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Adrian Vasiu

Publications and source records attributed to Adrian Vasiu.

At least 19 recordsLinked to original sources
arXiv2026

On endomorphisms of affine spaces and the Jacobian problem

Let $p$ be a prime. We provide examples which show that \'etale endomorphisms of affine planes over an algebraically closed field $k$ of characteristic $p$ can have fibers of arbitrary finite cardinal. Let $(l,m)\in\mathbb N\times\mathbb N^{\ast}$. We provide examples of such \'etale endomorphisms whose images have complements of cardinality $l$ and whose geometric degrees are $pm$. Several conjectures are disproved, and in particular we provide an analog over $k$ of Kulikov's counterexample to the `Generalized Jacobian Conjecture' of Bass for surfaces. Surjective (resp.\ surjective and non-surjective) counterexamples to Adjamagbo's Jacobian Conjecture over each $k$ are included in dimension $2$ (resp.\ in all dimensions at least $3$); for instance, if $p=2$, then we show for each $m$ there exist surjective \'etale endomorphisms of the affine spaces over $k$ of dimension at least $3$ of geometric degree $m$. If $e:X\rightarrow X$ is an endomorphism of a variety over an algebraically closed field $K$, then we show that there exists $n\in\mathbb N$ such that $\Imm(e^n)=\Imm(e^{n+1})$ provided either (i) $e$ is quasi-finite or (ii) $\dim(X)\le 2$ and $X\setminus\Imm(e)$ is finite. We provide examples of endomorphisms of affine planes of dimensions at least $3$ over $K$ whose images have complements of cardinality $l$ and for all $n\in\mathbb N$ we have $\Imm(e^n)\neq\Imm(e^{n+1})$. We prove that all affine moduli schemes of \'etale endomorphisms of affine spaces with Jacobian matrices of determinant 1 over $K$ are connected and classify all those that are smooth. We prove that the Jacobian Conjecture over $\mathbb C$ and Adjamagbo's analog of it over $k$ hold for \'etale endomorphisms of affine spaces that are composites $g\circ f$, where $f$ is quasi-finite and a locally closed embedding in codimension $1$ outside a specific finite subset and $g$ is a projection that omits one coordinate.

math.AG↗
arXiv2026

On lifting representations and actions on curves of the metacyclic groups $C_{p^s}\rtimes C_m$

For a prime $p$, a pair $(s,m)\in\mathbb{N}^2$ with $m$ relatively prime to $p$, a homomorphism $\chi:C_m\rightarrow\operatorname{Aut}(C_{p^s})$, and an algebraically closed field $k$ of characteristic $p$, we consider the semidirect product $G=C_{p^s}\rtimes_{\chi} C_m$, denote its $p$-Sylow subgroup $C_{p^s}$ by $H$, and consider a $k[G]$-module $V$. Let $R$ be a complete discrete valuation ring of residue field $k$ and mixed characteristic $(0,p)$ that contains a primitive $p^s$-th root of unity. If $\chi$ is injective, we present two necessary and sufficient criteria for lifting $V$ to an $R[G]$-module $\widetilde{V}$ which is a free $R$-module: (i) when no extra requirement is made on $\widetilde{V}$ and (ii) when we require $\widetilde{V}^{C_{p^s}}=\{0\}$. The criteria correct several results in the literature and we use them to prove that, if $\chi$ is injective and $G$ acts faithfully on a connected smooth projective curve $X$ over $k$, then, under mild hypotheses satisfied if $X\rightarrow X/G$ is a Harbater--Katz--Gabber cover, the $k[G]$-module $H^0(X,\Omega_X)$ has a lift $\widetilde{V}$ to $R$ with $\widetilde{V}^{C_{p^s}}=\{0\}$. With $B$ as the field of fractions of $R$, we prove the following obstruction when $p$ is odd, $G/\operatorname{Ker}(\chi)$ has even order, and $X/C_{p^s}\cong\mathbb{P}^1_k$: if no such lift $\widetilde{V}$ exists with the $B[H]$-module $\widetilde{V}\otimes_R B$ defined over $\mathbb{Q}$, then the action of $G$ on $X$ does not lift to $R$.

math.NT↗
arXiv2026

Infinite transitivity of tame groups of automorphisms of affine spaces

For positive integers $n$ and $m$, we study the actions of the groups of tame automorphisms of the $n$-dimensional affine spaces over finite fields on ordered subsets of $m$ points. Our primary interest lies in constructing tame automorphisms that take one ordered sequence to another and in proving upper and lower bounds on the maximal complexity of such automorphisms. Our preferred measure of complexity of an automorphism is the maximum of the degrees of the polynomials that define it and its inverse. Using methods and results from various branches of mathematics, including the theory of symmetric groups, affine geometry over finite fields, polynomial interpolation, combinatorics of projective spaces over fields, and polynomial automorphisms, we obtain a wide variety of qualitative and quantitative results.

