Suman Ahmed, Chandrakant Aribam, Sudhanshu Shekhar
Given two elliptic curves $E_1$ and $E_2$ defined over the field of rational numbers, $\mathbb{Q}$, with good reduction at an odd prime $p$ and equivalent mod $p$ Galois representation, we compare the $p$-Selmer rank, global and local root numbers of $E_1$ and $E_2$ over number fields.
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Iwasawa invariants and class number parity of multi-quadratic number fields
本文基于Iwasawa和Kida的方法，通过详细研究Hasse单位指标和素理想分歧，得到数域分圆Z_2扩张的Iwasawa不变量λ_2的显式结果，并在Greenberg猜想下给出虚多二次域λ_2的显式公式，进而提出判定多二次域类数奇偶性的准则。
In this article, we study the Iwasawa theory for Hilbert modular forms over the anticyclotomic extension of a CM field. We prove a one sided divisibility result toward the Iwasawa main conjecture. The proof relies on the first and second reciprocity law relating theta elements to Heegner points Euler system. As a by-product we also prove certain Bloch-Kato type result in the rank $0$ case and a parity conjecture
Let $Φ$ be a root system and let $Φ(\Zp)$ be the standard Chevalley $\Zp$-Lie algebra associated to $Φ$. For any integer $t\geq 1$, let $G$ be the uniform pro-$p$ group corresponding to the powerful Lie algebra $p^t Φ(\Zp)$ and suppose that $p\geq 5$. Then the Iwasawa algebra $Ω_G$ has no nontrivial reflexive two-sided ideals. This was previously proved by the authors for the root system $A_1$.
Construction of graph coverings with prescribed Iwasawa invariants
本文研究了图覆盖的Iwasawa不变量，证明在满足条件的情况下，任何(λ, μ)对均可作为无分支覆盖的不变量，且允许分支覆盖时无需奇偶条件。
Iwasawa invariants and exceptional pairs of certain abelian fields
本文针对Q(√d,ζₚ)研究岩泽不变量与例外对，确定岩泽模的Λ同构类及例外对结构差异，在指定范围内发现15个新例外对，包括λ不变量为2的特殊情形。
The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves
本文利用弱于 Iwasawa 主猜想的结论刻画任意秩椭圆曲线的 Selmer 群结构，并推出 Kato Kolyvagin 系统非平凡、BSD 结构精细化、秩上界及 Kurihara 猜想等结果。
Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes
本文推广Darmon-Iovita的正负Iwasawa主猜想至$\mathrm{GL}_2$型阿贝尔簇（非普通素数，分裂情形）及椭圆曲线（惰性情形），并证明相应的单侧包含关系。
The purpose of the paper is to complete several global and local results concerning parity of ranks of elliptic curves. Primarily, we show that the Shafarevich-Tate conjecture implies the parity conjecture for all elliptic curves over number fields, give a formula for local and global root numbers of elliptic curves and complete the proof of a conjecture of Kramer and Tunnell in characteristic 0. The method is to settle the outstanding local formulae by deforming from local fields to totally real number fields and then using global parity results.
The parity conjecture has a long and distinguished history. It gives a way of predicting the existence of points of infinite order on elliptic curves without having to construct them, and is responsible for a wide range of unexplained arithmetic phenomena. It is one of the main consequences of the Birch and Swinnerton-Dyer conjecture and lets one calculate the parity of the rank of an elliptic curve using root numbers. In this handbook, we explain how to use local root numbers of elliptic curves to realise some of these phenomena, with an emphasis on explicit calculations. The text is aimed at…
Iwasawa theory and ranks of elliptic curves in quadratic twist families
本研究用岩泽理论方法研究二次扭族椭圆曲线秩的分布，借助Matsuno的Kida型公式构造二次扭，得到相关秩的结果与扭数量的渐近下界，支撑Goldfeld猜想并补充Smith的相关成果。
Anticyclotomic Iwasawa theory of CM elliptic curves at ramified primes
该研究针对CM域中分歧素数处的CM椭圆曲线，构建反分圆岩泽理论的整框架，定义带符号塞尔默群，证明相关岩泽主猜想，填补了无非三角几何形变的对应理论空白。
For an elliptic curve E over a number field K, we prove that the algebraic rank of E goes up in infinitely many extensions of K obtained by adjoining a cube root of an element of K. As an example, we briefly discuss E=X_1(11) over Q, and how the result relates to Iwasawa theory.
