Adrian Vasiu
We prove surjectivity criteria for $p$-adic representations and we apply them to abelian varieties over number fields. In particular, we provide examples of Jacobians over $\dbQ$ of dimension $d\in\{1,2,3\}$ whose 2-adic representations have as images $GSp_{2d}(\dbZ_2)$.
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I prove the surjectivity of the $p$-adic regulator from Quillen's $K_2$ of Tate curve to the $p$-adic etale cohomology group when the base field is contained in a cyclotomic extension of $Q_p$. This implies the finiteness of torsion part of $K_1$ of Tate curves thanks to Suslin's exact sequence.
We prove several surjectivity criteria for $p$-adic representations. In particular, we classify all adjoint and simply connected group schemes $G$ over the Witt ring $W(k)$ of a finite field $k$ such that the epimorphism $G(W_2(k))\twoheadrightarrow G(k)$ has a section.
On the surjectivity of $(T)$-adic Galois Representations of Drinfeld $A$-Modules of Rank 2 and 3: Density results
本文研究了秩为2和3的Drinfeld A-模的(T)-adic伽罗瓦表示的满射性条件，并计算了此类模的密度。
On the Surjectivity of the Geometric Sen Map of Perfectoid Torsors
本文构造了一个在$\mathbb C_p$上的光滑连通刚性解析曲线及其完美拟平展$\mathbb Z_p^2$-环，其几何Sen态射非满射，否证了Rodríguez Camargo的猜想，并依赖拉回定理保持完美塔。
In this paper, we prove the $\ell$-adic abelian class field theory for henselian regular local rings of equi-characteristic assuming the surjectivity of Galois symbol maps, which is a $\ell$-adic variant of a result of Matsumi [13].
We generalize the surjectivity result of the $p$-adic monodromy for the ordinary locus of a Siegel moduli space by Faltings and Chai (independently by Ekedahl) to that for any $p$-rank stratum. We discuss irreducibility and connectedness of some $p$-rank strata of the moduli spaces with parahoric level structure. Finer results are obtained on the Siegel 3-fold with Iwahori level structure.
On the surjectivity of Galois representations attached to Drinfeld $A$-modules of rank $2$
本文研究了秩为2的Drinfeld A-模所附带的Galois表示的满射性，通过水平和垂直方向的方法给出条件并获得新的例子，证明了在特定条件下表示的满射性。
We explore Tate-type conjectures over $p$-adic fields. We study a conjecture of Raskind that predicts the surjectivity of $$ ({\rm NS}(X_{\bar{K}}) \otimes_{\mathbb{Z}}\mathbb{Q}_p)^{G_K} \longrightarrow H^2_{\rm et}(X_{\bar{K}},\mathbb{Q}_p(1))^{G_K} $$ if $X$ is smooth and projective over a $p$-adic field $K$ and has totally degenerate reduction. Sometimes, this is related to $p$-adic uniformisation. For abelian varieties, Raskind's conjecture is equivalent to the question whether $$ {\rm Hom}(A,B)\otimes{\mathbb{Q}}_p \,\to\, {\rm Hom}_{G_K}(V_p(A),V_p(B)) $$ is surjective if $A$ and $B$ ar…
We use the theta lift to study the multiplicity with which certain automorphic representations of cohomological type occur in a family of congruence covers of an arithmetic manifold. When the family of covers is a so-called `p-adic congruence tower' we obtain sharp asymptotics for the number of representations which occur as lifts. When combined with theorems on the surjectivity of the theta lift due to Howe and Li, and Bergeron, Millson and Moeglin, this allows us to verify certain cases of a conjecture of Sarnak and Xue.
Pointwise criteria of p-adic local systems
本文通过引入晶体 Riemann–Hilbert 函子，建立了 p-进局部系在光滑刚性空间上为晶体或半稳定的逐点判别准则，并应用于相对情形的半稳定比较定理。
We prove that the irreducible components of the space of framed deformations of the trivial 2-dimensional mod 2 representation of the absolute Galois group of Q_2 are in natural bijection with those of the trivial character, confirming a conjecture of Böckle. We deduce from this result that crystalline points are Zariski dense in that space: this provides the missing ingredient for the surjectivity of the p-adic local Langlands correspondence for GL_2(Q_p) in the case p=2 (the result was already known for p\geq 3).
We prove a Neron--Ogg--Shafarevich type criterion for good reduction of K3 surfaces, which states that a K3 surface over a complete discrete valuation field has potential good reduction if its $l$-adic cohomology group is unramified. We also prove a $p$-adic version of the criterion. (These are analogues of the criteria for good reduction of abelian varieties.) The model of the surface will be in general not a scheme but an algebraic space. As a corollary of the criterion we obtain the surjectivity of the period map of K3 surfaces in positive characteristic.