math.AG↗
arXiv2026

On the factorizations of integers via division algorithms for polynomials

We introduce and study several conditions related to the factorization problem of composite numbers. For this purpose, we employ cyclotomic polynomials, Sylvester resultants, and the Fermat equation. For instance, we show that for $m\in\mathbb N$ and distinct primes $p$ and $q$ with $p$ not dividing $m$, the existence of a solution to the Fermat equation $X^p+Y^p=Z^p$ in positive characteristic $q$ such that $X+Y\neq Z$ and $X, Y$ and $Z$ are $m$-th roots of unity implies the factorization of a composite natural number $N$ that is a multiple of $pq$ at the cost of $O\bigl( \phi(m)[m^3 + m^2(\log N)^2] M(\lfloor \log_2 N\rfloor +1) \bigr)$, where $\phi$ is the Euler's function and $M$ is the multiplication time function for $\mathbb Z$. We also show that such solutions do not exist for many semiprime integers $N$, provided that $m$ is required to have a fixed polynomial upper bound in $\log N$.

math.NT↗
arXiv2024

Matrix invertible extensions over commutative rings. Part III: Hermite rings

We reobtain and often refine prior criteria due to Kaplansky, McGovern, Roitman, Shchedryk, Wiegand, and Zabavsky--Bilavska and obtain new criteria for a Hermite ring to be an \textsl{EDR}. We mention three criteria: (1) a Hermite ring $R$ is an \textsl{EDR} iff for all pairs $(a,c)\in R^2$, the product homomorphism $U(R/Rac)\times U\bigl(R/Rc(1-a)\bigr)\to U(R/Rc)$ between groups of units is surjective; (2) a reduced Hermite ring $R$ is an \textsl{EDR} iff it is a pre-Schreier ring and for each $a\in R$, every zero determinant unimodular $2\times 2$ matrix with entries in $R/Ra$ lifts to a zero determinant matrix with entries in $R$; (3) a B\'{e}zout domain $R$ is an \textsl{EDD} iff for all triples $(a,b,c)\in R^3$ there exists a unimodular pair $(e,f)\in R^2$ such that $(a,e)$ and $(be+af,1-a-bc)$ are unimodular pairs. We use these criteria to show that each B\'{e}zout ring $R$ that is an $(SU)_2$ ring (as introduced by Lorenzini) such that for each nonzero $a\in R$ there exists no nontrivial self-dual projective $R/Ra$-module of rank $1$ generated by $2$ elements (e.g., all its elements are squares), is an \textsl{EDR}.

math.AC↗
arXiv2024

Matrix invertible extensions over commutative rings. Part II: determinant liftability

A unimodular $2\times 2$ matrix $A$ with entries in a commutative ring $R$ is called weakly determinant liftable if there exists a matrix $B$ congruent to $A$ modulo $R\det(A)$ and $\det(B)=0$; if we can choose $B$ to be unimodular, then $A$ is called determinant liftable. If $A$ is extendable to an invertible $3\times 3$ matrix $A^+$, then $A$ is weakly determinant liftable. If $A$ is simple extendable (i.e., we can choose $A^+$ such that its $(3,3)$ entry is $0$), then $A$ is determinant liftable. We present necessary and/or sufficient criteria for $A$ to be (weakly) determinant liftable and we use them to show that if $R$ is a $\Pi_2$ ring in the sense of Part I (resp.\ is a pre-Schreier domain), then $A$ is simply extendable (resp.\ extendable) iff it is determinant liftable (resp.\ weakly determinant liftable). As an application we show that each $J_{2,1}$ domain (as defined by Lorenzini) is an elementary divisor domain.

math.AC↗
arXiv2024

Matrix invertible extensions over commutative rings. Part I: general theory

A unimodular $2\times 2$ matrix with entries in a commutative $R$ is called extendable (resp.\ simply extendable) if it extends to an invertible $3\times 3$ matrix (resp.\ invertible $3\times 3$ matrix whose $(3,3)$ entry is $0$). We obtain necessary and sufficient conditions for a unimodular $2\times 2$ matrix to be extendable (resp.\ simply extendable) and use them to study the class $E_2$ (resp.\ $SE_2$) of rings $R$ with the property that all unimodular $2\times 2$ matrices with entries in $R$ are extendable (resp.\ simply extendable). We also study the larger class $\Pi_2$ of rings $R$ with the property that all unimodular $2\times 2$ matrices of determinant $0$ and with entries in $R$ are (simply) extendable (e.g., rings with trivial Picard groups or pre-Schreier domains). Among Dedekind domains, polynomial rings over $\mathbb Z$ and Hermite rings, only the EDRs belong to the class $E_2$ or $SE_2$. If $as(R)\le 2$, then $R$ is an $E_2$ ring iff it is an $SE_2$ ring.