Let $K/F$ be a finite Galois extension of number fields and $σ$ be an absolutely irreducible, self-dual representation of $\mathrm{Gal}(K/F)$. Let $p$ be an odd prime and consider two elliptic curves $E_1, E_2$ with good, ordinary reduction at primes above $p$ and equivalent mod-$p$ Galois representations. In this article, we study the variation of the parity of the multiplicities of $σ$ in the representation space associated to the $p^\infty$-Selmer group of $E_i$ over $K$. We also compare the root numbers for the twist of $E_i/F$ by $σ$ and show that the $p$-parity conjecture holds for the t…
The parity of the analytic rank of an elliptic curve is given by the root number in the functional equation L(E,s). Fixing an elliptic curve over any number field and considering the family of its quadratic twists, it is natural to ask what the average analytic rank in this family is. A lower bound on this number is given by the average root number. In this paper, we investigate the root number in such families and derive an asymptotic formula for the proportion of curves in the family that have even rank. Our results are then used to support a conjecture about the average analytic rank in thi…
For an elliptic curve $E$ over a number field $K$, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-Weil rank. Assuming finiteness of $\mathrm{Sha}(E/K)[p^\infty]$ for a prime $p$ this is equivalent to the $p$-parity conjecture: the global root number matches the parity of the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group. We complete the proof of the $p$-parity conjecture for elliptic curves that have a $p$-isogeny for $p > 3$ (the cases $p \le 3$ were known). T. and V. Dokchitser have showed …
The aim of the present paper is to establish relations between Iwasawa and Bernoulli numbers based on some results by M. Kervaire and M. P. Murthy about the structure of the $K_0$ groups of the integer group rings of cyclic groups of prime power order $p^n .$ In particular, we will prove that $λ_{i}\leq p-1$ under assumption that the generalized Bernoulli number $B_{1,ω^{-i}}$ is not divisible by $p^2$. Here $ω$ is the Teichmüller character of $\mathbb{Z}/(p-1)\mathbb{Z}$. $λ_{i}=1$ if $B_{1,ω^{-i}}$ is divisible by $p^2$. We will prove that $S_{n,i}\cong \mathbb{Z}/(p^{n+k_i})$, where $S_n$ i…
Anticyclotomic Iwasawa theory of abelian varieties of $\mathrm{GL}_2$-type at non-ordinary primes II
本文将Castella和Wan的超奇异素数处椭圆曲线反分圆Heegner点主猜想结果推广至$\u200b\mathrm{GL}_2$-型阿贝尔簇的Sprung型猜想及$p$惰性情形的正负猜想，证明了部分结果并给出Gross–Zagier与Kolyvagin定理的$p$-逆命题。
Let $p$ be an odd prime number. Let $K$ be the $p$-th cyclotomic field and $F$ its maximal real subfield. We give general formulae of the root numbers of the Jacobian varieties of the Fermat curves $X^p+Y^p=δ$ where $δ$ is an integer. As an application of these general formulae, we derive the equidistribution of the root numbers for the families of Jacobian varieties of the Fermat curves. When $p\nmid δ$, we bound the Selmer groups of these Jacobian varieties. Moreover, if $p$ is regular and all prime ideals of $K$ dividing $δ$ are inert in $K/F$, the Selmer groups are explicitly determined an…
Let $p$ be an odd prime. We attach appropriate signed Selmer groups to an elliptic curve $E$, where $E$ is assumed to have semistable reduction at all primes above $p$. We then compare the Iwasawa $λ$-invariants of these signed Selmer groups for two congruent elliptic curves over the cyclotomic $\mathbb{Z}_p$-extension in the spirit of Greenberg-Vatsal and B. D. Kim. As an application of our comparsion formula, we show that if the $p$-parity conjecture is true for one of the congruent elliptic curves, then it is also true for the other elliptic curve. In the midst of proving this latter result…
By considering a one-parameter family of elliptic curves defined over $\mathbb{Q}$, we might ask ourselves if there is any bias in the distribution (or parity) of the root numbers at each specialization. From the work of Helfgott, we know (at least conjecturally) that the average root number of an elliptic curve defined over $\mathbb{Q}(T)$ is zero as soon as there is a place of multiplicative reduction over $\mathbb{Q}(T)$ other than $-$deg. Recently, Helfgott's work was extended by Desjardins, where she relaxes some of Helfgott's hypotheses and is able to provide unconditional results on the…
Let E be a one-parameter family of elliptic curves over a number field. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain degeneracy condition. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any non-degenerate family E has average root number 0, provided that two classical arithmetical conjectures hold for two homogeneous polynomials with integral coefficients constructed ex…
Conjecturally, the parity of the Mordell-Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers, so the rank modulo 2 is conjecturally the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any n>2, so this is purely a parity phenomenon.
