Research


Interests: p-adic Hodge theory, moduli spaces of abelian varieties, Shimura varieties.




Here is the list of my papers, published or in progress. The pfd files could be (very) different from the published version:


Together with Adrian Iovita and inspired by work of E. Urban and by discussions with H. Darmon, we have constructed p-adic families of de Rham classes, providing the notion of nearly overconvergent elliptic modular forms. As applications we can define triple product p-adic L-functions for finite slope forms and p-adic L-functions for imaginary quadratic fileds at non-split primes..

- ANDREATTA F., IOVITA A., Katz type p-adic L-funztions for primes p non-split in the CM field , Submitted PadicL.pdf.

Warning The paper has been revised, notably we corrected an error occurred in the previous version in the comparison with classical L-values. In particular, we need to assume that some power of p (eg p^2 if p\geq 5) divides the conductor of the relevant Hecke characters. In fact, we allow arbitrary powers p^n (for n\geq 2 for p\geq 5) to divide the conductor. We also construct the two variable p-adic L-function and prove that it is locally analytic.


- ANDREATTA F., IOVITA A., Triple product p-adic L-functions associated to finite slope p-adic modular forms. , To appear in Duke Math. Journal LLL.pdf.




Some global results on the arithmetic of Shimura varieties of orthogonal type in the spirit of Kudla's programme, with applications to conjectures of Bruinier-Yang and Bruinier-Kudla-Yang, yielding a proof of an avaraged form of a conjecture of Colmez.

- ANDREATTA F. The height of CM points on orthogonal Shimura varieties and Colmez’s conjecture Ch. XII of Lecture Notes in Mathematics, vol. 2276 (2020) Grenoble.pdf.

These are the notes of a summer school organized on Arakelov Geometry and Diophantine Applications organized by by Emmanuel Peyre and Gaël Rémond at the Institut Fourier (Grenoble) in 2017, based on the following paper:

- ANDREATTA F., GOREN E.Z, HOWARD B., MADAPUSI PERA K. Faltings heights of abelian varieties with complex multiplication , Annals of Mathematics, vol. 187 (2018), pp. 391–531. Colmez.pdf.

- ANDREATTA F., GOREN E.Z, HOWARD B., MADAPUSI PERA K. Height pairings on orthogonal Shimura varieties , Compositio Math., vol. 153 (2017), pp. 474-534. GSpin.pdf.



Inspired by computations and conjectures by Robert Coleman we have constructed adic elliptic and Hilbert eigenvarities i.e., eigenvarieties over the adic weight space (in the sense of Huber), adding a still mysterious charcateristic p fiber

- ANDREATTA F., IOVITA A., PILLONI V. The adic, cuspidal, Hilbert eigenvarieties , Research in the Mathematical Sciences, vol. 3 (2016), 3:34. A special volume in honor of Robert F. Coleman, Hilbertadic.pdf.

- ANDREATTA F., IOVITA A., PILLONI V. Spectral halo, To appear in the Ann. Scient. Éc. Norm. Sup., SpectralHalo.pdf.




In recent years we have been able to construct p-adic families and eigenvarieties in a geometric fashion using the theory of the canonical subgroup:

- ANDREATTA F., IOVITA A., PILLONI V. p-Adic variation of automorphic sheaves, Proceedings of the ICM 2018 , Vol 1, pp. 291--318 ICM.pdf.

- ANDREATTA F., IOVITA A., STEVENS G. A 0.5 (half) overconvergent Eichler-Shimura isomorphism. Annales mathématiques du Québec, vol. 40 (2016), pp. 121-148. A volume in honor of Glenn Stevens' 60th birthday GlennFest.pdf

- ANDREATTA F., IOVITA A., STEVENS G. Overconvergent Eichler-Shimura isomorphisms , Journal of the Institute of Mathematics of Jussieu, vol. 14 (2015), pp. 221-274. EichlerShimura.pdf.

- ANDREATTA F., IOVITA A., PILLONI V. On overconvergent Hilbert modular cusp forms, Astérisque, vol. 382 (2016), pp. 163–193 Hilbert.pdf.

- ANDREATTA F., IOVITA A., PILLONI V. p-Adic families of Siegel modular cuspforms, Annals of Mathematics, vol. 181 (2015), pp. 623-697 Siegel.pdf.

- ANDREATTA F., IOVITA A., STEVENS G. Overconvergent modular sheaves and modular forms for GL_{2/F} , Israel Journal of Mathematics, vol. 201 (2014), pp. 299--359.AIS.pdf.



Here are some papers where we revisit and improve Faltings' method to prove the comparison isomorphisms C_cris and C_st:

- ANDREATTA F., IOVITA A. Semistable sheaves and comparison isomorphisms in the semistable case. A volume in honor of Francesco Baldassarri 60th birthday. Rend. Sem. Mat. univ. Padova n. 128 (2012) pp. 131--285. SemistableComparison.pdf.

