We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for $\mathit{GSp}_{4}/\mathbb{Q}$ in various aspects. A main tool is Arthur’s invariant trace formula. While Shin [Automorphic Plancherel density theorem, Israel J. Math.192(1) (2012), 83–120] and Shin–Templier [Sato–Tate theorem for families and low-lying zeros of automorphic $L$-functions, Invent. Math.203(1) (2016) 1–177] used Euler–Poincaré functions at infinity in the formula, we use a pseudo-coefficient of a holomorphic discrete series to extract holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms $A,B_{1}$ in Theorem 1.1 which have not been studied and a mysterious second term $B_{2}$ also appears in the second main term coming from the semisimple elements. Furthermore our explicit study enables us to treat more general aspects in the weight. We also give several applications including the vertical Sato–Tate theorem, the unboundedness of Hecke fields and low-lying zeros for degree 4 spinor $L$-functions and degree 5 standard $L$-functions of holomorphic Siegel cusp forms.

In a family of S d+1-fields (d = 2, 3, 4), we obtain the conjectured upper and lower bounds of the residues of Dedekind zeta functions except for a density zero set. For S 5-fields, we need to assume the strong Artin conjecture. We also show that there exists an infinite family of number fields with the upper and lower bounds, resp.

Let $\mathbf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let ${\rm\Gamma}=\mathbf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\geqslant 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to ${\rm\Gamma}$ from a Hecke cusp form in $S_{2k-8}(\text{SL}_{2}(\mathbb{Z}))$. We follow Ikeda’s idea of using Siegel’s Eisenstein series, their Fourier–Jacobi expansions, and the compatible family of Eisenstein series.

We construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group ${{A}_{4}},\,{{S}_{4}}$ , and ${{S}_{5}}$ . We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying the zero density result of Kowalski–Michel, we choose subfamilies of $L$ -functions that are zero-free close to 1. For these subfamilies, the $L$ -functions have the extremal value at $s\,=\,1$ , and by the class number formula, we obtain the extreme class numbers.

Let $K$ be a number field of degree $n$, and let $d_K$ be its discriminant. Then, under the Artin conjecture, the generalized Riemann hypothesis and a certain zero-density hypothesis, we show that the upper and lower bounds of the logarithmic derivatives of Artin $L$-functions attached to $K$ at $s=1$ are $\log \log |d_K|$ and $-(n-1) \log \log |d_K|$, respectively. Unconditionally, we show that there are infinitely many number fields with the extreme logarithmic derivatives; they are families of number fields whose Galois closures have the Galois group $C_n$ for $n=2,3,4,6$, $D_n$ for $n=3,4,5$, $S_4$ or $A_5$.

We assess the conditions under which majority status generates benefits for incumbent legislators and how these benefits are distributed among members of the majority party. We argue that majority status is valuable only in procedurally partisan chambers; that is, when the majority party monopolizes chamber leadership positions and control of the legislative agenda. Contrary to the existing literature, we also posit that these rewards should be distributed broadly across the majority party. To test our expectations, we utilize 10 recent transitions in the partisan control of U.S. state legislatures and data on campaign contributions. Consistent with our expectations, majority status is valuable, but only in procedurally partisan chambers. Furthermore, the premium in campaign contributions enjoyed by the majority party is primarily distributed to backbenchers, although top party leaders also benefit. These results provide important insights into the distribution of power and influence in U.S. state legislatures.

In this paper we make explicit all $L$ -functions in the Langlands–Shahidi method which appear as normalizing factors of global intertwining operators in the constant term of the Eisenstein series. We prove, in many cases, the conjecture of Shahidi regarding the holomorphy of the local $L$ -functions. We also prove that the normalized local intertwining operators are holomorphic and non-vaninishing for $\operatorname{Re}\left( s \right)\,\ge \,1/2$ in many cases. These local results are essential in global applications such as Langlands functoriality, residual spectrum and determining poles of automorphic $L$ -functions.

In this paper we use Langlands-Shahidi method and the result of Langlands which says that non self-conjugate maximal parabolic subgroups do not contribute to the residual spectrum, to prove the holomorphy of several completed automorphic $L$ -functions on the whole complex plane which appear in constant terms of the Eisenstein series. They include the exterior square $L$ -functions of $\text{G}{{\text{L}}_{\text{n}}},\,n$ odd, the Rankin-Selberg $L$ -functions of $\text{G}{{\text{L}}_{n}}\times \,\text{G}{{\text{L}}_{m}},\,n\,\ne \,m$ , and $L$ -functions $L\left( s,\,\sigma ,\,r \right)$ , where $\sigma $ is a generic cuspidal representation of $\text{S}{{\text{O}}_{10}}$ and $r$ is the half-spin representation of GSpin $\left( 10,\,\mathbb{C} \right)$ . The main part is proving the holomorphy and non-vanishing of the local normalized intertwining operators by reducing them to natural conjectures in harmonic analysis, such as standard module conjecture.

We completely determine the residual spectrum of the split exceptional group of type G 2, thus completing the work of Langlands and Moeglin-Waldspurger on the part of the residual spectrum attached to the trivial character of the maximal torus. We also give the Arthur parameters for the residual spectrum coming from Borel subgroups. The interpretation in terms of Arthur parameters explains the “bizarre” multiplicity condition in Moeglin-Waldspurger's work. It is related to the fact that the component group of the Arthur parameter is non-abelian.