math.AC↗
arXiv2024

On matrix invertible extensions over commutative rings

We introduce the class E2 (resp. SE2) of commutative rings R with the property that each unimodular 2 x 2 matrix with entries in R extends to an invertible 3 x 3 matrix (resp. invertible 3 x 3 matrix whose (3, 3) entry is 0). Among noetherian domains of dimension 1, polynomial rings over Z or Hermite rings, only EDRs belong to the class. Using this, stable ranges, and units and projective modules interpretations, we reobtain (often refine) criteria due to Kaplansky, McGovern, Roitman, Shchedryk, Wiegand and obtain new criteria for a Hermite ring to be an EDR. For instance, we characterize Hermite rings which are EDRs by (1) means of equations involving unimodular triples and (2) surjectivity of some maps involving units of factor rings by principal ideals. We use these criteria to show that each Bezout ring R that is either a J2,1 domain or an (SU)2 ring (as introduced by Lorenzini) such that for each nonzero a 2 R there exists no nontrivial self-dual projective R/Ra-module of rank 1 generated by 2 elements (e.g., all its elements are squares), is an EDR, thus solving or partially solving problems raised by Lorenzini. Many other classes of rings are introduced and studied. To ease the reading, this version concentrates only on extendability properties of unimodular 2 x 2 matrices and hence does not consider the case n > 2.

math.AC↗
arXiv2024

Waring Problem for Matrices over Finite Fields

We prove that for all integers $k \geq 1$, $q\ge (k-1)^4+ 6k$, and $m \geq 1$, every matrix in $ M_m(\mathbb F_q)$ is a sum of two kth powers: $M_m(\mathbb F_q)=\{A^k+B^k|A,B\in M_m(\mathbb F_q)\}$. We further generalize and refine this result in the cases when both $B$ and $C$ can be chosen to be invertible, cyclic, or split semisimple, when $k$ is coprime to $p$, or when $m$ is sufficiently large. We also give a criterion for the Waring problem in terms of stabilizers.

math.NT↗
arXiv2022

The $δ$-invariant theory of Hecke correspondences on $\mathcal A_g$

Let $p$ be a prime, let $N\geq 3$ be an integer prime to $p$, let $R$ be the ring of $p$-typical Witt vectors with coefficients in an algebraic closure of $\mathbb F_p$, and consider the correspondence $\mathcal A'_{g,1,N,R}\rightrightarrows \mathcal A_{g,1,N,R}$ obtained by taking the union of all prime to $p$ Hecke correspondences on Mumford's moduli scheme of principally polarized abelian schemes of relative dimension $g$ endowed with symplectic similitude level-$N$ structure over $R$-schemes. It is well-known that the coequalizer $\mathcal A_{g,1,N,R}/\mathcal A'_{g,1,N,R}$ of the above correspondence exists and is trivial in the category of schemes, i.e., is $\text{Spec}(R)$. We construct and study in detail such a coequalizer (categorical quotient) in a more refined geometry (category) referred to as {\it $δ$-geometry}. This geometry is in essence obtained from the usual algebraic geometry by equipping all $R$-algebras with {\it $p$-derivations}. In particular, we prove that our substitute of $\mathcal A_{g,1,N,R}/\mathcal A'_{g,1,N,R}$ in $δ$-geometry has the same `dimension' as $\mathcal A_{g,1,N,R}$, thus solving a main open problem in the work of Barcău--Buium. We also give applications to the study of various Zariski dense loci in $\mathcal A_{g,1,N,R}$ such as of isogeny classes and of points with complex multiplication. To prove our results we develop a Serre--Tate expansion theory for {\it Siegel $δ$-modular forms} of arbitrary genus which we then combine with old and new results from the geometric invariant theory of multiple quadratic forms and of multiple endomorphisms.

math.NT↗
arXiv2022

On Lie algebra modules which are modules over semisimple group schemes

Let $p$ be a prime. Given a split semisimple group scheme $G$ over a normal integral domain $R$ which is a faithfully flat $\mathbb Z_{(p)}$-algebra, we classify all finite dimensional representations $V$ of the fiber $G_K$ of $G$ over $K:=\text{Frac}(R)$ with the property that the set of lattices of $V$ with respect to $R$ which are $G$-modules is as well the set of lattices of $V$ with respect to $R$ which are $\text{Lie}(G)$-modules. We apply this classification to get a general criterion of extensions of homomorphisms between reductive group schemes over $\text{Spec} K$ to homomorphisms between reductive group schemes over $\text{Spec} R$. We also show that for a simply connected semisimple group scheme over a reduced $\mathbb Q$--algebra, the category of its representations is equivalent to the category of representations of its Lie algebra.

math.AG↗
arXiv2020

Good reductions of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic. Part I

