In 2007, B. Poonen (unpublished) studied the $p$--adic closure of a subgroup of rational points on a commutative algebraic group. More recently, J. Bellaïche asked the same question for the special case of Abelian varieties. These problems are $p$--adic analogues of a question raised earlier by B. Mazur on the density of rational points for the real topology. For a simple Abelian variety over the field of rational numbers, we show that the actual $p$--adic rank is at least the third of the expected value.
In this paper, we formulate and prove the so-called $p$-adic non-commutative analytic subgroup theorem. This result is seen as the $p$-adic analogue of a recent theorem given by Yafaev.
For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete characterization of the freeness of the ring of integers $\mathcal{O}_L$ over its associated order $\mathfrak{A}_{L/K}$ in the unique Hopf-Galois structure on $L/K$, which is analogous to the one already known for cyclic degree $p$ extensions of $p$-adic fields. We shall derive positive and negative results on criteria for the freeness of $\mathcal{O}_L$ as $\mat…
We use a $p$-adic analogue of the analytic subgroup theorem of Wüstholz to deduce the transcendence and linear independence of some new classes of $p$-adic numbers. In particular we give $p$-adic analogues of results of Wüstholz contained in [G. Wüstholz, Some remarks on a conjecture of Waldschmidt, Diophantine approximations and transcendental numbers, Progress in Mathematics 31, Birkhäuser Boston, Boston, MA, (1983), 329-336] and generalizations of results obtained by Bertrand in [D. Bertrand, Sous-groupes à un paramètre $p$-adique de variétés de groupe, Invent. Math. 40 (1977), no. 2, 171-1…
It is well-known that the Wüstholz' analytic subgroup theorem is one of the most powerful theorems in transcendence theory. The theorem gives in a very systematic and conceptual way the transcendence of a large class of complex numbers, e.g. the transcendence of $π$ which is originally due to Lindemann. In this paper we revisit the $p$-adic analogue of the analytic subgroup theorem and present a proof based on the method described and developed by the authors in a recent related paper.
For a curve $X$ over a $p$-adic field $k$, using the class field theory of $X$ due to S. Bloch and S. Saito we study the abelian geometric fundamental group $π_1^{\mathrm{ab}}(X)^{\mathrm{geo}}$ of $X$. In particular, it is investigated a subgroup of $π_1^{\mathrm{ab}}(X)^{\mathrm{geo}}$ which classifies the geometric and abelian coverings of $X$ which allow possible ramification over the special fiber of the model of $X$. Under the assumptions that $X$ has a $k$-rational point, $X$ has good reduction and its Jacobian variety has good ordinary reduction, we give some upper and lower bounds of …
Without using the $p$-adic Langlands correspondence, we prove that for many finite length smooth representations of $\mathrm{GL}_2(\mathbf{Q}_p)$ on $p$-torsion modules the $\mathrm{GL}_2(\mathbf{Q}_p)$-linear morphisms coincide with the morphisms that are linear for the normalizer of a parahoric subgroup. We identify this subgroup to be the Iwahori subgroup in the supersingular case, and $\mathrm{GL}_2(\mathbf{Z}_p)$ in the principal series case. As an application, we relate the action of parahoric subgroups to the action of the inertia group of $\mathrm{Gal}(\overline{\mathbf{Q}}_p/\mathbf{Q…
Iwasawa algebras of compact $p$-adic Lie groups are completed group algebras with applications in number theory in studying class numbers of towers of number fields and representation theory of $p$-adic Lie groups. In our earlier work, we determined an explicit presentation of the Iwasawa algebra for the first principal congruence kernel of Chevalley groups over $\mathbb{Z}_p$. In this paper, for prime $p>n+1$, we extend our result to determine the explicit presentation, in the form of generators and relations, of the Iwasawa algebra of the pro-$p$ Iwahori subgroup of $GL_n(\mathbb{Z}_p)$.
A refined non-vanishing of the $p$-adic logarithm of a rational point on an abelian variety
University of Texas at Austin（德克萨斯大学奥斯汀分校）；Princeton University（普林斯顿大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Let $p$ be a prime number and $M$ the extension field of a $p$-adic field $K$ obtained by adjoining all $p$-power roots of all elements of $K$. In this paper, we show that there exists a constant $C$, depending only on $K$ and an integer $g>0$, which satisfies the following property:If $A_{/K}$ is a $g$-dimensional CM abelian variety, then the order of the torsion subgroup of $A(M)$ is bounded by $C$.
We prove that the set of CM points on the Shimura curve associated to an Eichler order inside an indefinite quaternion $\mathbb{Q}$-algebra, is in bijection with the set of certain classes of $p$-adic binary quadratic forms, where $p$ is a prime dividing the discriminant of the quaternion algebra. The classes of $p$-adic binary quadratic forms are obtain by the action of a discrete and cocompact subgroup of $\mathrm{PGL}_{2}(\mathbb{Q}_{p})$ arising from the $p$-adic uniformization of the Shimura curve. We finally compute families of $p$-adic binary quadratic forms associated to an infinite fa…
We show that, for an abelian variety defined over a $p$-adic field $K$ which has potential good reduction, its torsion subgroup with values in the composite field of $K$ and a certain Lubin-Tate extension over a $p$-adic field is finite.
