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Fabrizio Andreatta

Publications and source records attributed to Fabrizio Andreatta.

16 recordsLinked to original sources
arXiv2026

BGG-decomposition for de Rham Banach sheaves

In this article we refer to "the BGG method" as a method in which by using Lie-algebra techniques one produces a complex of coherent sheaves on a Shimura variety, which is quasi-isomorphic to the de Rham complex of an automorphic vector bundle with integrable connection, on that Shimura variety. These BGG-complexes are in many ways "smaller" than the de Rham complex and have better cohomological properties. These BGG complexes contribute to the understanding of automorphic representations. In this article, we extend the BGG technique and apply it to complexes of Banach-modules with integrable connection on relevant open subspaces of Shimura varieties, seen as adic analytic spaces. Furthermore, we explain how this method can be used to obtain a new computation of the de Rham cohomology of certain p-adic automorphic representations in the GL_2 case. In future work, the authors intend to apply the techniques developed here to the Hilbert and Siegel cases.

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arXiv2026

Syntomic formalism with coefficients

This paper provides the technical tools needed in ongoing work of the authors to compute p-adic étale Abel-Jacobi maps in order to obtain explicit reciprocity laws for GSp4. In particular, we define and study syntomic polynomial cohomology for filtered Frobenius log-isocrystals over proper and semistable schemes over the ring of integers of a local field, with smooth generic fiber, endowed with horizontal divisors. We introduce syntomic polynomial cohomology with support, we define Kunneth morphisms, trace maps and cup products, Gysin maps with respect to divisors and we study some of their properties. We establish the relation with Hyodo-Kato cohomology of the special fiber and de Rham cohomology of the generic fiber. We also introduce overconvergent variants with and without support by restricting to open smooth formal subschemes. Most of all, in case that the filtered log-isocrystal is associated to a p-adic local system on the generic fiber, we establish comparison morphisms between étale and syntomic cohomology and compatibilities with Hochschild-Serre morphisms and between Gysin morphisms.

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arXiv2025

On a p-adic version of Narasimhan and Seshadri's theorem

Consider a smooth projective curve C of genus g over a complete discrete valuation field of characteristic 0 and residue field \Fbar_p. Motivated by Narasimhan and Seshadri's theorem, Faltings asked whether all semistable vector bundles of degree 0 over C_{\C_p} are in the image of the p-adic Simpson correspondence. Works of Deninger-Werner and Xu show that this is equivalent for the vector bundle to having potentially strongly semistable reduction. We prove that if C has good reduction, p>r(r-1) (g-1) and we consider a vector bundle of rank r with stable reduction, the conditions of having potentially strongly semistable reduction and of having strongly semistable reduction are equivalent. In particular, we provide a negative answer to Faltings' question

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arXiv2024

Katz type p-adic L-functions for primes p non-split in the CM field

For every triple F,K,p where F is a classical elliptic eigenform, K is a quadratic imaginary field and p> 3 is a prime integer which is not split in K, we attach a p-adic L function which interpolates the algebraic parts of the special values of the complex L-functions of F twisted by certain algebraic Hecke characters of K. This construction extends a classical construction of N. Katz, for F an Eisenstein series and of Bertolini-Darmon-Prasana, for F a cuspform, when p is split in K. Moreover we prove a Kronecker limit formula, respectively p-adic Gross-Zagier formulae for our newly defined p-adic L-functions.

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arXiv2021

On two mod p period maps: Ekedahl--Oort and fine Deligne--Lusztig stratifications

Consider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime p>3. Consider its perfectoid cover S(p^\infty) and the Hodge-Tate period map introduced by A. Caraiani and P. Scholze. We compare the pull-back to S(p^\infty) of the Ekedahl-Oort stratification on the mod p special fiber of S and the pull back to S(p^\infty) of the fine Deligne-Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge-Tate period map. An application to the non-emptiness of Ekedhal-Oort strata is provided.

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arXiv2020

Overconvergent de Rham Eichler-Shimura morphisms

This article represents our attempt to improve the previous results on defining and understanding overconvergent Eichler-Shimura maps for overconvergent modular symbols (in the elliptic case).

