Weight Reduction for Mod l Bianchi Modular Forms

Bianchi 模形式的 $p$-进 $L$-函数的函数方程

Functional equation of the $p$-adic $L$-function of Bianchi modular forms

本文研究类数为 1 的虚二次域上 Bianchi 模形式的 $p$-进 $L$-函数，证明小斜率 $p$-稳定化情形的函数方程，并借助 $p$-进族推广至 $\Sigma$-光滑基变换形式。

Bianchi 模符号与 $p$-进 $L$-函数

Bianchi modular symbols and $p$-adic $L$-functions

研究通过特殊 Bianchi 模符号刻画第一同调群并构造 Bianchi 模形式的 $p$-进 $L$-函数，证明正比例普通素数下相关 $\mu$-不变量消失及积分 $L$-值剩余非零。

非尖点 Bianchi 模形式与 Katz $p$-进 $L$-函数

Non-cuspidal Bianchi modular forms and Katz $p$-adic $L$-functions

本文引入 $C$-尖点性质与部分 Bianchi 模符号，构造非尖点 Bianchi 模形式的 $p$-进 $L$-函数，并在 $p$ 分裂情形证明基变换形式的相应函数分解为两个 Katz $p$-进 $L$-函数之积。

A finiteness result for $p$-adic families of Bianchi modular forms

We study $p$-adic families of cohomological automorphic forms for ${\mathrm{GL}}(2)$ over imaginary quadratic fields and prove that families interpolating a Zariski-dense set of classical cuspidal automorphic forms only occur under very restrictive conditions. We show how to computationally determine when this is not the case and establish concrete examples of families interpolating only finitely many Bianchi modular forms.

与Bianchi模形式相关的Stark-Heegner循环的合理性——基变换情形

The rationality of Stark-Heegner cycles attached to Bianchi modular forms -- The base-change scenario

本文证明了当Bianchi本征形式为经典椭圆尖点形式的基变换时，关于其Stark-Heegner循环为全局Bloch-Kato Selmer类限制的猜想成立。

Theta Lifts of Bianchi Modular Forms and Applications to Paramodularity

We explain how the work of Johnson-Leung and Roberts on lifting Hilbert modular forms for real quadratic fields to Siegel modular forms can be adapted to imaginary quadratic fields. For this we use archimedean results from Harris, Soudry, Taylor and replace the global arguments of Roberts by the non-vanishing result of Takeda. As an application of our lifting result, we exhibit an abelian surface $B$ defined over $\mathbb{Q}$, which is not restriction of scalars of an elliptic curve and satisfies the Brumer-Kramer Paramodularity Conjecture.

P-adic Asai L-functions of Bianchi modular forms

The Asai (or twisted tensor) $L$-function of a Bianchi modular form $Ψ$ is the $L$-function attached to the tensor induction to $\mathbb{Q}$ of its associated Galois representation. In this paper, when $Ψ$ is ordinary at $p$ we construct a $p$-adic analogue of this $L$-function: that is, a $p$-adic measure on $\mathbb{Z}_p^\times$ that interpolates the critical values of the Asai $L$-function twisted by Dirichlet characters of $p$-power conductor. The construction uses techniques analogous to those used by Lei, Zerbes and the first author in order to construct an Euler system attached to the A…

Bianchi 模形式与周期的有理性

Bianchi Modular Forms and the Rationality of Periods

本文通过显式 Eichler-Shimura-Harder 同构建立 Bianchi 周期的 Manin 有理性定理，从而在欧几里得虚二次域情形下给出 Hida 结果的新简短证明，并利用 Karabulut 的周期多项式空间显式描述 Hecke 算子作用。

提升以太 Bianchi 模形式

Lifting ethereal Bianchi modular forms

本文研究模 p Bianchi 模形式的提升问题，开发高效算法验证特征值同余，并借助模性定理证明某些情形下提升存在，实验表明多数模形式可提升。

Bianchi 模形式的 Ramanujan 与 Sato-Tate 猜想

The Ramanujan and Sato-Tate Conjectures for Bianchi modular forms

本文证明了 Bianchi 模形式及更一般的平行权正则代数尖点自守表示的 Ramanujan 与 Sato-Tate 猜想，其核心方法是通过一个新的潜在自同构定理处理二维相容 Galois 表示系的对称幂。

Bianchi 模群的同调和上同调群的 Eisenstein 部分的两个结果

Two results on Eisenstein part of homology and cohomology groups of Bianchi modular groups

