K. Fujiwara
In this paper, congruences between holomorphic Hilbert modular forms are studied. We show the best possible level optimization result outside l for l > 2 by solving the remaining case of Mazur principle when the degree of the totally real field is even.
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We show how to construct Darmon points on elliptic curves over totally real fields.
We prove explicit formulas for the $p$-adic $L$-functions of totally real number fields and show how these formulas can be used to compute values and representations of $p$-adic $L$-functions.
Some time ago we extended our monogenity investigations and calculations of generators of power integral bases to the relative case. Up to now we considered (usually totally real) extensions of complex quartic fields. In the present paper we consider power integral bases in relative extensions of totally real fields. Totally complex quartic extensions of totally real number fields seems to be the most simple case, that we detail here. As we shall see, even in this case we have to overcome several unexpected difficulties, which we can, however solve by properly (but not trivially) adjusting sta…
We enumerate all totally real number fields F with root discriminant delta_F <= 14. There are 1229 such fields, each with degree [F:QQ] <= 9.
C.J. Smyth and later Flammang studied the spectrum of the Weil height in the field of all totally real numbers, establishing both lower and upper bounds for the limit infimum of the height of all totally real integers and determining isolated values of the height. We remove the hypothesis that we consider only integers and establish an lower bound on the limit infimum of the height for all totally real numbers. Our proof relies on a quantitative equidistribution theorem for numbers of small height.
The addition on totally positive integers uniquely determines the totally real number field
本文证明全实数域的全正代数整数作为加法半群唯一确定该域，并给出其以不可分解元为生成元的表示，推广至半格类。
We investigate modularity of elliptic curves over a general totally real number field, establishing a finiteness result for the set non-modular $j$-invariants. By analyzing quadratic points on some modular curves, we show that all elliptic curves over certain real quadratic fields are modular.
In this article, we construct differential modular forms for compact Shimura curves over totally real fields bigger than rational of non-zero integral weights that is not classical (of order zero) generalizing the construction of Buium [8].
We describe an efficient algorithm to calculate generators of power integral bases in composites of totally real fields with imaginary quadratic fields. We show that the calculation can be reduced to solving index form equations in the original totally real fields. We illustrate our method by investigating monogenity in the infinite parametric family of imaginary quadratic extensions of the simplest quartic fields.
We construct p-adic L-functions associated with triples of finite slope p-adic families of quaternionic automorphic eigenforms over totally real fields on Shimura curves. These results generalize a previous construction, joint work with D.Barrera, performed in the ordinary setting.
In a recent paper, Freitas and Siksek proved an asypmtotic version of Fermat's Last Theorem for many totally real fields. We prove an extension of their result to generalized Fermat equations of the form $A x^p+B y^p+ C z^p=0$, where $A$, $B$, $C$ are odd integers belonging to a totally real field.
We prove that every elliptic curve defined over a totally real number field of degree 4 not containing $\sqrt{5}$ is modular. To this end, we study the quartic points on four modular curves.
We prove that all elliptic curves defined over totally real cubic fields are modular. This builds on previous work of Freitas, Le Hung and Siksek, who proved modularity of elliptic curves over real quadratic fields, as well as recent breakthroughs due to Thorne and to Kalyanswamy.
The Sylvester Theorem and the Rogers-Ramanujan Identities over Totally Real Number Fields
本文将西尔维斯特定理与罗杰斯-拉马努金恒等式推广到全实代数数域上的全正代数整数分拆，并进一步给出广义罗杰斯-拉马努金恒等式的另一版本。
We construct $p$-adic measures which interpolate the special values of reciprocals of $p$-adic $L$-functions of totally real number fields $K$ at negative integers. These measures are defined by analyzing the non-constant term of partial Eisenstein series of the Hilbert modular group of $K$.
We study and compute an infinite family of Hurwitz spaces parameterizing covers of P_C branched at four points and deduce explicit regular S_n and A_n-extensions over Q(T) with totally real fibers.
We prove a unicity result for the $L$-functions appearing in the non-commutative Iwasawa main conjecture over totally real fields. We then consider continuous representations $ρ$ of the absolute Galois group of a totally real field $F$ on adic rings in the sense of Fukaya and Kato. Using our unicity result, we show that there exists a unique sensible definition of a non-commutative $L$-function for any such $ρ$ that factors through the Galois group of a possibly infinite totally real extension. We also consider the case of CM-extensions and discuss the relation with the equivariant main conjec…
We propose and justify a generalised approach to prove the polynomial reduction of the RLWE to the PLWE problem attached to the ring of integers of a monogenic number field. We prove such equivalence in the case of the maximal totally real subextension of the 4p-th cyclotomic field, with p arbitrary prime.
