Masanori Asakura, Masataka Chida
Restricting ourselves to elliptic curves over $\mathbb{Q}$, we reformulate the $p$-adic Beilinson conjecture due to Perrin-Riou, which is customized to our computational approach. We then develop a new algorithm for numerical verifications of the $p$-adic Beilinson conjecture, which is based on the theory of rigid cohomology and $F$-isocrystals.
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The purpose of this article is to give an overview of the series of papers [BK1], [BK2] concerning the $p$-adic Beilinson conjecture of motives associated to Hecke characters of an imaginary quadratic field $K$, for a prime $p$ which splits in $K$.
We give a new proof of Howard's $Λ$-adic Gross-Zagier formula, which we extend to the context of indefinite Shimura curves over $\mathbf{Q}$ attached to nonsplit quaternion algebras. This formula relates the cyclotomic derivative of a two-variable $p$-adic $L$-function restricted to the anticyclotomic line to the cyclotomic $p$-adic heights of Heegner points over the anticyclotomic tower, and our proof, rather than inspired by the original approaches of Gross-Zagier and Perrin-Riou, is via Iwasawa theory, based on the connection between Heegner points, Beilinson-Flach elements, and their expli…
Our objective in this series of two articles, of which the present article is the first, is to give a Perrin-Riou-style construction of $p$-adic $L$-functions (of Bellaïche and Stevens) over the eigencurve. As the first ingredient, we interpolate the Beilinson-Kato elements over the eigencurve (including the neighborhoods of $θ$-critical points). Along the way, we prove étale variants of Bellaïche's results describing the local properties of the eigencurve. We also develop the local framework to construct and establish the interpolative properties of these $p$-adic $L$-functions away from $θ$-…
In this short note, we show a p-adic analogue of Beilinson's equivalence comparing two derived categories: the derived category of holonomic modules and derived category of modules whose cohomologies are holonomic.
A detailed presentation of Beilinson's approach to p-adic Hodge theory.
We consider certain CM elliptic curves which are related to Fermat curves, and express the values of $L$-functions at $s=2$ in terms of special values of generalized hypergeometric functions. We compare them and a similar result of Rogers-Zudilin with Otsubo's regulator formulas, and give a new proof of the Beilinson conjectures originally due to Bloch.
We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of he conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in some other cases.
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…
This is a sequel of the paper "A generalization of the Ross symbols in higher K-groups and hypergeometric functions I" where we introduced higher Ross symbols in higher $K$-groups of the hypergeometric schemes, and discussed the Beilinson regulators. In this paper we give its p-adic counterpart and an application to the $p$-adic Beilinson conjecture for K3 surfaces of Picard number 20.
In the first section of his seminal paper on height pairings, Beilinson constructed an $\ell$-adic height pairing for rational Chow groups of homologically trivial cycles of complementary codimension on smooth projective varieties over the function field of a curve over an algebraically closed field, and asked about an generalization to higher dimensional bases. In this paper we answer Beilinson's question by constructing a pairing for varieties defined over the function field of a smooth variety $B$ over an algebraically closed field, with values in the second $\ell$-adic cohomology group of …
This paper studies the derived de Rham cohomology of F_p and p-adic schemes, and is inspired by Beilinson's recent work. Generalising work of Illusie, we construct a natural isomorphism between derived de Rham cohomology and crystalline cohomology for lci maps of such schemes, as well logarithmic variants. These comparisons give derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory. Placing these ideas in the skeleton of Beilinson's construction leads to a new proof of Fontaine's crystalline conjecture and Fontaine-Jannsen's semistable conjecture.