In this article I define and study the overconvergent rigid fundamental group of a variety over an equicharacteristic local field. This is a non-abelian $(φ,\nabla)$-module over the bounded Robba ring $\mathcal{E}_K^\dagger$, whose underlying unipotent group (after base changing to the Amice ring $\mathcal{E}_K$) is exactly the classical rigid fundamental group. I then use this to prove an equicharacteristic, $p$-adic analogue of Oda's theorem that a semistable curve over a $p$-adic field has good reduction iff the Galois action on its $\ell$-adic unipotent fundamental group is unramified.
We establish the solvability criteria for the equation $x^q=a$ in the field of $p$-adic numbers, for any $q$ in two cases: (i) $q$ is not divisible by $p$; (ii) $q=p$. Using these criteria we show that any $p$-adic number can be represented in finitely many different forms and we describe the algorithms to obtain the corresponding representations. Moreover it is showed that solvability problem of $x^q=a$ for any $q$ can be reduced to the cases (i) and (ii).
This paper is devoted to the problem of ergodicity of $p$-adic dynamical systems. Our aim is to present criteria of ergodicity in terms of coordinate functions corresponding to digits in the canonical expansion of $p$-adic numbers. The coordinate representation can be useful, e.g., for applications to cryptography. Moreover, by using this representation we can consider non-smooth $p$-adic transformations. The basic technical toolsare van der Put series and usage of algebraic structure (permutations) induced by coordinate functions with partially frozen variables. We illustrate the basic theore…
In this note we give a brief survey of the most elementary criteria used to determine the surjectivity of the trace operator on the ring of integers of a number field $K$. Furthermore, we introduce an easy to state yet unknown surjectivity criterion depending only on the prime factorization of the degree $n$ of $K$ and on the squarefree part of the discriminant $d_K$.
In this paper we study the $p$-adic equation $x^q=a$ over the field of $p$-adic numbers. We construct an algorithm of calculation of criteria of solvability in the case of $q=p^m$ and present a computer program to compute the criteria for fixed value of $m \leq p-1$. Moreover, using this solvability criteria for $q=2,3,4,5,6$, we classify $p$-adic 6-dimensional filiform Leibniz algebras.
We consider a product $X=E_1\times\cdots\times E_d$ of elliptic curves over a finite extension $K$ of $\mathbb{Q}_p$ with a combination of good or split multiplicative reduction. We assume that at most one of the elliptic curves has supersingular reduction. Under these assumptions, we prove that the Albanese kernel of $X$ is the direct sum of a finite group and a divisible group, extending work of Raskind and Spiess to cases that include supersingular phenomena. Our method involves studying the kernel of the cycle map $CH_0(X)/p^n\rightarrow H^{2d}_{\text{ét}}(X, μ_{p^n}^{\otimes d})$. We give…
This paper is devoted to (discrete) $p$-adic dynamical systems, an important domain of algebraic and arithmetic dynamics. We consider the following open problem from theory of $p$-adic dynamical systems. Given continuous function $f:Z_p > Z_p.$ Let us represent it via special convergent series, namely van der Put series. How can one specify whether this function is measure-preserving or not for an arbitrary $p$? In this paper, for any prime $p$ we present a complete description of all compatible measure-preserving functions in the additive form representation. In addition we prove the criterio…
Several new estimates for the 2-adic valuations of Stirling numbers of the second kind are proved. These estimates, together with criteria for when they are sharp, lead to improvements in several known theorems and their proofs, as well as to new theorems. The estimates and criteria all depend on our previous analysis of powers of 2 in the denominators of coefficients of higher order Bernoulli polynomials. The corresponding estimates for Stirling numbers of the first kind are also proved. Some attention is given to asymptotic cases, which will be further explored in subsequent publications.
We study the functor of Weil restriction in the category of Huber's adic spaces and in the category of Berkovich spaces. We prove criteria for the representability of these functors in the respective categories. As an application of adic Weil restrictions, we prove that adic Néron models behave well with respect to tamely ramified base change.
Let $K$ be a local field, $X$ the Drinfel'd symmetric space $X$ of dimension $d$ over $K$ and ${\mathfrak X}$ the natural formal ${\mathcal O}_K$-scheme underlying $X$; thus $G={\rm GL}\sb {d+1}(K)$ acts on $X$ and ${\mathfrak X}$. Given a $K$-rational $G$-representation $M$ we construct a $G$-equivariant subsheaf ${\mathcal M}^0_{{\mathcal O}_{\dot{K}}}$ of ${\mathcal O}_K$-lattices in the constant sheaf $M$ on ${\mathfrak X}$. We study the cohomology of sheaves of logarithmic differential forms on $X$ (or ${\mathfrak X}$) with coefficients in ${\mathcal M}^0_{{\mathcal O}_{\dot{K}}}$. In the…
In this note, we give a criteria whether given two Eisenstein polynomials over a padic field define the same extension (Proposition 1.6). In particular, we completely identify Eisenstein polynomials of degree p (Theorem 1.16). This note is an English translation of a part of the original paper [12].