One of the basic questions in number theory is to determine semi-simple l-adic representations of the absolute Galois group of a number field. In this paper, we discuss the question for two dimensional representations over a totally real number field.
As a continuation of previous work, we establish sum expressions for $p$-adic Hecke $L$-functions of totally real fields in the sense of Delbourgo, assuming a totally real analog of Heegner hypothesis. This is done by finding explicit formulas of the periods of the corresponding $p$-adic measures. As an application, we extend the Ferrero-Greenberg formula of derivatives of $p$-adic $L$-functions to this setting.
Elliptic curve over totally real fields: A Survey
本文综述了全实域上椭圆曲线模性猜想的研究进展，并介绍了理解这些成果所需的背景知识及其应用。
Let ${\{K_m\}_{m\geq 4}}$ be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial $f_m(x)=x^3-mx^2-(m+1)x-1$, where $m$ is an integer with $m\geq 4$. In this paper, we will give a class number one criterion for $K_m$.
We refine and extend previous constructions of $p$-adic $L$-functions for Rankin-Selberg convolutions on $\GL(n)\times\GL(n-1)$ for regular algebraic representations over totally real fields. We also prove an intrinsic functional equation for this $p$-adic $L$-function, which will be of interest in further study of its arithmetic properties.
Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive integers of given trace in a general totally real number field of any degree. When the field is cubic, we show that the asymptotic behavior of a weighted Diophantine sum is related to the structure of the unit group. The main term can be expressed in terms of Grössencharacter $L$-functions.
We significantly improve our results of Glas. Mat., III. Ser. 53(2018), No. 2, 229-238, reducing relative Thue inequalities to absolute ones.
On Weyl's Subconvex Bound for Cube-Free Hecke characters: Totally Real Case
本文证明了全实域上无立方级Hecke特征的Weyl型次凸界，通过显式反演Motohashi公式并利用Schwartz函数与分布不变性。
We interpolate cohomology classes attached to families of Hilbert modular forms. Using this we construct a two variable $p$-adic L-function which interpolates one variable $p$-adic L-functions.
The goal of this paper is to illustrate different approaches to understand Euler characteristics in the setting of totally real commutative and non-commutative Iwasawa theory. In addition to this, and in the spirit of Hesselholt and Mitchell, we show how these objects interact with homotopy theoretic invariants such as ($K(1)$-local) $K$-theory and Topological Cyclic homology.
Generalised quadratic forms over totally real number fields
本文引入全实数域上一类新的广义二次型，并通过数域上的Hardy-Littlewood圆法证明其能刻画有理数上任意二次曲面组的算术性质。
This version improves the old version entitled "On the modularity of elliptic curves with a residually irreducible representation". Let $E$ be an elliptic curve over an abelian totally real field $K$ unramified at 3,5, and 7. We prove that $E$ is modular.
On zeta elements and functional equations for Tate motives over totally real fields
本文研究了完全实数域上塔特动机的伊瓦萨理论，构造了插值L函数在正整数处值的zeta元素，并建立了与L函数值相关的双 dual 上同调群中的canonical元素。
The conical zeta values are a generalization of the multiple zeta values which are defined by certain multiple sums over convex cones. In this paper, we present a relation between the values of the Dedekind zeta functions for totally real fields and the conical zeta values for certain algebraic cones. More precisely, we show that the values of the partial zeta functions for totally real fields can be expressed as a rational linear combination of the conical zeta values associated with certain algebraic cones up to the square root of the discriminant.
We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse--Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over Q with End_C(A)=Z. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.
Most totally real fields do not have universal forms or Northcott property
本文证明了在配备可构造拓扑的全实域空间中，具有泛型二次型或Northcott性质的域集合是贫集，核心工具是关于二次格表示全正单位平方类数量的新定理。
We show that if F is a totally real field in which p splits completely and f is a mod p Hilbert modular form with parallel weight 2<k<p, which is totally ordinary at p and has tamely ramified Galois representation at all primes dividing p, then there is a "companion form" of parallel weight k':=p+1-k. This work generalises results of Gross and Coleman-Voloch for modular forms over Q.