Our first goal in this note is to explain that a weak form of Perrin-Riou's conjecture on the non-triviality of Beilinson-Kato classes follows as an easy consequence of the Iwasawa main conjectures, and deduce its refined versions in the supersingular case from this fact and a variety of Gross-Zagier formulae. Our second goal is to set up a conceptual framework in the context of $Λ$-adic Kolyvagin systems to treat analogues of Perrin-Riou's conjectures for higher motives of higher rank. We apply this general discussion in order to establish a link between Heegner points on a general class of C…
Let G be a connected split reductive group over a p-adic field. In the first part of the paper we prove, under certain assumptions on G and the prime p, a localization theorem of Beilinson-Bernstein type for admissible locally analytic representations of principal congruence subgroups in the rational points of G. In doing so we take up and extend some recent methods and results of Ardakov-Wadsley on completed universal enveloping algebras to a locally analytic setting. As an application we prove, in the second part of the paper, a locally analytic version of Smith's theorem on the canonical di…
We propose a refined version of the Beilinson-Bloch conjecture for the motive associated with a modular form of even weight. This conjecture relates the dimension of the image of the relevant p-adic Abel-Jacobi map to certain combinations of Heegner cycles on Kuga-Sato varieties. We prove theorems in the direction of the conjecture and, in doing so, obtain higher weight analogues of results for elliptic curves due to Darmon.
Kings, Lei, Loeffler and Zerbes constructed a three-variable Euler system $κ({\bf g},{\bf h})$ of Beilinson-Flach elements associated to a pair of Hida families $({\bf g},{\bf h})$ and exploited it to obtain applications to the arithmetic of elliptic curves by specializing the Euler system to points of weights $(2,1,1)$. The aim of this article is showing that this Euler system also encodes arithmetic information at points of weights $(1,1,0)$, concerning the group of units of the associated number fields. The setting becomes specially novel and intriguing when ${\bf g}$ and ${\bf h}$ speciali…
In this paper, we study an analogue of the Tate conjecture for $K_2$ of U, the complement of split multiplicative fibers in an elliptic surface. A main result is to give an upper bound of the rank of the Galois fixed part of the etale cohomology $H^2(\bar{U},Q_p(2))$. As an application, we give an elliptic K3 surface $X$ over a p-adic field for which the torsion part of the Chow group $CH_0(X)$ of 0-cycles is finite. This would be the first example of a surface $X$ over a p-adic field whose geometric genus is non-zero and for which the torsion part of $CH_0(X)$ is finite.
A user's guide to Beilinson-Kato's zeta elements
本文是 Beilinson-Kato zeta 元素的使用指南，旨在解释 Kato 的 Euler 系统在椭圆曲线算术和 Iwasawa 理论中的框架，以帮助读者阅读原始论文。
We give a new proof of a conjecture of Darmon, Lauder and Rotger regarding the computation of the $\mathcal L$-invariant of the adjoint of a weight one modular form in terms of units and $p$-units. While in our previous work with Rotger the essential ingredient was the use of Galois deformations techniques following the computations of Bellaïche and Dimitrov, we propose a new approach exclusively using the properties of Beilinson--Flach classes. One of the key ingredients is the computation of a cyclotomic derivative of a cohomology class in the framework of Perrin-Riou theory, which can be se…
P-adic generalizations of hyper-elliptic functions
本研究聚焦超椭圆Mumford曲线分支点交比的表示问题，提出将其表为p进周期矩阵处取值的p进θ函数之商，推进了超椭圆函数的p进数推广研究。
In this paper, using p-adic integration with values in spaces of modular forms, we construct the p-adic analogue of Weil's elliptic functions according to Eisenstein in the book "Elliptic functions according to Eisenstein and and Kronecker". This construction extends Serre's p-adic family of Eisenstein series in "Formes modulaires et fonctions zêta p-adiques". We show that the power series expansion of Weil's elliptic functions also exists in the p-adic case.
In this thesis we construct a modified version of Karoubi's relative Chern character for smooth varieties over the complex numbers or the ring of integers in a p-adic number field. Comparison results with the Deligne-Beilinson Chern character and the p-adic Borel regulator constructed by Huber and Kings are proven. As a corollary we obtain a new proof of Burgos' theorem that Borel's regulator is twice Beilinson's regulator.