Let E be an elliptic curve over a number field K which admits a cyclic p-isogeny with p odd and semistable at primes above p. We determine the root number and the parity of the p-Selmer rank for E/K, in particular confirming the parity conjecture for such curves. We prove the analogous results for p=2 under the additional assumption that E is not supersingular at primes above 2.
Anticyclotomic $p$-adic $L$-functions for Rankin--Selberg product
本文在两种根数下沿反分圆方向为 hermitian 型自守形式的 Rankin--Selberg 乘积构造了 $p$-adic $L$-functions，并提出相应的非消失猜想和主猜想。
This is an expository article, based on a lecture course given at CRM Barcelona in December 2009. The purpose of these notes is to prove, in a reasonably self-contained way, that finiteness of the Tate-Shafarevich group implies the parity conjecture for elliptic curves over number fields. Along the way, we review local and global root numbers of elliptic curves and their classification, and discuss some peculiar consequences of the parity conjecture.
The Birch and Swinnerton-Dyer conjecture predicts that the parity of the algebraic rank of an abelian variety over a global field should be controlled by the expected sign of the functional equation of its $L$-function, known as the global root number. In this paper, we give explicit formulae for the local root numbers as a product of Jacobi symbols. This enables one to compute the global root number, generalising work of Rohrlich who studies the case of elliptic curves. We provide similar formulae for the root numbers after twisting the abelian variety by a self-dual Artin representation. As …
A note on the parity conjecture and base change
本文假设Shafarevich-Tate猜想，证明了若半稳定主极化阿贝尔簇在Q及所有二次域上满足奇偶性猜想，则该猜想对所有数域成立，并建立了基变换奇偶性由基域及小扩张蕴含的判据。
For a prime $\ell$ and an abelian variety $A$ over a global field $K$, the $\ell$-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the $\mathbb{Z}_{\ell}$-corank of the $\ell^{\infty}$-Selmer group and the analytic rank agree modulo $2$. Assuming that $\mathrm{char} K > 0$, we prove that the $\ell$-parity conjecture holds for the base change of $A$ to the constant quadratic extension if $\ell$ is odd, coprime to $\mathrm{char} K$, and does not divide the degree of every polarization of $A$. The techniques involved in the proof include the étale cohomo…
A Lehmer number modulo a prime $p$ is an integer $a$ with $1 \leq a \leq p-1$ whose inverse $\bar{a}$ within the same range has opposite parity. Lehmer numbers that are also primitive roots have been discussed by Wang and Wang in an endeavour to count the number of ways $1$ can be expressed as the sum of two primitive roots that are also Lehmer numbers (an extension of a question of S. Golomb). In this paper we give an explicit estimate for the number of Lehmer primitive roots modulo $p$ and prove that, for all primes $p \neq 2,3,7$, Lehmer primitive roots exist. We also make explicit the know…
In section 2, we show a $p$-parity result in a $D_{2p^{n}}$-extension of number fields $L/K$ ($p\geq 5$) for the twist $1\oplus η\oplus τ$: W(E/K,1\oplus η\oplus τ)=(-1)^{< 1\oplusη\oplus τ, X_{p}(E/L)>}, where $E$ is an elliptic curve over $K,$ $η$ and $τ$ are respectively the quadratic character and an irreductible representation of degree 2 of $Gal(L/K)=D_{2p^{n}},$ and $X_{p}(E/L)$ is the $p$-Selmer group. The main novelty is that we use a congruence result between $% ε_{0}$-factors (due to Deligne) for the determination of local root numbers in bad cases (places of additive reduction abov…
The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at 2 and 3, K/Q is abelian and K^\infty is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of Gal(K^\infty/Q). We also give analogous results when K/Q is non-abelian, the base field is not Q and E is replaced by an abelian variety. The heart of the paper is a study of relations b…
