Comparison Isomorphism Research Articles

Plectic structures in p-adic de Rham cohomology

Given a Hilbert modular form for a totally real field F, and a prime p split completely in F, the f-eigenspace in p-adic de Rham cohomology has a family of partial filtrations and partial Frobenius maps, indexed by the primes of F above p. The general plectic conjectures of Nekovář and Scholl suggest a “plectic comparison isomorphism” comparing these structures to étale cohomology. We prove this conjecture in the case [F:Q]=2 under some mild assumptions; and for general F we prove a weaker statement which is strong evidence for the conjecture, showing that the plectic Hodge filtration has a canonical splitting given by intersecting with simultaneous eigenspaces for the partial Frobenii.

Comparison of prismatic cohomology and derived de Rham cohomology

We establish a comparison isomorphism between prismatic cohomology and derived de Rham cohomology respecting various structures, such as their Frobenius actions and filtrations. As an application, when X is a proper smooth formal scheme over \mathcal{O}_K with K being a p -adic field, we improve Breuil–Caruso’s theory on comparison between torsion crystalline cohomology and torsion étale cohomology.

De Rham Comparison and Poincaré Duality for Rigid Varieties

Over any smooth algebraic variety over a p-adic local field k, we construct the de Rham comparison isomorphisms for the étale cohomology with partial compact support of de Rham \({\mathbb {Z}}_p\)-local systems, and show that they are compatible with Poincaré duality and with the canonical morphisms among such cohomology. We deduce these results from their analogues for rigid analytic varieties that are Zariski open in some proper smooth rigid analytic varieties over k. In particular, we prove finiteness of étale cohomology with partial compact support of any \({\mathbb {Z}}_p\)-local systems, and establish the Poincaré duality for such cohomology after inverting p.

A remark on singular cohomology and sheaf cohomology

We prove a comparison isomorphism between singular cohomology and sheaf cohomology.

On the subring of special cycles

Abstract By old results with Millson, the generating series for the cohomology classes of special cycles on orthogonal Shimura varieties over a totally real field are Hilbert–Siegel modular forms. These forms arise via theta series. Using this result and the Siegel–Weil formula, we show that the products in the subring of cohomology generated by the special cycles are controlled by the Fourier coefficients of triple pullbacks of certain Siegel–Eisenstein series. As a consequence, there are comparison isomorphisms between special subrings for different Shimura varieties. In the case in which the signature of the quadratic space V is ( m , 2 ) (m,2) at an even number d + d_{+} of archimedean places, the comparison gives a “combinatorial model” for the special cycle ring in terms of the associated totally positive definite space.

Single-valued integration and double copy

Abstract In this paper, we study a single-valued integration pairing between differential forms and dual differential forms which subsumes some classical constructions in mathematics and physics. It can be interpreted as a p-adic period pairing at the infinite prime. The single-valued integration pairing is defined by transporting the action of complex conjugation from singular to de Rham cohomology via the comparison isomorphism. We show how quite general families of period integrals admit canonical single-valued versions and prove some general formulae for them. This implies an elementary “double copy” formula expressing certain singular volume integrals over the complex points of a smooth projective variety as a quadratic expression in ordinary period integrals of half the dimension. We provide several examples, including non-holomorphic modular forms, archimedean Néron–Tate heights on curves, single-valued multiple zeta values and polylogarithms. The results of the present paper are used in [F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, preprint 2019, https://arxiv.org/abs/1910.01107] to prove a recent conjecture of Stieberger which relates the coefficients in a Laurent expansion of two different kinds of periods of twisted cohomology on the moduli spaces of curves ℳ 0 , n {\mathcal{M}_{0,n}} of genus zero with n marked points. We also study a morphism between certain rings of “motivic” periods, called the de Rham projection, which provides a bridge between complex periods and single-valued periods in many situations of interest.

Special functions and Gauss–Thakur sums in higher rank and dimension

Abstract Anderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 1980s where they were at the heart of comparison isomorphisms, further important applications, e.g., to transcendence theory have only been discovered recently. The Anderson–Thakur special function interpolates L-values via Pellarin-type identities, and its values at algebraic elements recover Gauss–Thakur sums, as shown by Anglès and Pellarin. For Drinfeld–Hayes modules, generalizations of Anderson generating functions have been introduced by Green–Papanikolas and – under the name of “special functions” – by Anglès, Ngo Dac and Tavares Ribeiro. In this article, we provide a general construction of special functions attached to any Anderson A-module. We show direct links of the space of special functions to the period lattice, and to the Betti cohomology of the A-motive. We also undertake the study of Gauss–Thakur sums for Anderson A-modules, and show that the result of Anglès–Pellarin relating values of the special functions to Gauss–Thakur sums holds in this generality.