Generalised Heegner cycles are associated to a pair of an elliptic Hecke eigenform and a Hecke character over an imaginary quadratic extension $K/\Q$. Let $p$ be an odd prime split in $K/\Q$ and $l\neq p$ an odd unramified prime. We prove the non-triviality of the $p$-adic Abel-Jacobi image of generalised Heegner cycles modulo $p$ over the $\Z_l$-anticylotomic extension of $K$. The result is an evidence for the refined Bloch-Beilinson and the Bloch-Kato conjecture. In the case of two, it provides a refinement of the results of Cornut and Vatsal on the non-triviality of Heegner points over the …
In this article, we discuss Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms. After the overview on the situation of the ordinary case of Hida family, we will introduce a Coleman map for Coleman family for the non-ordinary case (Coleman family) which was obtained as a joint work with Filippo Nuccio [NO16] and we give some results on Iwasawa Theory for Coleman families as applications of . First, we give a construction of a two-variable $p$-adic $L$-function for a Coleman family thanks to the ingredients given in [NO16] (Theorem 5.1). Combining this result and Coleman…
We formulate integral Iwasawa main conjectures for suitable twists of a newform $f$ that is non-ordinary at $p$, over the cyclotomic $\mathbb{Z}_p$-extension, the anticyclotomic $\mathbb{Z}_p$-extensions (in both the definite and the indefinite cases) as well as the $\mathbb{Z}_p^2$-extension of an imaginary quadratic field $K$ where $p$ splits. In order to do so, we define Kobayashi-Sprung-style signed Coleman maps, which we use to introduce doubly-signed Selmer groups. In the same spirit, we construct signed (integral) Beilinson-Flach elements (out of the collection of unbounded Beilinson-Fl…
This is a completely rewritten version of the paper formerly entitled "Sheaves of Iwasawa modules, moment maps and the $\ell$-adic elliptic polylogarithm". The proof of the main result is also simplified. In the paper we study systematically the $\ell$-adic realization of the elliptic polylogarithm in the context of sheaves of Iwasawa modules. This leads to a description of the elliptic polylogarithm in terms of elliptic units. As an application we prove a precise relation between $\ell$-adic Eisenstein classes and elliptic Soulé elements. This allows to give a new proof of the formula for the…
We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over $p$-adic rings extends uniquely to a cohomology theory for varieties over $p$-adic fields that satisfies $h$-descent. This new cohomology - syntomic cohomology - is a Bloch-Ogus cohomology theory, admits period map to étale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on the étale side. In relative dimension zero we recover the potentiall…
Congruent elliptic curves over some $p$-adic Lie extensions
本文研究奇素数p下p进李扩张中p-同余椭圆曲线的Iwasawa不变量变化，同时分析经典Selmer群与精细Selmer群。
Let $E_{-D}$ be the elliptic curve $y^2=x^3+Dx$ defined over $K=\mathbb{Q}(i)$ for $D\in K$ which is coprime to 2. In this paper, we give a lower bound for the 2-adic valuation of the algebraic part of the central value of Hecke $L$-function associated to $E_{-D}$.
Let $K$ be a $p$-adic field and $E$ an elliptic curve over $K$ with potential good reduction. For some large Galois extensions $L$ of $K$ containing all $p$-power roots of unity, we show the vanishing of certain Galois cohomology groups of $L$ with values in the $p$-adic representation associated with $E$. This generalizes some results of Coates, Sujatha and Wintenberger.
BF path integrals for elliptic curves and $p$-adic $L$-functions
本文借助 Iwasawa 主猜想与 Mazur 控制定理，证明良好通常约化椭圆曲线的 $p$-进 $L$-函数逆绝对值的 BF 算术路径积分公式。
We prove that the first two coefficients in the series expansion around $s=1$ of the $p$-adic $L$-function of an elliptic curve over $\mathbb{Q}$ are related by a formula involving the conductor of the curve. This is analogous to a recent result of Wuthrich for the classical $L$-function, which makes use of the functional equation. We present a few other consequences for the $p$-adic $L$-function and a generalisation to the base-change to an abelian number field.
