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Marco Adamo Seveso

Publications and source records attributed to Marco Adamo Seveso.

5 recordsLinked to original sources
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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arXiv2016

Triple product p-adic L-functions for balanced weights

We construct $p$-adic triple product $L$-functions that interpolate (square roots of) central critical $L$-values in the balanced region. Thus, our construction complements that of M. Harris and J. Tilouine. There are four central critical regions for the triple product $L$-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three $p$% -adic $L$-functions, one for each region of interpolation (this is the Harris-Tilouine setting). In the other setting there is only one region of interpolation, called the balanced region. An especially interesting feature of our construction is that we get three different $p$-adic triple product $% L $-functions with the same (balanced) region of interpolation. To the best of the authors' knowledge, this is the first case where an interpolation problem is solved on a single critical region by different $p$-adic $L$% -functions at the same time. This is possible due to the structure of the Euler-like factors at $p$ arising in the interpolation formulas, the vanishing of which are related to the dimensions of certain Nekovar period spaces. Our triple product $p$-adic $L$-functions arise as specializations of $p$-adic period integrals interpolating normalizations of the local archimedean period integrals. The latter encode information about classical representation theoretic branching laws. The main step in our construction of $p$-adic period integrals is showing that these branching laws vary in a $% p$-adic analytic fashion. This relies crucially on the Ash-Stevens theory of highest weight representations over affinoid algebras.

math.NT↗
arXiv2010

L-invariants and Darmon cycles attached to modular forms

Let f be a modular eigenform of even weight k>0 and new at a prime p dividing exactly the level, with respect to an indefinite quaternion algebra. The theory of Fontaine-Mazur allows to attach to f a monodromy module D_FM(f) and an L-invariant L_FM(f). The first goal of this paper is building a suitable p-adic integration theory that allows us to construct a monodromy module D(f) and an L-invariant L(f) in the spirit of Darmon. We conjecture both monodromy modules are isomorphic, and in particular the two L-invariants are equal. For the second goal of this note we assume the conjecture is true. Let K be a real quadratic field and assume the sign of the functional equation of the L-series of f over K is -1. The Bloch-Beilinson conjectures suggest that there should be a supply of elements in the Mordell-Weil group of the motive attached to f over the tower of narrow ring class fields of K. Generalizing work of Darmon for k=2, we give a construction of local cohomology classes which we expect to arise from global classes and satisfy an explicit reciprocity law, accounting for the above prediction.

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