We prove $p$-adic versions of a classical result in arithmetic geometry stating that an irreducible subvariety of an abelian variety with dense torsion has to be the translate of a subgroup by a torsion point. We do so in the context of certain rigid analytic spaces and formal groups over a $p$-adic field $K$ or its ring of integers $R$, respectively. In particular, we show that the rigidity results for algebraic functions underlying the so-called Manin-Mumford Conjecture generalize to suitable $p$-adic analytic functions. In the formal setting, this approach leads us to uncover purely $p$-adi…
We investigate generalizations along the lines of the Mordell--Lang conjecture of the author's $p$-adic formal Manin--Mumford results for $n$-dimensional $p$-divisible formal groups $\mathcal{F}$. In particular, given a finitely generated subgroup $Γ$ of $\mathcal{F}(\overline{\mathbb{Q}}_p)$ and a closed subscheme $X\hookrightarrow \mathcal{F}$, we show under suitable assumptions that for any points $P\in X(\mathbb{C}_p)$ satisfying $nP\inΓ$ for some $n\in\mathbb{N}$, the minimal such orders $n$ are uniformly bounded whenever $X$ does not contain a formal subgroup translate of positive dimens…
Let $k$ be a number field, $Ω$ be a finite symmetric subset of $\mathbb{GL}_{n_0}(k)$, and $Γ=\langle Ω\rangle$. Let \[ C(Γ):=\{\mathfrak{p}\in V_f(k)|\hspace{1mm} Γ\text{is a bounded subgroup of} \mathbb{GL}_{n_0}(k_{\mathfrak{p}})\}, \] and $Γ_{\mathfrak{p}}$ be the closure of $Γ$ in $\mathbb{GL}_{n_0}(k_{\mathfrak{p}})$. Assuming that the Zariski-closure of $Γ$ is semisimple, we prove that the family of left translation actions $\{Γ\curvearrowright Γ_{\mathfrak{p}}\}_{\mathfrak{p}\in C(Γ)}$ has {\em uniform spectral gap}. As a corollary we get that the left translation action $Γ\curvearrowr…
Let $p$ be a prime, $D$ a finite dimensional noncommutative division $\mathbb{Q}_p$-algebra, and $SL_1(D)$ the group of elements of $D$ of reduced norm $1$. When the center of $D$ is $\mathbb{Q}_p$, we prove that no open subgroup of $SL_1(D)$ admits self-similar actions on regular rooted trees. Moreover, we prove results on $\mathbb{Z}_p$-Lie lattices that allow to deal with the case where the center of $D$ is bigger than $\mathbb{Q}_p$, and lead to the classification of the torsion-free $p$-adic analytic pro-$p$ groups $G$ of dimension less than $p$ with the property that all the nontrivial c…
Let $F$ be a local field with residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. In a previous paper, Florian Herzig and the authors classified irreducible admissible $C$-representations of $G=\mathbf{G}(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. Here, for a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an irreducible admissible $C$-representation $τ$ of $M$, we determine the lattice of subrepresentations of $\mathrm{Ind}_P^G τ$ and we show that $\mathrm{Ind…
We consider the question of unicity of types on maximal compact subgroups for supercuspidal representations of $\mathbf{SL}_2$ over a nonarchimedean local field of odd residual characteristic. We introduce the notion of an archetype as the $\mathbf{SL}_2$-conjugacy class of a typical representation of a maximal compact subgroup, and go on to show that any archetype in $\mathbf{SL}_2$ is restricted from one in $\mathbf{GL}_2$. From this it follows that any archetype must be induced from a Bushnell--Kutzko type. Given a supercuspidal representation $π$, we give an additional explicit description…
Let $p \geq 2$ be a prime number and let $\mathbb{C}_p$ be the completion of an algebraic closure of the $p$-adic rational field $\mathbb{Q}_p$. Let $f_c(z)$ be a one-parameter family of rational functions of degree $d\geq 2$, where the coefficients are meromorphic functions defined at all parameters $c$ in some open disk $D\subseteq\mathbb{C}_p$. Assuming an appropriate stability condition, we prove that the parameters $c$ for which $f_c$ is postcritically finite (PCF) are isolated from one another in the $p$-adic disk $D$, except in certain trivial cases. In particular, all PCF parameters of…
Let $F$ be a non-Archimedean locally compact field, let $G$ be a split connected reductive group over $F$. For a parabolic subgroup $Q\subset G$ and a ring $L$ we consider the $G$-representation on the $L$-module$$(*)\quad\quad\quad\quad C^{\infty}(G/Q,L)/\sum_{Q'\supsetneq Q}C^{\infty}(G/Q',L).$$Let $I\subset G$ denote an Iwahori subgroup. We define a certain free finite rank $L$-module ${\mathfrak M}$ (depending on $Q$; if $Q$ is a Borel subgroup then $(*)$ is the Steinberg representation and ${\mathfrak M}$ is of rank one) and construct an $I$-equivariant embedding of $(*)$ into $C^{\infty}…