Given an elliptic curve $E$ over a local field $K$ with residue characteristic $3$, we investigate the action of the absolute Galois group of $K$ in the case of potentially good reduction. In particular the only not completely known case is that of the $\ell$-adic Galois representation attached to an elliptic curve such that the image of inertia is non-cyclic, and isomorphic to $C_3 \rtimes C_4$. In this work we describe such a representation explicitly.
In \cite{mccarthy2}, McCarthy defined a function $_{n}G_{n}[\cdots]$ using Teichmüller character of finite fields and quotients of $p$-adic gamma function, and expressed the trace of Frobenius of elliptic curves in terms of special values of $_{2}G_{2}[\cdots]$. We establish two different expressions for the traces of Frobenius of elliptic curves in terms of the function $_{2}G_{2}[\cdots]$. As a result, we obtain two relations between special values of the function $_{2}G_{2}[\cdots]$ with different parameters.
Fix $p$ an odd prime. Let $E$ be an elliptic curve over $\mathbb{Q}$ with semistable reduction at $p$. We show that the adjoint $p$-adic $L$-function of $E$ evaluated at infinitely many integers prime to $p$ completely determines up to a quadratic twist the isogeny class of $E$. To do this, we prove a result on the determination of isobaric representations of $\text{GL}(3, \mathbb{A}_\mathbb{Q})$ by certain $L$-values of $p$-power twists.
For primes $p>3$ we produce a new derivation of the universal $p$-adic sigma function and $p$-adic Weierstrass zeta functions of Mazur and Tate for ordinary elliptic curves by a method that highlights congruences among coefficients in Laurent expansions of elliptic functions, and works simultaneously for generalized elliptic curves defined by Weierstrass equations.
Weighted averages of $p$-adic hypergeometric functions and traces of Frobenius of elliptic curves
研究一类一参数椭圆曲线的弗罗贝尼乌斯迹及其与p-进超几何函数的关系，通过加权平均特殊值和特殊值表达，推导出p-进超几何函数的求和恒等式和p-进欧拉-普夫夫变换。
We define a function in terms of quotients of the $p$-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over finite fields to the $p$-adic setting. We prove, for primes $p > 3$, that the trace of Frobenius of any elliptic curve over $\mathbb{F}_p$, whose $j$-invariant does not equal 0 or 1728, is just a special value of this function. This generalizes results of Fuselier and Lennon which evaluate the trace of Frobenius in terms of hypergeometric functions over $\mathbb{F}_p$ when $p \equiv 1 \pmod {12}$.
Models of CM elliptic curves with a prescribed $\ell$-adic Galois image
该研究针对素数ℓ与类数1或2的虚二次序，刻画了对应复乘椭圆曲线在最小定义域上的所有ℓ-进伽罗瓦表示，并确定了具有指定ℓ-进伽罗瓦表示的椭圆曲线同构类。
We study the special value at 2 of L-functions of modular forms of weight 2 on congruence subgroups of the modular group. We prove an explicit version of Beilinson's theorem for the modular curve X_1(N). When N is prime, we deduce that the target space of Beilinson's regulator map is generated by the images of Milnor symbols associated to modular units of X_1(N). We also suggest a reformulation of Zagier's conjecture on L(E,2) for the jacobian J_1(N) of X_1(N), where E is an elliptic curve of conductor N. In this direction we define an analogue of the elliptic dilogarithm for any jacobian J : …
$L-$series attached to two classical families of elliptic curves with complex multiplications are studied over number fields, formulae for their special values at $s=1, $ bound of the values, and criterion of reaching the bound are given. Let $ E_1: y^{2}=x^{3}-D_1 x $ be elliptic curves over the Gaussian field $K=\Q(\sqrt{-1}), $ with $ D_1 =π_{1} ... π_{n} $ or $ D_1 =π_{1} ^{2}... π_{r} ^{2} π_{r+1} ... π_{n}$, where $π_{1}, ..., π_{n}$ are distinct primes in $K$. A formula for special values of Hecke $L-$series attached to such curves expressed by Weierstrass $\wp-$function are given; a lo…
Consider an elliptic curve defined over an imaginary quadratic field $K$ with good reduction at the primes above $p\geq 5$ and has complex multiplication by the full ring of integers $\mathcal{O}_K$ of $K$. In this paper, we construct $p$-adic analogues of the Eisenstein-Kronecker series for such elliptic curve as Coleman functions on the elliptic curve. We then prove $p$-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function.
