Tung T. Nguyen
K. Kato has recently constructed certain syntomic complexes associated with a uniform $F$-crystal over a smooth project variety $X$ and related their cohomology groups to special values of the $L$-function attached to $F$. In this paper, we generalize his results to a $p$-adic family of $F$-crystals.
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We conjecture that special elements associated with rank-one motives are obtained $p$-adically from Rubin-Stark elements by means of a precise `higher-rank Soulé twist' construction. We show this conjecture incorporates a variety of known results and existing predictions and also gives rise to a concrete strategy for proving the equivariant Tamagawa Number Conjecture for rank-one motives. We then use this approach to obtain new evidence in support of the equivariant Tamagawa Number Conjecture in the setting of CM abelian varieties.
Let $E/K$ be a finite Galois extension of totally real number fields with Galois group $G$. Let $p$ be an odd prime and let $r>1$ be an odd integer. The $p$-adic Beilinson conjecture relates the values at $s=r$ of $p$-adic Artin $L$-functions attached to the irreducible characters of $G$ to those of corresponding complex Artin $L$-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the `$p$-part' of the equivariant Tamagawa number conjecture for the pair $(h^0(\mathrm{Spec}(E))(r), \mathbb Z[G])$. If $r>1$ is even we obtain a sim…
Hearing Tamagawa Factors modulo $q - 1$
该研究将阿贝尔簇的塔马加瓦因子与 $p$ 进拉普拉斯积分算子的小波特征值联系，通过局部 $L$ 因子、Serre 不变量的相关结果，证明了二者模 $q - 1$ 的同余关系。
The equivariant Tamagawa number conjecture (hereinafter called the eTNC) predicts close relationships between algebraic and analytic aspects of motives. In this paper, we prove a lot of new cases of the minus component of the eTNC for $\mathbb{G}_m$ and for CM abelian extensions. One of the main results states that the $p$-component of the eTNC is true when there exists at least one $p$-adic prime that is tamely ramified. The fundamental strategy is inspired by the work of Dasgupta and Kakde on the Brumer-Stark conjecture.
Let $E$ be an elliptic curve over a number field $K$, $c_v$ the Tamagawa number of $E$ at $v$, and let $c_E=\prod_{v}c_v$. Lorenzini proved that $v_{13}(c_E)$ is postive for all elliptic curves over quadratic fields with a point of order $13$. Krumm conjectured, based on extensive computation, that the $13$-adic valuation of $c_E$ is even for all such elliptic curves. In this note we prove this conjecture and furhtermore prove that there is an unique such curve satisfying $v_{13}(c_E)=2$.
This is essentially the author's thesis submited to The University of Chicago (May 1997). I prove the validity of Tamagawa number conjecture of Bloch-Kato for certain Hecke characters. I study the exponential map and local Tamagawa number for all odd primes (both ordinary and supersingular), using Kato's explicit reciprocity law for one dimensional Lubin-Tate formal group. I also study p-part of Shafarevich-Tate group for motives associated to Hecke characters, using Rubin's Main Conjecture in Iwasawa theory. Interestingly a congruence property between p-adic periods of elliptic curve and weig…
On the Tamagawa number conjecture for newforms at Eisenstein primes
本文推广[CGLS22]至更高权模形式，在周期假设下证明Eisenstein素数处秩0的Tamagawa数公式，并讨论秩1部分结果及条件性p-逆定理。
We prove a uniform version of a finiteness conjecture due to Rasmussen and Tamagawa in the case of CM abelian varieties. This extends recent results concerning CM elliptic curves to CM abelian varieties of arbitrary dimension.
Let L/K be a finite Galois extension of number fields with Galois group G. Let p be a rational prime and let r be a non-positive integer. By examining the structure of the p-adic group ring Z_p[G], we prove many new cases of the p-part of the equivariant Tamagawa number conjecture (ETNC) for the pair (h^0(Spec(L)(r),Z[G])). The same methods can also be applied to other conjectures concerning the vanishing of certain elements in relative algebraic K-groups. We then prove a conjecture of Burns concerning the annihilation of class groups as Galois modules for a wide class of interesting extension…
We express the order of the pole and the leading coefficient of the L-function of a (large class of) -adic coefficients (any prime) over a quasi-projective variety over a finite field of characteristic p. This is a generalization of the result of Milne-Ramachandran with coefficients. The new key ingredient is the use of F-gauges and their equivalence in the derived category with Raynaud modules proved by Ekedahl.