本文明确构造了 Bianchi 模群同调群的 Eisenstein 环，并研究上同调部分，进而得到尖点上同调群的渐近维数公式。

Examples of genuine QM abelian surfaces which are modular

Let $K$ be an imaginary quadratic field. Modular forms for GL(2) over $K$ are known as Bianchi modular forms. Standard modularity conjectures assert that every weight 2 rational Bianchi newform has either an associated elliptic curve over $K$ or an associated abelian surface with quaternionic multiplication over $K$. We give explicit evidence in the way of examples to support this conjecture in the latter case. Furthermore, the quaternionic surfaces given correspond to genuine Bianchi newforms, which answers a question posed by J. Cremona as to whether this phenomenon can happen.

$\mathcal{L}$-invariants and exceptional zeros of Bianchi modular forms

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F. In this paper, we prove an exceptional zero conjecture in the case where f is new at a prime above p. More precisely, for each prime $\mathfrak{p}$ of F above p we prove the existence of an L-invariant $\mathcal{L}_{\mathfrak{p}}$, depending only on $\mathfrak{p}$ and f, such that when the p-adic L-function of f has an exceptional zero at $\mathfrak{p}$, its derivative can be related to the classical L-value multiplied by $\mathcal{L}_{\mathfrak{p}}$. The proof uses cohomological method…

Stark-Heegner cycles attached to Bianchi modular forms

Let f be a Bianchi modular form, that is, an automorphic form for GL(2) over an imaginary quadratic field F, and let P be a prime of F at which f is new. Let K be a quadratic extension of F, and L(f/K,s) the L-function of the base-change of f to K. Under certain hypotheses on f and K, the functional equation of L(f/K,s) ensures that it vanishes at the central point. The Bloch--Kato conjecture predicts that this should force the existence of non-trivial classes in an appropriate global Selmer group attached to f and K. In this paper, we use the theory of double integrals developed by Barrera Sa…

Bianchi模形式在虚二次域上的研究：任意类数的情况

Bianchi Modular Forms over Imaginary Quadratic Fields with arbitrary class group

本文研究了虚二次域上Bianchi模形式的计算方法，针对任意类数的情况，通过实例证明了椭圆曲线的模性，并将结果整理至LMFDB数据库。

关于Bianchi模形式在非普通素数处的p-进AsaiL函数及其分解为有界p-进L函数

On $p$-adic Asai $L$-functions of Bianchi modular forms at non-ordinary primes and their decomposition into bounded $p$-adic $L$-functions

本文研究了Bianchi模形式在非普通素数处的p-进AsaiL函数的构造及其分解为有界p-进L函数的方法。

Bianchi 模形式重数的尖锐界

Sharp bounds for multiplicities of Bianchi modular forms

本文证明了非全实数域上 $\mathrm{SL}_2$ 的固定级、增长权上同调自守形式空间维数的一次节省界，建立了尖 Bianchi 模形式增长的尖锐界，并通过代数微局部化将问题转化为完备泛包络代数上的表示论问题予以解决。

Families of Bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism

Let $K$ be an imaginary quadratic field. In this article, we study the eigenvariety for $\mathrm{GL}_2/K$, proving an étaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this; let $f$ be a $p$-stabilised newform of weight $k \geq 2$ without CM by $K$. Suppose $f$ has finite slope at $p$ and its base-change $f_{/K}$ to $K$ is $p$-regular. Then: (1) We construct a two-variable $p$-adic $L$-function attached to $f_{/K}$ under assumptions on $f$ that conjecturally always hold, in particular…

Bianchi and Hilbert-Blumenthal quaternionic orbifolds

In a series of papers, published in Mathematische Annalen, Bianchi and Blumenthal introduced the notions of Bianchi orbifolds and Hilbert-Blumnethal surfaces as generalizations of modular curves associated to quadratic fields. In this paper, in the same spirit, and following a similar line of reasoning, we introduce the concept of Bianchi and Hilbert-Blumenthal quaternionic orbifolds as generalizations of the Lipschitz and Hurwitz quaternionic modular orbifolds defined recently by Díaz, Vlacci and the first author. In particular, we describe the cusp cross-sections of the Hilbert-Blumenthal qu…

带有Bianchi模形式的有符号p-adic L-函数

Signed $p$-adic $L$-functions of Bianchi modular forms

本文研究了Bianchi模形式的有符号p-adic L-函数，通过p-稳定化形式和局部解析分布，将双变量幂级数分解为有界系数的线性组合。

Modular symbols for Q-rank one groups and Voronoi reduction

We present an algorithm to compute the action of the Hecke operators on the top dimensional integral cohomology of certain torsion-free arithmetic subgroups of algebraic groups of Q-rank one. This generalizes the modular symbol algorithm to a setting including Bianchi groups and Hilbert modular groups. In addition, we generalize some results of Voronoi for real positive-definite quadratic forms to self-adjoint homogeneous cones of arbitrary Q-rank.