We prove that the canonical dimension of a coadmissible representation of a semisimple $p$-adic Lie group in a $p$-adic Banach space is either zero or at least half the dimension of a non-zero coadjoint orbit. To do this we establish analogues for $p$-adically completed enveloping algebras of Bernstein's inequality for modules over Weyl algebras, the Beilinson-Bernstein localisation theorem and Quillen's Lemma about the endomorphism ring of a simple module over an enveloping algebra.
Let $A/\mathbb{Q}$ be an elliptic curve with split multiplicative reduction at a prime $p$. We prove (an analogue of) a conjecture of Perrin-Riou, relating $p$-adic Beilinson$-$Kato elements to Heegner points in $A(\mathbb{Q})$, and a large part of the rank-one case of the Mazur$-$Tate$-$Teitelbaum exceptional zero conjecture for the cyclotomic $p$-adic $L$-function of $A$. More generally, let $f$ be the weight-two newform associated with $A$, let $f_{\infty}$ be the Hida family of $f$, and let $L_{p}(f_{\infty},k,s)$ be the Mazur$-$Kitagawa two-variable $p$-adic $L$-function attached to $f_{\…
We discuss abelian equivariant Iwasawa theory for elliptic curves over $\mathbb{Q}$ at good supersingular primes and non-anomalous good ordinary primes. Using Kobayashi's method, we construct equivariant Coleman maps, which send the Beilinson-Kato element to the equivariant $p$-adic $L$-functions. Then we propose equivariant main conjectures and, under certain assumptions, prove one divisibility via Euler system machinery. As an application, we prove a case of a conjecture of Mazur-Tate.
Consider the elliptic curve $E$ given by the Weierstrass equation $y^2 = x^3 - 11x - 14$, which has complex multiplication by the order of conductor $2$ inside $\mathbb{Z}[i]$. It was recently observed in a paper of Daniels and Lozano-Robledo that, for each $n \geq 2$, $\mathbb{Q}(μ_{2^{n+1}}) \subseteq \mathbb{Q}(E[2^n])$. In this note, we prove that this (a priori surprising) ``tower of vertical entanglements'' is actually more a feature than a bug: it holds for any elliptic curve $E$ over $\mathbb{Q}$ with complex multiplication by any order of even discriminant.
In this paper we present a description of the Galois representation attached to an elliptic curve defined over a $2$-adic field $K$, in the case where the image of inertia is non-abelian. There are two possibilities for the image of inertia, namely $Q_8$ and $SL_2(\mathbb{F}_3)$, and in each case we need to distinguish whether the inertia degree of $K$ over $\mathbb{Q}_2$ is even or odd. The result presented here can be implemented in an algorithm to compute explicitly the Galois representation in these four cases.
We give a classification of all possible $2$-adic images of Galois representations associated to elliptic curves over $\mathbb{Q}$. To this end, we compute the 'arithmetically maximal' tower of 2-power level modular curves, develop techniques to compute their equations, and classify the rational points on these curves.
On the integral $2$-adic Tate module of elliptic curves
本文研究奇剩余特征完全离散赋值域上椭圆曲线的整2-adic Tate模，利用显式对半公式确定伽罗瓦群在2幂挠塔的作用，证明其由2-挠表示、c的平方类及根差的局部信息决定。
Complete classification of $7$-adic Galois images for non-CM elliptic curves over $\Q$
本文解决了Furio和Lombardo的猜想1.6，从而完成了非CM椭圆曲线在有理数域上的7-adic Galois像的完全分类，证明基于作者2026年6月至9月的AI训练，且为独立完成。
For modular elliptic curves over number fields of narrow class number one, and with multiplicative reduction at a collection of p-adic primes, we define new p-adic invariants. Inspired by Nekovar and Scholl's plectic conjectures, we believe these invariants control the Mordell--Weil group of higher rank elliptic curves and we support our expectations with numerical experiments.
