arXiv subjects

Séverin Philip

Publications and source records attributed to Séverin Philip.

9 recordsLinked to original sources

arXiv2026

Finite monodromy groups in degenerating families and the stability degree of curves

We compute the stability degree of curves of genus g and show that it is equal to the classical Minkowsky bound. We also compute the stability degree of semi-abelian varieties with prescribed toric and abelian ranks. The proof uses a specialization result for finite monodromy groups in degenerating families, obtained from a variant of Krasner's lemma, together with an explicit construction at the prime 2.

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arXiv2026

Anabelian geometry for Deligne-Mumford curves

We develop an anabelian framework for general Deligne-Mumford curves, showing that their stack and orbifold structures are encoded in the group-theoretic properties of their étale fundamental groups. After establishing the required properties for profinite F-groups, we prove that fundamental geometric features, including hyperbolicity, affineness, and inertia data, can already be detected from low-level solvable quotients of the associated profinite groups, namely at the optimal 3-step level. As a consequence, we obtain some anabelian reconstruction results for Deligne-Mumford curves, their rigidifications, and their coarsification. While the m-step Grothendieck conjecture doesn't hold for Deligne-Mumford curves, we establish a 5-step anabelian theorem for the rigidification of affine Deligne-Mumford curves, namely affine stacky curves. A certain emphasis is given to the role of stack inertia groups.

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arXiv2025

The local obstruction to semi-stable reduction for abelian varieties

Grothendieck defined a group that represents the local obstruction for an abelian variety to have semi-stable reduction. These groups were studied by Silverberg and Zarhin and more recently by the author in order to give a group theoretic characterization of them depending only on the dimension. We give an overview of the developments since Grothendieck's definition with the added novelty of the case of equal characteristic local fields.

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arXiv2025

The $(p,t,a)$-inertial groups as finite monodromy groups

Silverberg and Zarhin introduced the notion of a $(p,t,a)$-inertial group in the hope of having a group theoretic characterization of the finite groups that appear as finite monodromy groups -- the groups that represent the local obstruction to semi-stable reduction -- of abelian varieties in fixed dimension $t+a$. In this text, we provide a positive answer to their question, that is, every $(p,t,a)$-inertial group is the finite monodromy group of an abelian variety in dimension $t+a$. To prove this, we show a structure theorem on the rational group algebra $\mathbf{Q}[G]$ of ramification groups, refining a theorem of Serre and generalizing results on $p$-groups of Roquette and Ford.

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arXiv2024

Groupes de monodromie finie des variétés abéliennes

The finite monodromy groups of abelian varieties over number fields have been introduced by Grothendieck. They represent the local obstruction to semi-stable reduction. In this paper we prove a criteria for finite groups to be realized as finite monodromy groups in given dimension. An application to the degree of semi-stability gives an effective version of Grothendieck's semi-stable reduction theorem in terms of the degree of the extension with regards to the dimension of the variety.

math.NT↗

arXiv2023

On Oda's problem and special loci

Oda's problem, which deals with the fixed field of the universal monodromy representation of moduli spaces of curves and its independence with respect to the topological data, is a central question of anabelian arithmetic geometry. This paper emphasizes the stack nature of this problem by establishing the independence of monodromy fields with respect to finer special loci data of curves with symmetries, which we show provides a new proof of Oda's prediction.

math.AG↗

arXiv2021

On the degree of semi-stable reduction

For an abelian variety $A$ over a number field we study bounds depending only on the dimension of $A$ for the minimal degree $d(A)$ of a field extension over which $A$ acquires semi-stable reduction. We first compute $d(A)$ in terms of the cardinalities of the finite monodromy groups of $A$ which leads to a bound on $d(A)$ in terms of the classical Minkowski bound. We then show this bound is tight up to its $2$-part by considering $p$-adic coverings of the local points of a universal abelian scheme.

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arXiv2021

Variétés abéliennes CM et grosse monodromie finie sauvage

In this paper we study the wild part of the finite monodromy groups of abelian varieties over number fields. We solve Grunwald problems for groups of the form $\mathbf{Z}/p\mathbf{Z}\wr \mathfrak{S}_n$ over number fields to build CM abelian varieties with maximal wild finite monodromy in the odd prime case. For the even prime case we prove a new bound on the 2-part of the order of the finite monodromy group for CM abelian varieties and build varieties that reach it.

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arXiv2020

Fields of definition of abelian subvarieties

In this paper we study the field of definition of abelian subvarieties $B\subset A_{\overline{K}}$ for an abelian variety $A$ over a field $K$ of characteristic $0$. We show that, provided that no isotypic component of $A_{\overline{K}}$ is simple, there are infinitely many abelian subvarieties of $A_{\overline{K}}$ with field of definition $K_A$, the field of definition of the endomorphisms of $A_{\overline{K}}$. This result combined with earlier work of Rémond gives an explicit maximum for the minimal degree of a field extension over which an abelian subvariety of $A_{\overline{K}}$ is defined with varying $A$ of fixed dimension and $K$ of characteristic $0$.

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