Baskar Balasubramanyam, Eknath Ghate, Ravitheja Vangala
We establish a rationality result for the twisted Asai L-values attached to a Bianchi cusp form and construct distributions interpolating these L-values. Using the method of abstract Kummer congruences, we then outline the main steps needed to show that these distributions come from a measure.
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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…
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。
We give a construction of $p$-adic Asai $L$-functions for cohomological cuspidal automorphic representations of ${\rm GL}_2$ over CM fields. If the base field is imaginary quadratic, Loeffler-Williams recently constructed the $p$-adic Asai $L$-functions. We generalize their construction to the case that the base fields are general CM fields.
P-adic Asai L-functions for quadratic Hilbert eigenforms
本文提出利用更高Hida理论构造实二次域上GL2自守表示的p进AsaiL函数。
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…
In this paper, we present the outcome of vast computer calculations, locating several of the very rare instances of level one cuspidal Bianchi modular forms that are not lifts of elliptic modular forms.
Let $F$ be a totally real number field and $E/F$ a totally imaginary quadratic extension of $F$. Let $Π$ be a cohomological, conjugate self-dual cuspidal automorphic representation of $GL_n(\mathbb A_E)$. Under a certain non-vanishing condition we relate the residue and the value of the Asai $L$-functions at $s=1$ with rational structures obtained from the cohomologies in top and bottom degrees via the Whittaker coefficient map. This generalizes a result in Eric Urban's thesis when $n = 2$, as well as a result of the first two named authors, both in the case $F = \mathbb Q$.
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。
Poles of p-adic Asai L-functions and distinguished representations
本文通过p-进AsaiL函数的极点条件，研究实二次域上GL(2)自守表示的区分性质，提供复情况的p-进对应定理。
On $p$-adic adjoint $L$-functions for Bianchi cuspforms: the $p$-split case
本文通过 Bianchi 三维簇过收敛上同调上的 Hecke 等变配对，在 $p$ 分裂情形构造 Bianchi 尖点形式的 $p$-进伴随 $L$-函数，刻画特征簇分歧轨迹并证明其在特定点非零。
Bianchi modular forms are automorphic forms over an imaginary quadratic field, associated to a Bianchi group. Even though modern studies of Bianchi modular forms go back to the mid 1960's, most of the fundamental problems surrounding their theory are still wide open. Only for certain types of Bianchi modular forms, which we will call non-genuine, it is possible at present to develop dimension formulas: They are (twists of) those forms which arise from elliptic cuspidal modular forms via the Langlands Base-Change procedure, or arise from a quadratic extension of the imaginary quadratic field vi…
We prove three main results: all Langlands-Shahidi automorphic $L$-functions over function fields are rational; after twists by highly ramified characters our automorphic $L$-functions become polynomials; and, if $π$ is a globally generic cuspidal automorphic representation of a split classical group or a unitary group ${\bf G}_n$ and $τ$ a cuspidal (unitary) automorphic representation of a general linear group, then $L(s,π\times τ)$ is holomorphic for $\Re(s) > 1$ and has at most a simple pole at $s=1$. We also prove the holomorphy and non-vanishing of automorphic exterior square, symmetric s…
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.
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。
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。
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。
On the Second Moment of $ L (1/2, \mathrm{As}(f) \times ϕ)$
本文研究了实二次域上 Asai 抬升与 Hecke-Maass 自带形式卷积中心 $L$-值的二阶矩，并建立了相关的大筛不等式。
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。
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…
Complex continued fractions, Kleinian and extremal theory for cusp excursions
本文研究了五个复二次整数欧氏环上的复连分数算法，证明其最大数字服从Fréchet分布，并据此求出相关Bianchi轨形上尖点远足的极限分布。
We prove explicit rationality-results for Asai- $L$-functions, $L^S(s,Π',{\rm As}^\pm)$, and Rankin-Selberg $L$-functions, $L^S(s,Π\timesΠ')$, over arbitrary CM-fields $F$, relating critical values to explicit powers of $(2πi)$. Besides determining the contribution of archimedean zeta-integrals to our formulas as concrete powers of $(2πi)$, it is one of the advantages of our approach, that it applies to very general non-cuspidal isobaric automorphic representations $Π'$ of ${\rm GL}_n(\mathbb A_F)$. As an application, this enables us to establish a certain algebraic version of the Gan--Gross--…
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。
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…
Betti-Whittaker periods under duality: variations and applications
本文推广了Betti-Whittaker周期在对偶性下的行为结果，应用于三重积L函数和扭曲Asai L函数的特殊值有理性结果，并给出了正交群L函数特殊值的新证明。
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…
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。
In earlier work, the first named author generalized the construction of Darmon-style $\mathcal{L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial coefficient system and Steinberg at a fixed prime. In this paper, assuming that the Archimedean component of the group has discrete series we show that these automorphic $\mathcal{L}$-invariants can be computed in terms of derivatives of Hecke-eigenvalues in $p$-adic families. Our proof is novel even in the case of modular forms, which was established by Bert…
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