheme, flat and proper over the ring of integers of a $p$-adic field, with generic fiber $X$ and special fiber $\mathcal X_s$. We study the left kernel $Br(\mathcal X)$ of the Brauer-Manin pairing $Br(X)\times CH_0(X)\to \mathbb Q/\mathbb Z$. Our main result is that the kernel of the reduction map $Br(\mathcal X)\to Br(\mathcal X_s)$ is the direct sum of $(\mathbb Q/\mathbb Z[\frac{1}{p}])^s\oplus (\mathbb Q/\mathbb Z)^t$ and a finite $p$-group, where $s+t= ρ_{\mathcal X_s}-ρ_X-I+1$, for $ρ_{\mathcal X_s}$ and $ρ_X$ the Picard numbers of $\mathcal X_s$ and $X$, and $I$ the number of irreducible components of $\mathcal X_s$. Moreover, we show that $t>0$ implies $s>0$.

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arXiv2017

On the trace forms of Galois algebras

We study the trace form $q_L$ of $G$-Galois algebras $L/K$ when $G$ is a finite group and $K$ is a field of characteristic different from $2$. We introduce in this paper the category of $2$-reduced groups and, when $G$ is such a group, we use a formula of Serre to compute the second Hasse-Witt invariant of $q_L$. By combining this computation with work of Quillen we determine the isometry class of $q_L$ for large families of $G$-Galois algebras over global fields. We also indicate how our results generalize to Galois $G$-covers of schemes.

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arXiv2017

Weil-étale cohomology and Zeta-values of proper regular arithmetic schemes

We give a conjectural description of the vanishing order and leading Taylor coefficient of the Zeta function of a proper, regular arithmetic scheme $\mathcal{X}$ at any integer $n$ in terms of Weil-étale cohomology complexes. This extends work of Lichtenbaum \cite{Lichtenbaum05} and Geisser \cite{Geisser04b} for $\mathcal{X}$ of characteristic $p$, of Lichtenbaum \cite{li04} for $\mathcal{X}=\mathrm{Spec}(\mathcal{O}_F)$ and $n=0$ where $F$ is a number field, and of the second author for arbitrary $\mathcal{X}$ and $n=0$ \cite{Morin14}. We show that our conjecture is compatible with the Tamagawa number conjecture of Bloch, Kato, Fontaine and Perrin-Riou \cite{fpr91} if $\mathcal{X}$ is smooth over $\mathrm{Spec}(\mathcal{O}_F)$, and hence that it holds in cases where the Tamagawa number conjecture is known.

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arXiv2016

Milne's correcting factor and derived de Rham cohomology II

Milne's correcting factor, which appears in the Zeta-value at $s=n$ of a smooth projective variety $X$ over a finite field $\mathbb{F}_q$, is the Euler characteristic of the derived de Rham cohomology of $X/\mathbb{Z}$ modulo the Hodge filtration $F^n$. In this note, we extend this result to arbitrary separated schemes of finite type over $\mathbb{F}_q$ of dimension at most $d$, provided resolution of singularities for schemes of dimension at most $d$ holds. More precisely, we show that Geisser's generalization of Milne's factor, whenever it is well defined, is the Euler characteristic of the $eh$-cohomology with compact support of the derived de Rham complex relative to $\mathbb{Z}$ modulo $F^n$.

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arXiv2015

Tannakian twists of quadratic forms and orthogonal Nori motives

We revisit classical results of Serre, Fröhlich and Saito in the theory of quadratic forms. Given a neutral Tannakian category $(\mathcal{T},ω)$ over a field $k$ of characteristic $\neq 2$, another fiber functor $η$ over a $k$-scheme $X$ and an orthogonal object $(M,q)$ in $\mathcal{T}$, we show formulas relating the torsor $\bf{Isom}^{\otimes}(ω,η)$ to Hasse-Witt invariants of the quadratic space $ω(M,q)$ and the symmetric bundle $η(M,q)$. We apply this result to various neutral Tannakian categories arising in different contexts. We first consider Nori's Tannakian category of essentially finite bundles over an integral proper $k$-scheme $X$ with a rational point, in order to study an analogue of the Serre-Fröhlich embedding problem for Nori's fundamental group scheme. Then we consider Fontaine's Tannakian categories of $B$-admissible representations, in order to obtain a generalization of both the classical Serre-Fröhlich formula and Saito's analogous result for Hodge-Tate $p$-adic representations. Finally we consider Nori's category of mixed motives over a number field. These last two examples yield formulas relating the torsor of periods of an orthogonal motive to Hasse-Witt invariants of the associated Betti and de Rham quadratic forms and to Stiefel-Withney invariants of the associated local $l$-adic orthogonal representations. We give some computations for Artin motives and for the motive of a smooth hypersurface.

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arXiv2014

Milne's correcting factor and derived de Rham cohomology

Milne's correcting factor is a numerical invariant playing an important role in formulas for special values of zeta functions of varieties over finite fields. We show that Milne's factor is simply the Euler characteristic of the derived de Rham complex (relative to $\mathbb{Z}$) modulo the Hodge filtration.

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arXiv2013

Zeta functions of regular arithmetic schemes at s=0

Lichtenbaum conjectured the existence of a Weil-étale cohomology in order to describe the vanishing order and the special value of the Zeta function of an arithmetic scheme $\mathcal{X}$ at $s=0$ in terms of Euler-Poincaré characteristics. Assuming the (conjectured) finite generation of some étale motivic cohomology groups we construct such a cohomology theory for regular schemes proper over $\mathrm{Spec}(\mathbb{Z})$. In particular, we obtain (unconditionally) the right Weil-étale cohomology for geometrically cellular schemes over number rings. We state a conjecture expressing the vanishing order and the special value up to sign of the Zeta function $ζ(\mathcal{X},s)$ at $s=0$ in terms of a perfect complex of abelian groups $RΓ_{W,c}(\mathcal{X},\mathbb{Z})$. Then we relate this conjecture to Soulé's conjecture and to the Tamagawa number conjecture of Bloch-Kato, and deduce its validity in simple cases.