The modular approach to Diophantine equations over totally real fields
本文综述了将Wiles证明Fermat大定理的模方法推广至全实数域上丢番图方程时所面临的新障碍，并调查了该领域现有的方法与结果。
Quaternionic big Heegner points over totally real fields
本研究将Howard的相容Heegner点族构造推广到全实域上的Gross曲线与志村曲线塔，同时处理全定与不定四元数代数，定义平行权希尔伯特模形式族的大Heegner点，构建对应p进L函数类似物与大Heegner类系统。
In this paper, we construct a higher rank Euler system for the multiplicative group over a totally real field by using the Iwasawa main conjecture proved by Wiles. A key ingredient of the construction is to generalize the notion of the characteristic ideal. Under certain technical assumptions, we prove that all higher Fitting ideals of a certain $p$-ramified Iwasawa module are described by analytic invariants canonically associated with Stickelberger elements.
Lifting problem for universal quadratic forms over totally real cubic number fields
本文研究泛二次型的提升问题，证明全实三次域中仅$\boldsymbol{\rm Q}(\u03b6_7+\u03b6_7^{-1})$满足条件，且不存在满足条件的双二次域，推进了代数数域上泛二次型的分类研究。
In this paper, we compute constant terms of Eisenstein series defined over a totally real field, at various cusps. In his paper published in 2003, M. Ohta computed the constant terms of Eisenstein series of weight two over the field of rational numbers, at all equivalence classes of cusps. As for Eisenstein series defined over a totally real field, S. Dasgupta, H. Darmon and R. Pollack calculated the constant terms at particular (not all) equivalence classes of cusps in 2011. We compute constant terms of Eisenstein series defined over a general totally real field at all equivalence classes of …
We completely calculate the Fitting ideal of the classical $p$-ramified Iwasawa module for any abelian extension $K/k$ of totally real fields, using the shifted Fitting ideals recently developed by the second author. This generalizes former results by the first and third authors where we had to assume that only $p$-adic places may ramify in $K/k$. One of the important ingredients is the computation of some complexes in appropriate derived categories.
We study an analogue of Serre's modularity conjecture for projective representations $\overlineρ: \operatorname{Gal}(\overline{K} / K) \rightarrow \operatorname{PGL}_2(k)$, where $K$ is a totally real number field. We prove new cases of this conjecture when $k = \mathbb{F}_5$ by using the automorphy lifting theorems over CM fields established in previous work of the authors.
Let p be a prime number, F a totally real field such that [F(mu_p): F]=2 and [F:Q] is odd. For delta \in F^times, let [delta] denote its class in F^times/F^{times p}. In this paper, we show Main Theorem. There are infinitely many classes [delta]\in F^times/F^{times p} such that the twisted affine Fermat curves W_delta: X^p+Y^p=delta have no F-rational points.
The Euler-Glaisher Theorem over Totally Real Number Fields
本文在全实代数数域上推广Euler-Glaisher定理，通过建立分拆计数恒等式并引入链分拆概念，揭示了分拆与方程解之间的对应关系。
We study generalisations to totally real fields of methods originating with Wiles and Taylor-Wiles. In view of the results of Skinner-Wiles on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some applications of our results, for example proving the modularity of all semistable elliptic curves over $\mathbb{Q}(\sqrt{2})$.
Additively indecomposable quadratic forms over totally real number fields
本文给出全实域上加法不可分解全正定二次型行列式范数的上界，并应用于确定$n$-万有二次型最小秩的上下界，同时分类了五个二次域上的加法不可分解二元二次型。
W. Luo and P. Sarnak have proved the quantum unique ergodicity property for Eisenstein series on $\rm{PSL}(2,\mathbb{Z}) \backslash H$. We extend their result to Eisenstein series on $\rm{PSL}(2,O) \backslash H^n$, where $O$ is the ring of integers in a totally real field of degree $n$ over $Q$ with narrow class number one, using the Eisenstein series considered by I. Efrat. We also give an expository treatment of the theory of Hecke operators on non-holomorphic Hilbert modular forms.
We introduce a ray class invariant $X(C)$ for a totally real field, following Shintani's work in the real quadratic case. We prove a factorization formula $X=X_1... X_n$ where each $X_i=X_i(C)$ corresponds to a real place. Although this factorization depends a priori on some choices (especially on a cone decomposition), we can show that it is actually independent of these choices. Finally, we describe the behavior of $X_i(C)$ when the signature of $C$ at a real place is changed. This last result is also interpreted into an interesting behavior of the derivative $L'(0,χ)$ of $L$-functions.
We develop the topological polylogarithm which provides an integral version of Nori's Eisenstein cohomology classes for $GL_n(\mathbf{Z})$ and yields classes with values in an Iwasawa algebra. This implies directly the integrality properties of special values of $L$-functions of totally real fields and a construction of the associated $p$-adic $L$-function. Using a result of Graf, we also apply this to prove some integrality and $p$-adic interpolation results for the Eisenstein cohomology of Hilbert modular varieties.