Product formulas for periods of CM abelian varieties and the function field analog

We survey Colmez's theory and conjecture about the Faltings height and a product formula for the periods of abelian varieties with complex multiplication, along with the function field analog developed by the authors. In this analog, abelian varieties are replaced by Drinfeld modules and A-motives. We also explain the necessary background on abelian varieties, Drinfeld modules and A-motives, including their cohomology theories and comparison isomorphisms and their theory of complex multiplication.

RELATIVE UNITARY RZ-SPACES AND THE ARITHMETIC FUNDAMENTAL LEMMA

AbstractWe prove a comparison isomorphism between certain moduli spaces of$p$-divisible groups and strict${\mathcal{O}}_{K}$-modules (RZ-spaces). Both moduli problems are of PEL-type (polarization, endomorphism, level structure) and the difficulty lies in relating polarized$p$-divisible groups and polarized strict${\mathcal{O}}_{K}$-modules. We use the theory of relative displays and frames, as developed by Ahsendorf, Lau and Zink, to translate this into a problem in linear algebra. As an application of these results, we verify new cases of the arithmetic fundamental lemma (AFL) of Wei Zhang: The comparison isomorphism yields an explicit description of certain cycles that play a role in the AFL. This allows, under certain conditions, to reduce the AFL identity in question to an AFL identity in lower dimension.

Standard conjectures in model theory, and categoricity of comparison isomorphisms: A model theory perspective

We formulate two conjectures about étale cohomology and fundamental groups motivated by categoricity conjectures in model theory. One conjecture says that there is a unique -form of the étale cohomology of complex algebraic varieties, up to -action on the source category; put differently, each comparison isomorphism between Betti and étale cohomology comes from a choice of a topology on Another conjecture says that each functor to groupoids from the category of complex algebraic varieties which is similar to the topological fundamental groupoid functor in fact factors through up to a field automorphism of the complex numbers acting on the category of complex algebraic varieties. We also try to present some evidence towards these conjectures, and show that some special cases seem related to Grothendieck standard conjectures and conjectures about motivic Galois group.

Crystalline comparison isomorphisms in p-adic Hodge theory :the absolutely unramified case

We construct the crystalline comparison isomorphisms for proper smooth formal schemes over an absolutely unramified base. Such isomorphisms hold for \'etale cohomology with nontrivial coefficients, as well as in the relative setting, i.e. for proper smooth morphisms of smooth formal schemes. The proof is formulated in terms of the pro-\'etale topos introduced by Scholze, and uses his primitive comparison theorem for the structure sheaf on the pro-\'etale site. Moreover, we need to prove the Poincar\'e lemma for crystalline period sheaves, for which we adapt the idea of Andreatta and Iovita. Another ingredient for the proof is the geometric acyclicity of crystalline period sheaves, whose computation is due to Andreatta and Brinon.

On $p$-adic absolute Hodge cohomology and syntomic coefficients. I

We interpret syntomic cohomology defined in [50] as a p -adic absolute Hodge cohomology. This is analogous to the interpretation of Deligne–Beilinson cohomology as an absolute Hodge cohomology by Beilinson [8] and generalizes the results of Bannai [6] and Chiarellotto, Ciccioni, Mazzari [15] in the good reduction case. This interpretation yields a simple construction of the syntomic descent spectral sequence and its degeneration for projective and smooth varieties. We introduce syntomic coefficients and show that in dimension zero they form a full triangulated subcategory of the derived category of potentially semistable Galois representations. Along the way, we obtain p -adic realizations of mixed motives including p -adic comparison isomorphisms. We apply this to the motivic fundamental group generalizing results of Olsson and Vologodsky [56, 71].

The geometry of Hida families II: -adic -modules and -adic Hodge theory

We construct the $\unicode[STIX]{x1D6EC}$-adic crystalline and Dieudonné analogues of Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology, and employ integral $p$-adic Hodge theory to prove $\unicode[STIX]{x1D6EC}$-adic comparison isomorphisms between these cohomologies and the $\unicode[STIX]{x1D6EC}$-adic de Rham cohomology studied in Cais [The geometry of Hida families I:$\unicode[STIX]{x1D6EC}$-adic de Rham cohomology, Math. Ann. (2017), doi:10.1007/s00208-017-1608-1] as well as Hida’s $\unicode[STIX]{x1D6EC}$-adic étale cohomology. As applications of our work, we provide a ‘cohomological’ construction of the family of $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules attached to Hida’s ordinary $\unicode[STIX]{x1D6EC}$-adic étale cohomology by Dee [$\unicode[STIX]{x1D6F7}$–$\unicode[STIX]{x1D6E4}$modules for families of Galois representations, J. Algebra 235 (2001), 636–664], and we give a new and purely geometric proof of Hida’s finiteness and control theorems. We also prove suitable $\unicode[STIX]{x1D6EC}$-adic duality theorems for each of the cohomologies we construct.