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 show that Hida's families of $p$-adic elliptic modular forms generalize to $p$-adic families of Jacobi forms. We also construct $p$-adic versions of theta lifts from elliptic modular forms to Jacobi forms. Our results extend to Jacobi forms previous works by Hida and Stevens on the related case of half-integral weight modular forms.
On Zagier's conjecture for base extensions of elliptic curves
本文利用Beilinson定理于模曲线，证明了椭圆曲线基扩张的Zagier猜想弱版本，涉及L(E/F,2)值。
Let $E$ be a rational elliptic curve and let $p$ be an odd prime of additive reduction. Let $K$ be an imaginary quadratic field and fix a positive integer $c$ prime to the conductor of $E$. The main goal of the present article is to define an anticyclotomic $p$-adic $L$-function $Ł$ attached to $E/K$ when $E/\QQ_p$ attains semistable reduction over an abelian extension. We prove that $Ł$ satisfies the expected interpolation properties; namely, we show that if $χ$ is an anticyclotomic character of conductor $cp^n$ then $χ(Ł)$ is equal (up to explicit constants) to $L(E,χ,1)$ or $L'(E,χ,1)$.
Models of Elliptic Curves with Maximal 2-adic Image Subject to a Rational 2-Isogeny
本文给出在有理2-同源约束下具有最大2-adic像的非CM椭圆曲线的显式模型，利用模曲线与同源图分类使通用性显式化。
In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic analogue of the result of Beilinson and Levin expressing the complex elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. Our result is valid even if the elliptic curve has supersingular…
The 3-adic representations arising from elliptic curves over $\mathbb{Q}_3$ with potential good reduction
本文完整分类了同构于 $\mathbb{Q}_3$ 上椭圆曲线 Tate 模的潜在结晶 3-adic 表示，通过过滤 $(φ, \mathrm{Gal}(K/\mathbb{Q}_3))$-模描述，并重点关注野性潜在好约化的情况。
We prove the first cases of a conjecture by Darmon--Rotger on the non-vanishing of generalized Kato classes attached to elliptic curves $E$ over $\mathbf{Q}$ of rank $2$. Our method also shows that the non-vanishing of generalized Kato classes implies that the $p$-adic Selmer group of $E$ is $2$-dimensional. The main novelty in the proof is a formula for the leading term at the trivial character of an anticyclotomic $p$-adic $L$-function attached to $E$ in terms of the derived $p$-adic height of generalized Kato classes and an enhanced $p$-adic regulator.
Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined over $\mathbb{Q}$ without CM. The isogeny graph associated to $\mathcal{E}$ is a graph which has a vertex for each elliptic curve in $\mathcal{E}$ and an edge for each $\mathbb{Q}$-isogeny of prime degree that maps one elliptic curve in $\mathcal{E}$ to another elliptic curve in $\mathcal{E}$, with the degree recorded as a label of the edge. An isogeny-torsion graph is an isogeny graph where, in addition, we label each vertex with the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the correspond…
We formulate a multi-variable p-adic Birch and Swinnerton-Dyer conjecture for p-ordinary elliptic curves A over number fields K. It generalises the one-variable conjecture of Mazur-Tate-Teitelbaum, who studied the case K=Q and the phenomenon of exceptional zeros. We discuss old and new theoretical evidence towards our conjecture and in particular we fully prove it, under mild conditions, in the following situation: K is imaginary quadratic, A=E_K is the base-change to K of an elliptic curve over the rationals, and the rank of A is either 0 or 1. The proof is naturally divided into a few cases.…
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