Balancing numbers $n$ are originally defined as the solution of the Diophantine equation $1+2+\cdots+(n-1)=(n+1)+\cdots+(n+r)$, where $r$ is called the balancer corresponding to the balancing number $n$. By slightly modifying, $n$ is the cobalancing number with the cobalancer $r$ if $1+2+\cdots+n=(n+1)+\cdots+(n+r)$. Let $B_n$ denote the $n^{th}$ balancing number and $b_n$ denote the $n^{th}$ cobalancing number. Then $8B_n^2+1$ and $8b_n^2+8b_n+1$ are perfect squares. The $n^{th}$ Lucas-balancing number $C_n$ and the $n^{th}$ Lucas-cobalancing number $c_n$ are the positive roots of $8B_n^2+1$ …
Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of Sha, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of Sha[6^\infty] and some restrictions on the reduction at primes above 2 a…
Arithmetic invariants of torus links
本文从算术视角研究环面纽结与链环 $T_{p,q}$ 的亚历山大多项式零点分布，证明其渐近均匀分布于单位圆并给出本原单位根频率公式，同时确定相应分歧覆盖的岩泽不变量及同调增长的次指数性质。
Tamagawa numbers and positive rank of elliptic curves
本文探讨无需寻找无限阶点或计算L函数即可预测椭圆曲线正秩的方法，通过塔玛加瓦数乘积值预测正秩，并证明其与奇偶性猜想的关系。
We extend many results on Selmer groups for elliptic curves and modular forms to the non-ordinary setting. More precisely, we study the signed Selmer groups defined using the machinery of Wach modules over $\mathbf{Z}_p$-cyclotomic extensions. First, we provide a definition of residual and non-primitive Selmer groups at non-ordinary primes. This allows us to extend techniques developed by Greenberg (for $p$-ordinary elliptic curves) and Kim ($p$-supersingular elliptic curves) to show that if two $p$-non-ordinary modular forms are congruent to each other, then the Iwasawa invariants of their si…
Parity of ranks of Jacobians of curves
本文研究带自同构群作用的曲线 Jacobian 簇 Selmer 群，利用局部不变量刻画 Mordell–Weil 秩奇偶性，并据此给出椭圆曲线奇偶性猜想的新证明。
For a global field K and an elliptic curve E_eta over K(T), Silverman's specialization theorem implies that rank(E_eta(K(T))) <= rank(E_t(K)) for all but finitely many t in P^1(K). If this inequality is strict for all but finitely many t, the elliptic curve E_eta is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, …
We show the density of rational points on non-isotrivial elliptic surfaces by studying the variation of the root numbers among the fibers of these surfaces, conditionally to two analytic number theory conjectures (the squarefree conjecture and Chowla's conjecture). This is a weaker statement than the one found in a preprint of Helfgott which proves (under the same assumptions) that the average root number is $0$ when the surface admits a place of multiplicative reduction. However, we use a different technique. The conjectures involved impose a restriction on the degree of the irreducible facto…
In this paper, we study the theories of analytic and arithmetic local constants of elliptic curves, with the work of Rohrlich, for the former, and the work of Mazur and Rubin, for the latter, as a basis. With the Parity Conjecture as motivation, one expects that the arithmetic local constants should be the algebraic additive counterparts to ratios of local analytic root numbers. We calculate the constants on both sides in various cases, establishing this connection for a substantial class of elliptic curves. By calculating the arithmetic constants in some new cases, we also extend the class of…
Gaussian Universality of Products Over Split Reductive Groups and the Satake Isomorphism
该研究利用Satake同构与Hall-Littlewood多项式，刻画非阿基米德域上分裂约化群乘积的奇异数与角元，推广已有结果并证明其最小分离性，推导相关概率极限定理，建立代数分解与概率极限的统一框架。
$\infty$-Categorical Generalized Langlands Program I: Mixed-Parity Modules and Sheaves
本研究在小v-栈框架下构建混合奇偶性场景的p进霍奇理论基础，研究对应模v-栈，为广义朗兰兹纲领提供支撑，并给出Moduli_G上理论的p进上同调化实现。
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