- ANDREATTA F., IOVITA A. Comparison Isomorphisms for Smooth Formal Schemes. Journal de l'Institut de Mathématiques de Jussieu n. 12 (2013) pp. 77--151. SmoothComparison.pdf.

- ANDREATTA F., BRINON O. Acyclicité géométrique de B_cris relatif. Commentarii Mathematici Helvetici, vol. 88, p. 965–1022 (2013). Acyclicite.pdf.



Here is an application of our approach to the comparison isomorphisms to non-abelian geometry:

- ANDREATTA F., IOVITA A., KIM M. A p-adic non-abelian criterion for good reduction of curves, Duke Math. Journal. vol. 164, p. 2597-2642 (2015).Anabelian.pdf.



A paper whose interests date back to the previous millennium when I did my PhD.

- ANDREATTA F. Coleman–Oort’s conjecture for degenerate irreducible curves. Israel Journal of Mathematics, vol. 187, p. 231-285 (2012) ColemanOortConjectureFinal.pdf.



Papers on Motives and their periods, especially in mixed characteristics:

- ANDREATTA F., BARBIERI--VIALE L., BERTAPELLE A. Motivic periods and Grothendieck arithmetic invariants, with an appendix by appendix by B.Kahn . Advances in Math. 359, 50 pp. (2020).P1MAFin.pdf.

- ANDREATTA F., BARBIERI--VIALE L., BERTAPELLE A. Ogus realization of 1-motives.. Journal of Algebra 487, pp. 294-316 (2017).Ogus.pdf.

- ANDREATTA F., BERTAPELLE A. Universal extension crystals of 1-motives and applications. Journal of Pure and Applied Algebra, vol. 215, p. 1919-1944 (2011) Crystalline-1-Motives.pdf.

- ANDREATTA F., BARBIERI--VIALE L. Crystalline realizations of 1-motives. Mathematische Annalen, vol. 331, pp. 111-172 (2005).



Several works where we developed Fontaine's theory in the relative setting , assuming good reduction

- ANDREATTA F., BRINON O. B_dR-représentations dans le cas relatif. Ann. Scient. Éc. Norm. Sup., vol. 43, p. 279-339 (2010) SenBdR.pdf.

- ANDREATTA F., IOVITA A. Global applications of relative (phi, Gamma)-modules I. Astérisque vol. 319, p. 339-420 (2008) RelativephiGamma.pdf.
- ANDREATTA F., IOVITA A. Erratum to the article: Global applications to relative (phi,Gamma)-modules I. Astérisque, vol. 330, p. 543-554 Erratum.pdf.

- ANDREATTA F., BRINON O. Surconvergence des représentations p-adiques : le cas relatif. Astérisque, vol. 319, p. 39-116 (2008) TateSen.pdf.

- ANDREATTA F. Generalized ring of norms and generalized (phi,Gamma)-modules. Ann. Scient. Éc. Norm. Sup., vol. 39, pp. 599-647 (2006).





Papers related to the theory of the deformation of torsors and the theory of the canonical subgroup:

- ANDREATTA F., GASBARRI C. Deformation of torsors under monogenic group schemes . Journal de Théorie des Nombres de Bordeaux, vol 28 (2016), pp. 125–143. Torsors.pdf.

- ANDREATTA F., GASBARRI C. Torsors under some group schemes of order p^n. J. of Algebra, vol. 318, pp. 1057-1067 (2007).

- ANDREATTA F., GASBARRI C. The canonical subgroup for families of abelian varieties. Compositio Math., vol. 143, pp. 566--602 (2007).





Some papers on Hilbert modular varieties and Hilbert modular forms:

- ANDREATTA F., GOREN E.Z. Hilbert modular forms: mod p and p-adic aspects. Memoirs of the American Mathematical Society, vol. 173 n. 819 (2005).

- ANDREATTA F., GOREN E.Z. Hilbert modular varieties of low dimension. In ``Geometric Aspects of Dwork's Theory, A Volume in memory of Bernard Dwork." Edited by A. Adolphson, F. Baldassarri, P. Berthelot, N.M. Katz, F. Loeser. De Gruyter Proceedings in Mathematics, pp. 113-176 (2004).

- ANDREATTA F., GOREN E.Z. Geometry of Hilbert modular varieties over ramified primes. International Mathematics Research Notices, vol. 33 (2003), pp. 1785-1835.



My PhD thesis under the supervision of Frans Oort

- ANDREATTA F. The small Schottky--Jung locus in positive characteristics different from two. Annales de l'Institut Fourier, vol. 53 (2003), pp. 69-106.

- ANDREATTA F. On Mumford's uniformization and Néron models of Jacobians of semistable curves over complete rings. Moduli of abelian varieties, Progr. Math. 195 (2001), 11--126.