On the Tamagawa number conjecture for modular forms twisted by anticyclotomic Hecke characters
本文研究模形式的Tamagawa数猜想，针对满足特定条件的正规化新形式、虚二次域与反循环Hecke特征，证明临界值的Tamagawa数猜想可由Bertolini-Darmon-Prasanna $p$进$L$函数的Iwasawa主猜想推出。
Assuming specific instances of two general conjectures in arithmetic algebraic geometry (bijectivity of $p$-adic regulator maps, injectivity of $p$-adic Abel-Jacobi maps), we prove several cases of the $p$-part of the Tamagawa number conjecture ($p$-TNC) of Bloch-Kato and Fontaine-Perrin-Riou for (homological) motives of modular forms of even weight $\geq4$ in analytic rank $1$. More precisely, we prove our results for a large class of newforms $f$ and prime numbers $p$ that are ordinary for $f$ and such that the weight of $f$ is congruent to $2$ modulo $2(p-1)$. Inspired by work of W. Zhang i…
We prove the conjectured compatibility of $p$-adic fundamental lines with specializations at motivic points for a wide class of $p$-adic families of $p$-adic Galois representations (for instance, the families which arise from $p$-adic families of automorphic representations of the unit group of a quaternion algebra or of a totally definite unitary groups) and deduce the compatibility of the Equivariant Tamagawa Number Conjectures for them. However, we also show that fundamental lines are not compatible with arbitrary characteristic zero specializations with values in a domain in general. This …
Let $f$ be a cuspidal newform and $p \geq 3$ a prime such that the associated $p$-adic Galois representation has large image. We establish a new and refined "Birch and Swinnerton-Dyer type" formula for Bloch-Kato Selmer groups of the central critical twist of $f$ via Kolyvagin derivatives of $L$-values instead of complex analytic or $p$-adic variation of $L$-values only under the Iwasawa main conjecture localized at the augmentation ideal. Our formula determines the exact rank and module structure of the Selmer groups and is insensitive to weight, the local behavior of $f$ at $p$, and analytic…
We develop a framework to investigate conjectures on congruences between the algebraic part of special values of $L$-functions of congruent motives. We show that algebraic local Euler factors satisfy precise interpolation properties in $p$-adic families of motives and that algebraic $p$-adic $L$-functions exist in quite large generality for $p$-adic families of automorphic motives. We formulate two conjectures refining (and correcting) the currently existing formulation of the Equivariant Tamagawa Number Conjecture with coefficients in Hecke algebras and pointing out the links between conjectu…
In a previous paper we constructed a new class of Iwasawa modules as $\ell$--adic realizations of what we called abstract $\ell$--adic $1$--motives in the number field setting. We proved in loc. cit. that the new Iwasawa modules satisfy an equivariant main conjecture. In this paper we link the new modules to the $\ell$--adified Tate canonical class, defined by Tate in 1960 and give an explicit construction of (the minus part of) $\ell$--adic Tate sequences for any Galois CM extension $K/k$ of an arbitrary totally real number field $k$. These explicit constructions are significant and useful in…
Analytic Geometry and Hodge-Frobenius Structure
本文研究高维p进解析几何中的Frobenius结构及相关p进泛函分析，依托既有研究纲领与方法探究广义p进Hodge结构形变，为相关上同调、岩泽理论及非交换玉河数猜想研究奠基。
We give an introduction to generalisations of conjectures of Brumer and Stark on the annihilator of the class group of a number field. We review the relation to the equivariant Tamagawa number conjecture, the main conjecture of Iwasawa theory for totally real fields, and a conjecture of Gross on the behaviour of $p$-adic Artin $L$-functions at zero.