计算特征簇的切空间

Computing Tangent Spaces to Eigenvarieties

本文提出一种局部有效算法，计算Bianchi模形式在p进族中关于权空间的导数，并数值验证特征簇光滑性及族的方向猜想，同时应用于椭圆模形式的$\mathcal{L}$-不变量研究。

Hecke算子、Hecke特征系统与数字段上的形式模形式

Hecke operators, Hecke Eigensystems, and Formal Modular Forms over Number Fields

本文提出了一种数字段上形式模形式的显式理论，通过模点定义Hecke算子和特征系统，并展示了如何利用主Hecke算子计算自守形式。

Integral period relations and congruences

Under relatively mild and natural conditions, we establish an integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the {\it congruence number} controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch-Kato conjecture for adjoint modular Galois representations twisted by an even quadratic character. In the odd case, we formulate a conjecture linking the degree two topological period attached to the base change Bianchi modu…

On the Bloch-Kato conjecture for the Asai L-function

Following Ribet's seminal 1976 paper there have been many results employing congruences between stable cuspforms and lifted forms to construct non-split extensions of Galois representations. We show how this strategy can be extended to construct elements in the Bloch-Kato Selmer groups of +/--Asai (or tensor induction) representations associated to Bianchi modular forms. We prove, in particular, how the Galois representation associated to a suitable low weight Siegel modular form produces elements in the Selmer group for exactly the Asai representation (+ or -) that is critical in the sense of…

Arithmetic of p-irregular modular forms: families and p-adic L-functions

Let $f_{\mathrm{new}}$ be a classical newform of weight $\geq 2$ and prime to $p$ level. We study the arithmetic of $f_{\mathrm{new}}$ and its unique $p$-stabilisation $f$ when $f_{\mathrm{new}}$ is $p$-irregular, that is, when its Hecke polynomial at $p$ admits a single repeated root. In particular, we study $p$-adic weight families through $f$ and its base-change to an imaginary quadratic field $F$ where $p$ splits, and prove that the respective eigencurves are both Gorenstein at $f$. We use this to construct a two-variable $p$-adic $L$-function over a Coleman family through $f$, and a three…

On Elliptic Curves of prime power conductor over imaginary quadratic fields with class number one

The main result of this paper is to extend from $\Q$ to each of the nine imaginary quadratic fields of class number one a result of Serre (1987) and Mestre-Oesterlé (1989), namely that if $E$ is an elliptic curve of prime conductor then either $E$ or a $2$-, $3$- or $5$-isogenous curve has prime discriminant. For four of the nine fields, the theorem holds with no change, while for the remaining five fields the discriminant of a curve with prime conductor is either prime or the square of a prime. The proof is conditional in two ways: first that the curves are modular, so are associated to suita…

Motivic action for Siegel modular forms

We study the coherent cohomology of automorphic sheaves corresponding to Siegel modular forms $f$ of low weight on ${\rm GSp}(4)$ Shimura varieties. Inspired by the work of Prasanna--Venkatesh on singular cohomology of locally symmetric spaces, we propose a conjecture that explains all the contributions of a Hecke eigensystem to coherent cohomology in terms of the action of a motivic cohomology group. Under some technical conditions, we prove that our conjecture is equivalent to Beilinson's conjecture for the adjoint $L$-function of $f$. We also prove some unconditional results in special case…

具有$\u2124_2$对称性的Calabi-Yau流形的通量真空与模性

Flux Vacua and Modularity for $\mathbb{Z}_2$ Symmetric Calabi-Yau Manifolds

本研究在具有$\u2124_2$对称性的多参数Calabi-Yau流形的IIB紧致化中找到超对称通量真空连续族，通过扩展方法计算局部zeta函数，验证其分子二次因子对应权2模形式，支持通量模性猜想，并拓展至代数数域参数与相关模形式的研究。