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arXiv2013

The classifying topos of a group scheme and invariants of symmetric bundles

Let $Y$ be a scheme in which 2 is invertible and let $V$ be a rank $n$ vector bundle on $Y$ endowed with a non-degenerate symmetric bilinear form $q$. The orthogonal group ${\bf O}(q)$ of the form $q$ is a group scheme over $Y$ whose cohomology ring $H^*(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})\simeq A_Y[HW_1(q),..., HW_n(q)]$ is a polynomial algebra over the étale cohomology ring $A_Y:=H^*(Y_{et},{\bf Z}/2{\bf Z})$ of the scheme $Y$. Here the $HW_i(q)$'s are Jardine's universal Hasse-Witt invariants and $B_{{\bf O}(q)}$ is the classifying topos of ${\bf O}(q)$ as defined by Grothendieck and Giraud. The cohomology ring $H^*(B_{{\bf O}(q)},{\bf Z}/2{\bf Z})$ contains canonical classes $\mathrm{det}[q]$ and $[C_q]$ of degree 1 and 2 respectively, which are obtained from the determinant map and the Clifford group of $q$. The classical Hasse-Witt invariants $w_i(q)$ live in the ring $A_Y$. Our main theorem provides a computation of ${det}[q]$ and $[C_{q}]$ as polynomials in $HW_{1}(q)$ and $HW_{2}(q)$ with coefficients in $A_Y$ written in terms of $w_1(q),w_2(q)\in A_Y$. This result is the source of numerous standard comparison formulas for classical Hasses-Witt invariants of quadratic forms. Our proof is based on computations with (abelian and non-abelian) Cech cocycles in the topos $B_{{\bf O}(q)}$. This requires a general study of the cohomology of the classifying topos of a group scheme, which we carry out in the first part of this paper.

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arXiv2011

Orthogonal representations of affine group schemes and twists of symmetric bundles

Following Serre's initial work, a number of authors have considered twists of quadratic forms on a scheme Y by torsors of a finite group G, together with formulas for the Hasse-Witt invariants of the twisted form. In this paper we take the base scheme Y to be affine and consider non-constant groups schemes G. There is a fundamental new feature in this case - in that the torsor may now be ramified over Y. The natural framework for handling the case of a non-constant group scheme over the affine base is provided by the quadratic theory of Hopf-algebras.

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arXiv2010

On the Weil-étale topos of regular arithmetic schemes

We define and study a Weil-étale topos for any regular, proper scheme $X$ over $\Spec(Z)$ which has some of the properties suggested by Lichtenbaum for such a topos. In particular, the cohomology with $R$-coefficients has the expected relation to $ζ(X,s)$ at $s=0$ if the Hasse-Weil L-functions $L(h^i(X_Q),s)$ have the expected meromorphic continuation and functional equation. If $\X$ has characteristic $p$ the cohomology with $Z$-coefficients also has the expected relation to $ζ(X,s)$ and our cohomology groups recover those previously studied by Lichtenbaum and Geisser.

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arXiv2010

Sur l'analogie entre le système dynamique de Deninger et le topos Weil-étale

We express some basic properties of Deninger's conjectural dynamical system in terms of morphisms of topoi. Then we show that the current definition of the Weil-étale topos satisfies these properties. In particular, the flow, the closed orbits, the fixed points of the flow and the foliation in characteristic $p$ are well defined on the Weil-étale topos. This analogy extends to arithmetic schemes. Over a prime number $p$ and over the archimedean place of $\mathbb{Q}$, we define a morphism from a topos associated to Deninger's dynamical system to the Weil-étale topos. This morphism is compatible with the structure mentioned above.

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arXiv2010

The Weil-étale fundamental group of a number field II

We define the fundamental group underlying to Lichtenbaum's Weil-étale cohomology for number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced in \cite{Lichtenbaum}. We show that the (small) Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition given in \cite{Lichtenbaum-finite-field}. Then we compute the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring. Our fundamental group is a projective system of locally compact topological groups, which represents first degree cohomology with coefficients in locally compact abelian groups. We apply this result to compute the Weil-étale cohomology in low degrees and to prove that the Weil-étale topos of a number ring satisfies the expected properties of the conjectural Lichtenbaum topos.

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arXiv2010

The Weil-étale fundamental group of a number field I

Lichtenbaum has conjectured the existence of a Grothendieck topology for an arithmetic scheme $X$ such that the Euler characteristic of the cohomology groups of the constant sheaf $\mathbb{Z}$ with compact support at infinity gives, up to sign, the leading term of the zeta-function $ζ_X(s)$ at $s=0$. In this paper we consider the category of sheaves $\bar{X}_L$ on this conjectural site for $X=Spec(\mathcal{O}_F)$ the spectrum of a number ring. We show that $\bar{X}_L$ has, under natural topological assumptions, a well defined fundamental group whose abelianization is isomorphic, as a topological group, to the Arakelov Picard group of $F$. This leads us to give a list of topological properties that should be satisfied by $\bar{X}_L$. These properties can be seen as a global version of the axioms for the Weil group. Finally, we show that any topos satisfying these properties gives rise to complexes of étale sheaves computing the expected Lichtenbaum cohomology.

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