For a prime number p and a number field k, we first study certain etale cohomology groups with coefficients associated to a p-adic Artin representation of its Galois group, where we twist the coefficients using a modified Tate twist with a p-adic index. We show that those groups are cofinitely generated and explicitly compute an additive Euler characteristic. When k is totally real and the representation is even, we relate the order of vanishing of the p-adic L-function at a point of its domain and the corank of such a cohomology group with a suitable p-adic twist. If the groups are finite, th…
Tamagawa number divisibility of central $L$-values of twists of the Fermat elliptic curve
研究Fermat椭圆曲线扭曲的中心L值代数部分的3-adic赋值，揭示Tate-Shafarevich群3-部分与惰性素因子个数的关系，并证明在分裂素数乘积情形下BSD猜想预测的阶为完全平方。
An analogue of a conjecture of Rasmussen and Tamagawa for abelian varieties over function fields
本文将 Rasmussen-Tamagawa 关于数域上阿贝尔簇 ℓ-幂挠点稀少性的猜想推广到函数域上的非迷向阿贝尔簇，证明椭圆曲线情形，并在高维情形下将其归结为一致几何挠猜想，且对实乘法阿贝尔簇给出无条件证明。
The main new result is the computation of the degeneration of l-adic Eisenstein classes at the cusps. This is done by relating it to the degeneration of the elliptic polylog. These classes come from K-theory and their Hodge regulator can also be computed (see: Dirichlet motives via modula curves, on the K-theory server). This gives a new proof of a comparison conjecture of Bloch and Kato which was used in the proof of their Tamagawa number conjecture for the Riemann zeta-function. The paper contains appendices on the definition of the classical and elliptic polylog, its degeneration and the co…
Let E/F be a finite Galois extension of totally real number fields and let p be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at s=1 of p-adic Artin L-functions to those of the complex Artin L-functions attached to E/F. We prove this conjecture unconditionally when E/Q is abelian. We also show that for certain non-abelian extensions E/F the p-adic Stark conjecture at s=1 is implied by Leopoldt's conjecture for E at p. Moreover, we prove that for a fixed prime p, the p-adic Stark conjecture at s=1 for E/F implies Stark's conjecture at s=1 for E/F. This leads to a `prim…
A Base Change Version of Rasmussen-Tamagawa Conjecture
本文证明了Shafarevich猜想的一致版本，并作为推论，针对数域上处处具有潜在好约化的特定阿贝尔簇类证明了Rasmussen-Tamagawa猜想。
We give a criterion for two l-adic Galois representations of an algebraic number field to be isomorphic when restricted to a decomposition group, in terms of the global representations mod l. This is applied to prove a generalization of a conjecture of Rasmussen-Tamagawa under a semistablity condition, extending some results of one of the authors. It is also applied to prove a congruence result on the Fourier coefficients of modular forms.
Tamagawa Products for Elliptic Curves Over Number Fields
本文将Griffin等人的玉河积L-级数推广至任意数域，通过构造马尔可夫链精确计算玉河积分布，并给出了分布比例与平均值的界及渐近序列。
In this paper, we prove the non-existence of certain semistable Galois representations of a number field. Our consequence can be applied to some geometric problems. For example, we prove a special case of a Conjecture of Rasmussen and Tamagawa, related with the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion.
We construct the p-adic zeta function for a one-dimensional (as a p-adic Lie extension) non-commutative p-extension of a totally real number field such that the finite part of its Galois group is a pgroup with exponent p. We first calculate the Whitehead groups of the Iwasawa algebra and its canonical Ore localisation by using Oliver-Taylor's theory upon integral logarithms. This calculation reduces the existence of the non-commutative p-adic zeta function to certain congruence conditions among abelian p-adic zeta pseudomeasures. Then we finally verify these congruences by using Deligne-Ribet'…
We describe an explicit `higher rank' Iwasawa theory for zeta elements associated to the multiplicative group over abelian extensions of general number fields. We then show that this theory leads to a concrete new strategy for proving special cases of the equivariant Tamagawa number conjecture. As a first application of this approach, we use it to prove new cases of the conjecture for Tate motives over natural families of abelian CM extensions of totally real fields for which the relevant $p$-adic $L$-functions possess trivial zeroes.