Fundamental domains for quaternionic S-arithmetic groups over totally real fields
本文提出一种算法计算全实域上全定四元数代数中S-算术群在Bruhat-Tits树上的基本域，并据此列出了任意全实域上亏格至多为3且可p-进一致化的Shimura曲线。
A $p$-part Birch and Swinnerton-Dyer Formula over Totally Imaginary Quadratic Extension of Totally Real Fields
本文在Iwasawa主猜想及其他假设下，证明了全实域的全虚二次扩域上，模半稳定椭圆曲线$E$的Birch和Swinnerton-Dyer公式的$p$部分变体，给出了相关$p$进单位范围内的等式关系。
In the paper we study the distribution of the discriminant $D(P)$ of polynomials $P$ from the class $\mathcal{P}_{n}(Q)$ of all integer polynomials of degree $n$ and height at most $Q$. We evaluate the asymptotic number of polynomials $P\in \mathcal{P}_{n}(Q)$ having all the roots real and satisfying the inequality $|D(P)|\le X$ as $Q\to\infty$ and $X/Q^{2n-2}\to 0$.
Rationality results are proved for the ratios of critical values of Rankin-Selberg L-functions of GL(n) x GL(n') over a totally imaginary field F, by studying rank-one Eisenstein cohomology for the group GL(N)/F, where N = n+n', generalizing the methods and results of previous work with Guenter Harder where the base field was totally real. In contrast to the totally real situation, the internal structure of the totally imaginary base field has a delicate effect on the rationality results.
On the Torsion Congruence for Zeta Functions of Totally Real Fields
本研究利用Shintani锥分解探究全实域ζ函数的特殊值，证明素数次域扩张下其特殊值的一类同余关系，该结果蕴含Ritter-Weiss的挠同余，对全实域非交换岩泽主猜想的证明具有关键意义。
Let F be a totally real number field. Using a recent geometric approach developed by Andreatta and Iovita we construct several variables p-adic families of finite slope quaternionic automorphic forms over F. It is achieved by interpolating the modular sheaves defined over some auxiliary unitary Shimura curves. Secondly, we attach p-adic L-functions to triples of ordinary p-adic families of quaternionic automorphic eigenforms. This is done by relating trilinear periods to some trilinear products over unitary Shimura curves which can be interpolated adapting the work of Liu-Zhang-Zhang to our fa…
We show that the conjectural mod $p$ local Langlands correspondence can be realised in the mod $p$ cohomology of the Lubin-Tate towers. The proof utilizes a well known conjecture of Buzzard-Diamond-Jarvis \cite[Conj. 4.9]{BDJ10}, a study of completed cohomology of the ordinary and supersingular locus of the Shimura curves for a totally real field $F$ and of mod $l(\neq p)$ local Langlands correspondence as given by Emerton-Helm \cite{EmertonHelm14}. %And then we connect the completed cohomlgy with the cohomology of Lubin-Tate towers. In the case of modular curves a similar theorem was obtained…
Let $K$ be a totally real number field and consider a Fermat-type equation $Aa^p+Bb^q=Cc^r$ over $K$. We call the triple of exponents $(p,q,r)$ the signature of the equation. We prove various results concerning the solutions to the Fermat equation with signature $(p,p,2)$ and $(p,p,3)$ using a method involving modularity, level lowering and image of inertia comparison. These generalize and extend the recent work of Işik, Kara and Ozman. For example, consider $K$ a totally real field of degree $n$ with $2 \nmid h_K^+$ and $2$ inert. Moreover, suppose there is a prime $q\geq 5$ which totally ram…
Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if for every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers O_L is free as an O_K[G]-module. Let C_p denote the cyclic group of prime order p. We show that if p >= 7 (or p=5 and extra conditions are met) and K is totally real with K/Q ramified at p, then K is not Hilbert-Speiser of type C_p.
A survey on the generalized Fermat equation of various signatures over totally real fields
本文综述了在完全实数域上使用模性方法研究不同签名的推广费马方程的最新研究成果。
C.J. Smyth was among the first to study the spectrum of the Weil height in the field of all totally real numbers, establishing both lower and upper bounds for the limit infimum of the height of all totally real integers and determining isolated values of the height. Later, Bombieri and Zannier established similar results for totally p-adic numbers and, inspired by work of Ullmo and Zhang, termed this the Bogomolov property. In this paper, we use results on equidistribution of points of low height to generalize both Bogomolov-type results to a wide variety of heights arising in arithmetic dynam…
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