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arXiv2019

Triple product p-adic L-functions associated to finite slope p-adic families of modular forms, with an appendix by Eric Urban

We p-adically interpolate the relative de Rham cohomology of the universal elliptic curve over strict neighbourhoods of the ordinary locus of modular curves, together with the Hodge filtration and Gauss-Manin connection. Sections of these sheaves provide the so called nearly overconvergent modular forms. This extends previous work of Andreatta, Iovita and Pilloni where we p-adically interpolate powers of the Hodge bundle and in that case the sections coincide with Coleman overconvergent modular forms. We also show that, under suitable assumptions, one can p-adically interpolate the Gauss-Manin connection. This is used to define p-adic L-functions attached to a triple of p-adic finite slope families of modular forms, generalizing previous constructions for Hida families.

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arXiv2017

Faltings heights of abelian varieties with complex multiplication

Let M be the Shimura variety associated with the group of spinor similitudes of a rational quadratic space over of signature (n,2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain $L$-functions. As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties.

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arXiv2016

Height pairings on orthogonal Shimura varieties

Let $M$ be the Shimura variety associated to the group of spinor similitudes of a quadratic space over $\mathbb{Q}$ of signature $(n,2)$. We prove a conjecture of Bruinier and Yang, relating the arithmetic intersection multiplicities of special divisors and CM points on $M$ to the central derivatives of certain $L$-functions. Each such $L$-function is the Rankin-Selberg convolution associated with a cusp form of half-integral weight $n/2 +1 $, and the weight $n/2$ theta series of a positive definite quadratic space of rank $n$. When $n=1$ the Shimura variety $M$ is a classical quaternionic Shimura curve, and our result is a variant of the Gross-Zagier theorem on heights of Heegner points.

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arXiv2013

Overconvergent Eichler-Shimura isomorphisms

We provide a geometric Hodge-Tate map giving generic description of the overconvergent modular symbols of some p-adic (accessible) weight k, base-changed to C_p, in terms of overconvergent modular forms of weight k+2.

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arXiv2012

P-Adic families of Siegel modular cuspforms

Let p be an odd prime and g an integer greater or equal to 2. We prove that a finite slope Siegel cuspidal eigenform of genus g can be p-adically deformed over the g-dimensional weight space. The proof of this result relies on the construction of a family of sheaves of locally analytic overconvergent modular forms.

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arXiv2012

Comparison Isomorphisms for Smooth Formal Schemes

For a smooth proper scheme or formal scheme over an unramified, complete DVR of mixed characteristics we prove a comparison isomorphism relating etale cohomology of the generic fiber with values in a crystalline etale sheaf to the crystalline cohomology of its special fiber with values in the associated F-isocrystal.

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arXiv2006

Global applications of relative (phi-Gamma)-modules I

In this paper, given a smooth proper scheme X over a p-adic dvr and a p-power torsion etale local system L on it, we study a family of sheaves associated to the cohomology of local relative (Phi-Gamma)-modules of L and their cohomology. As applications we derive descriptions of the etale cohomology groups on the geometric generic fiber of X with values in L, as well as of their classical (Phi-Gamma)-modules, in terms of cohomology of the above mentioned sheaves.

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arXiv2003

Crystalline realizations of 1-motives

We consider the crystalline realization of Deligne's 1-motives in positive characteristics and prove a comparison theorem with the De Rham realization of liftings to zero characteristic. We then show that one dimensional crystalline cohomology of an algebraic variety, defined by universal cohomological descent via de Jong's alterations, coincide with the crystalline realization of the (cohomological) Picard 1-motive, over perfect fields.

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arXiv2003

Hilbert modular forms: mod $p$ and $p$-adic aspects

We study Hilbert modular forms in characteristic $p$ and over $p$-adic rings. In the characteristic $p$-theory we describe the kernel and image of the $q$-expansion map and prove the existence of filtration for Hilbert modular forms; we define operators $U$, $V$ and $Θ_χ$ and study the variation of the filtration under these operators. In particular, we prove that every ordinary eigenform has filtration in a prescribed box of weights. Our methods are geometric -- comparing holomorphic Hilbert modular forms with rational functions on a moduli scheme with level-$p$ structure, whose poles are supported on the non-ordinary locus. In the $p$-adic theory we study congruences between Hilbert modular forms. This applies to the study of congruences between special values of zeta functions of totally real fields. It also allows us to define $p$-adic Hilbert modular forms "à la Serre" as $p$-adic uniform limit of classical modular forms, and compare them with the $p$-adic modular forms "à la Katz" that are regular functions on a certain formal moduli scheme. We show that the two notions agree for cusp forms and for a suitable class of weights containing all the classical ones. We extend the operators $V$ and $Θ_χ$ to the $p$-adic setting.

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