On generalized Iwasawa main conjectures and $p$-adic Stark conjectures for Artin motives
本文提出关于Artin动机的广义Iwasawa主猜想和$p$-adic Stark猜想，并证明其与Rubin-Stark元素的等价性及虚二次域中相关猜想的无条件成立。
Functional equations of algebraic Rankin-Selberg $p$-adic $L$-functions
本文为 Selmer 复形建立代数函数方程，从而得到代数 $p$-进 $L$ 函数的函数方程，并证明其完美性，应用于 Iwasawa 理论中 Rankin-Selberg 乘积的研究。
A geometric approach to the uniform boundedness of $\ell$-primary torsion points
本文通过几何方法证明了阿贝尔方案上ℓ-原 torsion 点的统一有界性，并解决了Cadoret和Tamagawa的猜想。
Let $p$ be an odd prime number. We construct explicit uniformizers for the totally ramified extension $\mathbb{Q}_p(ζ_{p^2},\sqrt[p]{p})$ of the field of $p$-adic numbers $\mathbb{Q}_p$, where $ζ_{p^2}$ is a primitive $p^2$-th root of unity.
Let A be an abelian variety over a number field k and F a finite cyclic extension of k of p-power degree for an odd prime p. Under certain technical hypotheses, we obtain a reinterpretation of the equivariant Tamagawa number conjecture (eTNC) for A, F/k and p as an explicit family of p-adic congru- ences involving values of derivatives of the Hasse-Weil L-functions of twists of A, normalised by completely explicit twisted regulators. This reinterpretation makes the eTNC amenable to numerical verification and furthermore leads to explicit predictions which refine well-known conjectures of Mazur…
Let $N/K$ be a Galois extension of $p$-adic number fields and let $V$ be a $p$-adic representation of the absolute Galois group $G_K$ of $K$. The equivariant local $ε$-constant conjecture $C_{EP}^{na}(N/K, V)$ is related to the compatibility of the equivariant Tamagawa number conjecture with the functional equation of Artin $L$-functions and it can be formulated as the vanishing of a certain element $R_{N/K}$ in $K_0(\mathbb Z_p[G],\mathbb Q_p^c[G])$. One of the main technical difficulties in the computation of $R_{N/K}$ arises from the so-called cohomological term $C_{N/K}$, which requires th…
Let $L/K$ be a finite Galois extension of number fields with Galois group $G$. Let $p$ be an odd prime and $r>1$ be an integer. Assuming a conjecture of Schneider, we formulate a conjecture that relates special values of equivariant Artin $L$-series at $s=r$ to the compact support cohomology of the étale $p$-adic sheaf $\mathbb Z_p(r)$. We show that our conjecture is essentially equivalent to the $p$-part of the equivariant Tamagawa number conjecture for the pair $(h^0(\mathrm{Spec}(L))(r), \mathbb Z[G])$. We derive from this explicit constraints on the Galois module structure of Banaszak's $p…
Given two monic polynomials f and g with coefficients in a number field K, and some a in K, we examine the action of the absolute Galois group of K on the directed graph of iterated preimages of a under the correspondence g(y)=f(x), assuming that deg(f)>deg(g) and that gcd(deg(f), deg(g))=1. If a prime of K exists at which f and g have integral coefficients, and at which a is not integral, we show that this directed graph of preimages consists of finitely many Galois-orbits. We obtain this result by establishing a p-adic uniformization of such correspondences, tenuously related to Bottcher's u…
Let A be an abelian variety defined over a number field F. For a prime number $\ell$, we consider the field extension of F generated by the $\ell$-powered torsion points of A. According to a conjecture made by Rasmussen and Tamagawa, if we require these fields to be both a pro-$\ell$ extension of $F(μ_{\ell^{\infty}})$ and unramified away from $\ell$, examples are quite rare. Indeed, it is expected that for a fixed dimension and field of definition, there exists such an abelian variety for only a finite number of primes. We prove a uniform version of the conjecture in the case where the abelia…
Given an abelian, CM extension K of any totally real number field k, we consider two conjectures `of Stark type'. The `Integrality Conjecture' concerns the image of a p-adic map `\mathfrak{s}_{K/k,S}' determined by the minus-part of the S-truncated equivariant L-function for K/k at s=1. It is connected to the Equivariant Tamagawa Number Conjecture of Burns and Flach. The `Congruence Conjecture' says that \mathfrak{s}_{K/k,S} gives an explicit reciprocity law for the element predicted by the corresponding Rubin-Stark Conjecture for K^+/k. We study the general properties of these conjectures and…
The Langlands Programme predicts that a weight 2 newform f over a number field K with integer Hecke eigenvalues generally should have an associated elliptic curve E_f over K. In our previous paper, we associated, building on works of Darmon and Greenberg, a p-adic lattice to f, under certain hypothesis, and implicitly conjectured that this lattice is commensurable with the p-adic Tate lattice of E_f . In this paper, we present this conjecture in detail and discuss how it can be used to compute, directly from f, an explicit Weierstrass equation for the conjectural E_f . We develop algorithms to…
We prove a variant of Oesterlé's conjecture about $p$-adic periods of the modular curve $X_0(p)$, with an additional $Γ(2)$-structure. We use de Shalit's techniques and $p$-adic uniformization of curves whose reduction is semi-stable.
Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally stud…
- Let K be a totally imaginary number field. Denote by G ur K (2) the Galois group of the maximal unramified pro-2 extension of K. By comparing cup-products in {é}tale cohomology of SpecO K and cohomology of uniform pro-2 groups, we obtain situations where G ur K (2) has no non-trivial uniform analytic quotient, proving some new special cases of the unramified Fontaine-Mazur conjecture. For example, in the family of imaginary quadratic fields K for which the 2-rank of the class group is equal to 5, we obtain that for at least 33.12% of such K, the group G ur K (2) has no non-trivial uniform an…
In this paper we show that certain Shimura varieties, uniformized by the product of complex unit balls, can be p-adically uniformized by the product (of equivariant coverings) of Drinfeld upper half-spaces. We also extend a p-adic uniformization to automorphic vector bundles. It is a continuation of our previous work [V], and contains all cases (up to a central modification) of a uniformization by known p-adic symmetric spaces. The idea of the proof is to show that an arithmetic quotient of the product of Drinfeld upper half-spaces cannot be anything else than a certain unitary Shimura variety…
We introduce a notion of inertial equivalence for integral $\ell$-adic representation of the Galois group of a global field. We show that the collection of continuous, semisimple, pure $\ell$-adic representations of the absolute Galois group of a global field lifting a fixed absolutely irreducible residual representation and with given inertial type outside a fixed finite set of places is uniformly bounded independent of the inertial type.
This thesis contains some new results on the uniform rationality of Poincaré series of p-adic equivalence relations and Igusa's conjecture on exponential sums
Let $k$ be a finitely generated field of characteristic $p > 0$ and $\ell$ a prime. Let $X$ be a smooth, separated, geometrically connected curve of finite type over $k$ and $ρ: π_1(X)\rightarrow GL_r(\mathbb Z_{\ell})$ a continuous representation of the \etale fundamental group of $X$ with image $G$. Any $k$-rational point $x:Spec(k)\rightarrow X$ induces a local representation $ρ_x: π_1(Spec(k)) \rightarrow π_1(X) \rightarrow GL_r(\mathbb Z_{\ell})$ with image $G_x$. The goal of this paper is to study how $G_x$ varies with $x\in X(k)$. In particular we prove that if $\ell\neq p$ and every op…
Classical and uniform exponents of multiplicative $p$-adic approximation
本文研究乘法 $p$-进逼近的无理指数，证明乘法与经典无理指数之比可任意取 $[1,2]$ 中值，并给出一致乘法指数的新上界 $(5+\sqrt{5})/2$。
A p-adic Poissonian Pair Correlation Concept
本文提出 p-adic Poissonian 对关联概念，推广实数均匀分布理论中的对关联统计量至 p-adic 整数，证明其主要性质，并探讨与 p-adic 偏差理论的联系。
Uniform bounds and uncertainty for asymptotics of representations of $p$-adic ${\rm GL}_N$
该研究在Langlands和Zelevinsky分类框架下，证明了p进GL_N表示在主同余子群下固定向量维数增长的两个独立结果：基于GK维数的固定向量及Harish-Chandra-Howe系数一致界，以及酉表示中GK维数与矩阵系数衰减率的定量关系。
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 develop some of the foundations of affinoid pre-adic spaces without Noetherian or finiteness hypotheses. We give some explicit examples of non-adic affinoid pre-adic spaces (including a locally perfectoid one). On the positive side, we also show that if every affinoid subspace of an affinoid pre-adic space is uniform, then the structure presheaf is a sheaf; note in particular that we assume no finiteness hypotheses on our rings here. One can use our result to give a new proof that the spectrum of a perfectoid algebra is an adic space.