Galois representations are surjective for almost all Drinfeld modules
本文证明在函数域上，对于秩r≥2的Drinfeld模，T-adic Galois表示在密度为1的模集合上是满射的，利用Hilbert不可约性、Drinfeld均匀化理论和筛法。
We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of Borel-Dwork and Pólya-Bertrandias valid over the projective line to arbitrary algebraic curves over a number field. The formulation and the proof of these criteria involve some basic notions in Arakelov geometry, combined with complex and rigid analytic geometry (notably, potential theory over complex and $p$-adic curves). We also discuss geometric analogues, pe…
We provide a solvability criteria for a depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$. We show that, in principal, the Cardano method is not always applicable for such equations. Moreover, the numbers of solutions of the depressed cubic equation in domains $\bz_p^{*},\bz_p,\bq_p$ are provided. Since $\bbf_p\subset\bq_p,$ we generalize J.-P. Serre's \cite{JPSJ} and Z.H.Sun's \cite{ZHS1,ZHS3} results concerning with depressed cubic equations over the finite field $\bbf_p$. Finally, all depressed cubic equations, for which the Cardano method could be applied, are described and the $…
We study the relationship between potential equivalence and character theory; we observe that potential equivalence of a representation $ρ$ is determined by an equality of an $m$-power character $g\mapsto Tr(ρ(g^m))$ for some natural number $m$. Using this, we extend Faltings' finiteness criteria to determine the equivalence of two $\ell$-adic, semisimple representations of the absolute Galois group of a number field, to the context of potential equivalence. We also discuss finiteness results for twist unramified representations.
On generalized Fuchs theorem over relative $p$-adic polyannuli
本文在相对 $p$-进多环域上研究满足 Robba 条件的凝聚局部自由（对数-）$\ abla$-模，建立了绝对与相对情形下的广义 $p$-进 Fuchs 定理，推广了 Shiho 的结果，并证明了两种指数定义的一致性。
Given our set-up of a system of curves and maps between them satisfying certain assumptions, we prove a classicality criterion for overconvergent sections of line bundles over these curves. As a result, we prove such criteria for overconvergent modular forms over various Shimura curves. In particular, we provide a classicality criterion for overconvergent modular forms studied in [Kassaei: P-adic modular forms over Shimura curves over totally real fields, Compositio Math. 140 (2004), no 2, 359-395] and their higher-level generalizations.
Let F be a p-adic field and let G(n) and G`(n) be the metaplectic double covers of the general symplectic group and symplectic group attached to a 2n dimensional symplectic space over F. We show here that if n is odd then all the genuine irreducible representations of G(n) are induced from a normal subgroup of finite index closely related to G`(n). Thus, we reduce, in this case, the theory of genuine admissible representations of G(n) to the better understood corresponding theory of G`(n). For odd n we also prove the uniqueness of certain Whittaker functionals along with Rodier type of Heredit…
In this paper, we prove a `cut-by-curves criterion' for an overconvergent isocrystal on a smooth variety over a field of characteristic $p>0$ to extend logarithmically to its smooth compactification whose complement is a strict normal crossing divisor, under certain assumption. This is a $p$-adic analogue of a version of cut-by-curves criterion for regular singuarity of an integrable connection on a smooth variety over a field of characteristic 0. In the course of the proof, we also prove a kind of cut-by-curves criteria on solvability, highest ramification break and exponent of $\nabla$-modul…
A mod $p$ determinant criterion for Cohen--Lenstra convergence of random $p$-adic matrices with prescribed zero patterns
受 Cohen--Lenstra 启发，研究具有指定零模式的 Haar 随机 $p$-adic 矩阵的余核分布，证明其收敛于 Cohen--Lenstra 分布等价于模 $p$ 的渐近非奇异性条件。
Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove that if $E$ has non-split multiplicative reduction at at least one odd prime or split multiplicative reduction at at least two odd primes and if the rank of $E(\mathbb{Q})$ is one and the Tate-Shafarevich group of $E$ has finite order, then $\mathrm{ord}_{s=1}L(E,s)=1$. We also prove the corresponding result for the abelian variety associated with a weight two newform $f$ of trivial character. These, and other related results, are consequences of our main theorem, which establishes criteria for $f$ and $H^1_f(\mathbb{Q},V)$, wh…
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