An integrable connection on the configuration space of a Riemann surface of positive genus

Let X be a Riemann surface of positive genus. Denote by X(n) the configuration space of n distinct points on X. We use the Betti–de Rham comparison isomorphism on H1(X(n)) to define an integrable connection on the trivial vector bundle on X(n) with fiber the universal algebra of the Lie algebra associated with the descending central series of π1 of X(n). The construction is inspired by the Knizhnik–Zamolodchikov system in genus zero and its integrability follows from Riemann period relations.

The geometry of Hida families I: $$\Lambda $$-adic de Rham cohomology

We construct the $$\Lambda $$ -adic de Rham analogue of Hida’s ordinary $$\Lambda $$ -adic etale cohomology and of Ohta’s $$\Lambda $$ -adic Hodge cohomology, and by exploiting the geometry of integral models of modular curves over the cyclotomic extension of $$\mathbf {Q}_p$$ , we give a purely geometric proof of the expected finiteness, control, and $$\Lambda $$ -adic duality theorems. Following Ohta, we then prove that our $$\Lambda $$ -adic module of differentials is canonically isomorphic to the space of ordinary $$\Lambda $$ -adic cuspforms. In the sequel (Cais, Compos Math, to appear) to this paper, we construct the crystalline counterpart to Hida’s ordinary $$\Lambda $$ -adic etale cohomology, and employ integral p-adic Hodge theory to prove $$\Lambda $$ -adic comparison isomorphisms between all of these cohomologies. As applications of our work in this paper and (Cais, Compos Math, to appear), we will be able to provide a “cohomological” construction of the family of $$(\varphi ,\Gamma )$$ -modules attached to Hida’s ordinary $$\Lambda $$ -adic etale cohomology by Dee (J Algebra 235(2), 636–664, 2001), as well as a new and purely geometric proof of Hida’s finiteness and control theorems. We are also able to prove refinements of the main theorems in Mazur and Wiles (Compos Math 59(2):231–264, 1986) and Ohta (J Reine Angew Math 463:49–98, 1995).

Théorème de comparaison pour les cycles proches par un morphisme sans pente

The goal of this article is to prove the comparison theorem between algebraic and topological nearby cycles of a morphism without slopes. We prove in particular that for a family of holomorphic functions without slopes, if we iterate comparison isomorphisms for nearby cycles of each function the result is independent of the order of iteration.

Syntomic complexes and p-adic nearby cycles

We compute syntomic cohomology of semistable affinoids in terms of cohomology of $$(\varphi ,\Gamma )$$ -modules which, thanks to work of Fontaine–Herr, Andreatta–Iovita, and Kedlaya–Liu, is known to compute Galois cohomology of these affinoids. For a semistable scheme over a mixed characteristic local ring this implies a comparison isomorphism, up to some universal constants, between truncated sheaves of p-adic nearby cycles and syntomic cohomology sheaves. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. As an application, we combine this local comparison isomorphism with the theory of finite dimensional Banach Spaces and finiteness of étale cohomology of rigid analytic spaces proved by Scholze to prove a Semistable conjecture for formal schemes with semistable reduction.

On some special types of Fontaine sheaves and their properties

In this paper we construct special types of Fontaine sheaves \(\mathbb{A}_{\max }\) and \(\mathbb{A}_{\max }^\nabla\) and we study their properties, most importantly their localizations over small affines. They will be used in sequel work to prove in a different manner a comparison isomorphism theorem of Faltings [7]. We conclude with making several conjectures.

An alternative description of the Drinfeld $p$-adic half-plane

We show that the Deligne formal model of the Drinfeld p-adic halfplane relative to a local field F represents a moduli problem of polarized OF -modules with an action of the ring of integers in a quadratic extension E of F . The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of SL2(F ) and SU(C)(F ) for a two-dimensional split hermitian space C for E/F .

-ADIC HODGE THEORY FOR RIGID-ANALYTIC VARIETIES

AbstractWe give proofs of de Rham comparison isomorphisms for rigid-analytic varieties, with coefficients and in families. This relies on the theory of perfectoid spaces. Another new ingredient is the pro-étale site, which makes all constructions completely functorial.