We construct a non-sheafy uniform Banach algebra such that a rational localisation of the Berkovich spectrum does not preserve the uniformity. We also construct uniform affinoid rings in the sense of Roland Huber such that rational localisations of the adic spectra do not preserve the uniformity. One of them is an example of a non-sheafy uniform affinoid ring. We introduce the notion of local uniformity instead, and prove that the local uniformity implies the sheaf condition.
We study Euler systems for $\mathbb{G}_m$ over a number field $k$. Motivated by a distribution-theoretic idea of Coleman, we formulate a conjecture regarding the existence of such systems that is elementary to state and yet strictly finer than Kato's equivariant Tamagawa number conjecture for Dirichlet $L$-series at $s=0$. To investigate the conjecture, we develop an abstract theory of `Euler limits' and, in particular, prove the existence of canonical `restriction' and `localisation' sequences in this theory. By using this approach we obtain a variety of new results, ranging from a proof, mod…
We propose a conjecture for exponential sums which generalizes both a conjecture by Igusa and a local variant by Denef and Sperber, in particular, it is without the homogeneity condition on the polynomial in the phase, and with new predicted uniform behavior. The exponential sums have summation sets consisting of integers modulo $p^m$ lying $p$-adically close to $y$, and the proposed bounds are uniform in $p$, $y$, and $m$. We give evidence for the conjecture, by showing uniform bounds in $p$, $y$, and in some values for $m$. On the way, we prove new bounds for log-canonical thresholds which a…
Existential uniform $p$-adic integration and descent for integrability and largest poles
本文提出跟踪存在量词的 $p$-进胞腔分解变体，证明存在性函数可积性的子域下降，并推出 Serre-Poincaré 级数最大极点在域扩张中只增不减。
On non-commutative Iwasawa theory and derivatives of Euler systems
本文利用约化行列式函子描述非交换代数的相对 $K_0$-群，结合 zeta element 推广，提出了非交换 $p$-进岩泽理论的主猜想及 Rubin-Stark 欧拉系统的高阶导数公式，为等变 Tamagawa 数猜想提供了广泛推广的证据。
Real convergence and periodicity of $p$-adic continued fractions
本文研究了$p$-adic连续分数的实收敛与周期性关系，证明收敛是周期性的必要条件，并提出关于逆向的猜想，同时发展了非周期性的概率论论据。
Let E be a quadratic extension of a totally real number field. We construct Stickelberger elements for Hilbert modular forms of parallel weight 2 in anticyclotomic extensions of E. Extending methods developed by Dasgupta and Spieß from the multiplicative group to an arbitrary one-dimensional torus we bound the order of vanishing of these Stickelberger elements from below and, in the analytic rank zero situation, we give a description of their leading terms via automorphic L-invariants. If the field E is totally imaginary, we use the p-adic uniformization of Shimura curves to show the equality …
This paper originated as an appendix to the paper "Topology and Geometry of the Berkovich Ramification Locus for Rational Functions, II" by Xander Faber arXiv:1104.0943v2 [math.NT]. It may however be read independently. We prove a variant of Alain Robert's p-adic Rolle theorem, via the theory of the radius of convergence of p-adic connections and the theory of semistable reduction of p-adic curves. We carefully compare the present author's notion [Inv. Math. 182 (2010)] of radius of convergence, of a connection on a p-adic curve X, normalized by the choice of a semistable model of X, with Kedl…
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