We prove the existence of good smooth integral models of Shimura varieties of Hodge type in arbitrary unramified mixed characteristic $(0,p)$. As a first application we provide a smooth solution (answer) to a conjecture (question) of Langlands for Shimura varieties of Hodge type. As a second application we prove the existence in arbitrary unramified mixed characteristic $(0,p)$ of integral canonical models of projective Shimura varieties of Hodge type with respect to h--hyperspecial subgroups as pro-étale covers of Néron models; this forms progress towards the proof of conjectures of Milne and Reimann. Though the second application was known before in some cases, its proof is new and more of a principle.

math.NT↗
arXiv2020

Purity for Barsotti-Tate groups in some mixed characteristic situations

Let $p$ be a prime. Let $R$ be a regular local ring of dimension $d\ge 2$ whose completion is isomorphic to $C(k)[[x_1,\ldots,x_d]]/(h)$, with $C(k)$ a Cohen ring with the same residue field $k$ as $R$ and with $h\in C(k)[[x_1,\ldots,x_d]]$ such that its reduction modulo $p$ does not belong to the ideal $(x_1^p,\ldots,x_d^p)+(x_1,\ldots,x_d)^{2p-2}$ of $k[[x_1,\ldots,x_d]]$. We extend a result of Vasiu-Zink (for $d=2$) to show that each Barsotti-Tate group over $\text{Frac}(R)$ which extends to every local ring of $\text{Spec}(R)$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $R$. This result corrects in many cases several errors in the literature. As an application, we get that if $Y$ is a regular integral scheme such that the completion of each local ring of $Y$ of residue characteristic $p$ is a formal power series ring over some complete discrete valuation ring of absolute ramification index $e\le p-1$, then each Barsotti-Tate group over the generic point of $Y$ which extends to every local ring of $Y$ of dimension $1$, extends uniquely to a Barsotti-Tate group over $Y$.

math.NT↗
arXiv2018

Purity of Crystalline Strata

Let $p$ be a prime. Let $n\in\mathbb N-\{0\}$. Let $\mathcal C$ be an $F^n$-crystal over a locally noetherian $\mathbb F_p$-scheme $S$. Let $(a,b)\in\mathbb N^2$. We show that the reduced locally closed subscheme of $S$ whose points are exactly those $x\in S$ such that $(a,b)$ is a break point of the Newton polygon of the fiber $\mathcal C_x$ of $\mathcal C$ at $x$ is pure in $S$, i.e., it is an affine $S$-scheme. This result refines and reobtains previous results of de Jong--Oort, Vasiu, and Yang. As an application, we show that for all $m\in \mathbb N$ the reduced locally closed subscheme of $S$ whose points are exactly those $x\in S$ for which the $p$-rank of $\mathcal C_x$ is $m$ is pure in $S$; the case $n=1$ was previously obtained by Deligne (unpublished) and the general case $n\ge 1$ refines and reobtains a result of Zink.

math.AG↗
arXiv2015

Extension theorems for reductive group schemes

We prove several basic extension theorems for reductive group schemes. We also prove that each Lie algebra with a perfect Killing form over a commutative $\dbZ$-algebra, is the Lie algebra of an adjoint group scheme.

math.NT↗
arXiv2012

Stratifications of Newton polygon strata and Traverso's conjectures for p-divisible groups

The isomorphism number (resp. isogeny cutoff) of a p-divisible group D over an algebraically closed field is the least positive integer m such that D[p^m] determines D up to isomorphism (resp. up to isogeny). We show that these invariants are lower semicontinuous in families of p-divisible groups of constant Newton polygon. Thus they allow refinements of Newton polygon strata. In each isogeny class of p-divisible groups, we determine the maximal value of isogeny cutoffs and give an upper bound for isomorphism numbers, which is shown to be optimal in the isoclinic case. In particular, the latter disproves a conjecture of Traverso. As an application, we answer a question of Zink on the liftability of an endomorphism of D[p^m] to D.

math.AG↗
arXiv2012

On the Tate and Langlands--Rapoport conjectures for special fibres of integral canonical models of Shimura varieties of abelian type

We prove the isogeny property for special fibres of integral canonical models of compact Shimura varieties of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type. The approach used also shows that many crystalline cycles on abelian varieties over finite fields which are specializations of Hodge cycles, are algebraic. These two results have many applications. First, we prove a variant of the conditional Langlands--Rapoport conjecture for these special fibres. Second, for certain isogeny sets we prove a variant of the unconditional Langlands--Rapoport conjecture (like for many basic loci). Third, we prove that integral canonical models of compact Shimura varieties of Hodge type that are of $A_n$, $B_n$, $C_n$, and $D_n^{\dbR}$ type, are closed subschemes of integral canonical models of Siegel modular varieties.

math.NT↗