Congruences between abelian pseudomeasures, II

Finite abelian groups via congruences

For every finite abelian group $G$, there are positive integers $n$ and $d$ such that $G$ is isomorphic to the multiplicative group of $d$-th powers of reduced residues modulo $n$.

Inductive construction of the p-adic zeta functions for non-commutative p-extensions of totally real fields with exponent p

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'…

On Mordell-Weil groups and congruences between derivatives of twisted Hasse-Weil L-functions

Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair ( h^1(A/F)(1), Z[Gal(F/k)] ). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mor…

On the transfer congruence between $p$-adic Hecke $L$-functions

We prove the transfer congruence between $p$-adic Hecke $L$-functions for CM fields over cyclotomic extensions, which is a non-abelian generalization of the Kummer's congruence. The ingredients of the proof include the comparison between Hilbert modular varieties, the $q$-expansion principle, and some modification of Hsieh's Whittaker model for Katz' Eisenstein series. As a first application, we prove explicit congruence between special values of Hasse-Weil $L$-function of a CM elliptic curve twisted by Artin representations. As a second application, we prove the existence of a non-commutative…

Asymptotics of class number and genus for abelian extensions of an algebraic function field

Among abelian extensions of a congruence function field, an asymptotic relation of class number and genus is established. The proof is classical, employing well-known results from congruence function field theory.

Eisenstein同余在tame族中的性质与一个metabelian扩张的类群

Eisenstein congruences in tame families and the class group of a metabelian extension

本文利用tame族中Eisenstein同余存在性的新结果，解释了一个特定metabelian数域中素数分裂行为的计算观察，延续了Ribet开创的范式。

Comments on Y. O. Hamidoune's Paper "Adding Distinct Congruence Classes"

The main result in Y.~O.~Hamidoune's paper "Adding Distinct Congruence Classes" ({\em Combin.~Probab.~Comput.}~{\bf 7} (1998) 81-87) is as follows: If $S$ is a generating subset of a cyclic group $G$ such that $0 \not \in S$ and $|S| \geq 5$, then the number of sums of the subsets of $S$ is at least $\min (|G|, 2|S| )$. Unfortunately, argument of the author, who, sadly, passed away in 2011, relies on a lemma whose proof is incorrect; in fact, the lemma is false for all cyclic groups of even order. In this short note we point out this mistake, correct the proof, and discuss why the main result …

Congruences for critical values of higher derivatives of twisted Hasse-Weil L-functions

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…

Some congruences on prime factors of class number of algebraic extensions K/Q

This paper is an updated version of ANT-0372 (2002 dec 4) with the same title. Several errors are corrected in this version. An example of the kind of results obtained is: Let K/\Q be an abelian extension with N = [K:\Q] > 1, N odd. Let h(K) be the class number of K. Suppose that h(K) > 1. Let p be a prime dividing h(K). Let r_p be the rank of the p-class group of K. Then p \times (p^{r_p}-1) and N are not coprime. The paper is at elementary level and contains a lot of numerical examples.

Congruences for critical values of higher derivatives of twisted Hasse-Weil L-functions, III

Let $A$ be an abelian variety defined over a number field $k$, let $p$ be an odd prime number and let $F/k$ be a cyclic extension of $p$-power degree. Under not-too-stringent hypotheses we give an interpretation of the $p$-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of $A$ of the ${\rm Gal}(F/k)$-valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in…

关于模阿贝尔簇的D-新模次数与志村次数的比较

On the Comparison of the D-new Modular Degree and the Shimura Degree of Modular Abelian Varieties

研究有理数域上椭圆曲线E，在局部伽罗瓦表示假设下，证明其志村参数化次数等于D-新模次数，应用此结果证明相关猜想，建立志村雅可比簇一重性新情况，得到一重性不成立例子，否定特定问题并关联相关现象。

整体函数域上有效 A-动机的伽罗瓦表示的同余

Congruences of Galois representations attached to effective $A$-motives over global function fields

本文在整体函数域上建立了强半稳定有效 A-动机的 p-进伽罗瓦表示同余判别准则，是 Ozeki-Taguchi 数域结果的函数域类比，并据此证明了受 Rasmussen-Tamagawa 猜想启发的某类有效 A-动机的不存在性。

同余覆盖的基本维数

The Essential Dimension of Congruence Covers

本文利用p可除群和有限平坦群概形的形变理论，计算了主极化阿贝尔簇模空间同余覆盖的基本维数，并应用于曲线模空间及局部对称簇，回答了Kronecker-Klein问题。

Extensions of abelian varieties defined over a number field

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.

Proof of the Main Conjecture of Noncommutative Iwasawa Theory for Totally Real Number Fields in Certain Cases

Fix an odd prime $p$. Let $G$ be a compact $p$-adic Lie group containing a closed, normal, pro-$p$ subgroup $H$ which is abelian and such that $G/H$ is isomorphic to the additive group of $p$-adic integers $\mathbbZ_p$ . First we assume that $H$ is finite and compute the Whitehead group of the Iwasawa algebra, $Λ(G)$, of $G$. We also prove some results about certain localisation of $Λ(G)$ needed in Iwasawa theory. Let $F$ be a totally real number field and let $F_{\infty}$ be an admissible $p$-adic Lie extension of $F$ with Galois group $G$. The computation of the Whitehead groups are used to …

Modular abelian varieties over number fields

The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the zeta function $L(B/K;s)$ is equivalent to the product of zeta functions of non-CM newforms for congruence subgroups $Γ_1(N)$. The characterization involves the structure of End(B), isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$. We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illu…

关于 $\text{GL}_2$ 型阿贝尔簇的自同态代数及其丢番图应用

On endomorphism algebras of $\text{GL}_2$-type abelian varieties and Diophantine applications

本文证明两个无复乘新形式在素数同余下导致对应阿贝尔簇自同态代数同构，并应用于证明特定丢番图方程无解。

模符号与等变双有理不变量

Modular Symbols and Equivariant Birational Invariants

本文研究经典模符号与有限阿贝尔群相关的Kontsevich-Pestun-Tschinkel群之间的关系，探讨其等变双有理不变量。

Fine Selmer groups of congruent Galois representations

In this paper, we study the fine Selmer groups of two congruent Galois representations over an admissible $p$-adic Lie extension. We show that under appropriate congruence conditions, if the dual fine Selmer group of one is pseudo-null, so is the other. Our results also compare the $π$-primary submodules of the two dual fine Selmer groups. We then apply our results to compare the structure of Galois group of the maximal abelian unramified pro-$p$ extension of an admissible $p$-adic Lie extension and the structure of the dual fine Selmer group over the said admissible $p$-adic Lie extension.

Non-commutative Iwasawa theory for modular forms

The aim of the present paper is to give evidence, largely numerical, in support of the non-commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k>2 over the Galois extension of Q obtained by adjoining to Q all p-power roots of unity, and all p-power roots of a fixed integer m>1. The predictions of the main conjecture are rather intricate in this case because there is more than one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agrees perfectly with all aspects of the main conjecture, including Kat…