Let $K/F$ be an unramified quadratic extension of a non-Archimedean local field. In a previous work [1], we proved a formula for the intersection number on Lubin–Tate spaces. The main result of this article is an algorithm for computation of this formula in certain special cases. As an application, we prove the linear Arithmetic Fundamental Lemma for $ \operatorname {{\mathrm {GL}}}_4$ with the unit element in the spherical Hecke Algebra.

Let $E/\mathbb {Q}$ be a number field of degree $n$. We show that if $\operatorname {Reg}(E)\ll _n |\!\operatorname{Disc}(E)|^{1/4}$ then the fraction of class group characters for which the Hecke $L$-function does not vanish at the central point is $\gg _{n,\varepsilon } |\!\operatorname{Disc}(E)|^{-1/4-\varepsilon }$. The proof is an interplay between almost equidistribution of Eisenstein periods over the toral packet in $\mathbf {PGL}_n(\mathbb {Z})\backslash \mathbf {PGL}_n(\mathbb {R})$ associated to the maximal order of $E$, and the escape of mass of the torus orbit associated to the trivial ideal class.

We improve upon the local bound in the depth aspect for sup-norms of newforms on $D^\times$, where $D$ is an indefinite quaternion division algebra over ${\mathbb {Q}}$. Our sup-norm bound implies a depth-aspect subconvexity bound for $L(1/2, f \times \theta _\chi )$, where $f$ is a (varying) newform on $D^\times$ of level $p^n$, and $\theta _\chi$ is an (essentially fixed) automorphic form on $\textrm {GL}_2$ obtained as the theta lift of a Hecke character $\chi$ on a quadratic field. For the proof, we augment the amplification method with a novel filtration argument and a recent counting result proved by the second-named author to reduce to showing strong quantitative decay of matrix coefficients of local newvectors along compact subsets, which we establish via $p$-adic stationary phase analysis. Furthermore, we prove a general upper bound in the level aspect for sup-norms of automorphic forms belonging to any family whose associated matrix coefficients have such a decay property.

We show that if the zeros of an automorphic $L$-function are weighted by the central value of the $L$-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the $L$-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted $L$-values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

Since Rob Pollack and Glenn Stevens used overconvergent modular symbols to construct $p$-adic $L$-functions for non-critical slope rational modular forms, the theory has been extended to construct $p$-adic $L$-functions for non-critical slope automorphic forms over totally real and imaginary quadratic fields by the first and second authors, respectively. In this paper, we give an analogous construction over a general number field. In particular, we start by proving a control theorem stating that the specialisation map from overconvergent to classical modular symbols is an isomorphism on the small slope subspace. We then show that if one takes the modular symbol attached to a small slope cuspidal eigenform, then one can construct a ray class distribution from the corresponding overconvergent symbol, which moreover interpolates critical values of the $L$-function of the eigenform. We prove that this distribution is independent of the choices made in its construction. We define the $p$-adic $L$-function of the eigenform to be this distribution.

Katz and Sarnak predicted that the one level density of the zeros of a family of L-functions would fall into one of five categories. In this paper, we show that the one level density for L-functions attached to cubic Galois number fields falls into the category associated with unitary matrices.

Inspired by a construction by Bump, Friedberg, and Ginzburg of a two-variable integral representation on $\text{GS}{{\text{p}}_{4}}$ for the product of the standard and spin $L$ -functions, we give two similar multivariate integral representations. The first is a three-variable Rankin-Selberg integral for cusp forms on $\text{PG}{{\text{L}}_{4}}$ representing the product of the $L$ -functions attached to the three fundamental representations of the Langlands $L$ -group $\text{S}{{\text{L}}_{\text{4}}}\left( \text{C} \right)$ . The second integral, which is closely related, is a two-variable Rankin-Selberg integral for cusp forms on $\text{PGU}\left( 2,\,2 \right)$ representing the product of the degree $8$ standard $L$ -function and the degree $6$ exterior square $L$ -function.

This paper concerns the problem of algebraic differential independence of the gamma function and ${\mathcal{L}}$ -functions in the extended Selberg class. We prove that the two kinds of functions cannot satisfy a class of algebraic differential equations with functional coefficients that are linked to the zeros of the ${\mathcal{L}}$ -function in a domain $D:=\{z:0<\text{Re}\,z<\unicode[STIX]{x1D70E}_{0}\}$ for a positive constant $\unicode[STIX]{x1D70E}_{0}$ .

We give a simple proof and strengthening of a uniqueness theorem for functions in the extended Selberg class.

We generalise a result of Hilbert which asserts that the Riemann zeta-function ${\it\zeta}(s)$ is hypertranscendental over $\mathbb{C}(s)$. Let ${\it\pi}$ be any irreducible cuspidal automorphic representation of $\text{GL}_{m}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We establish a certain type of functional difference–differential independence for the associated $L$-function $L(s,{\it\pi})$. This result implies algebraic difference–differential independence of $L(s,{\it\pi})$ over $\mathbb{C}(s)$ (and more strongly, over a certain field ${\mathcal{F}}_{s}$ which contains $\mathbb{C}(s)$). In particular, $L(s,{\it\pi})$ is hypertranscendental over $\mathbb{C}(s)$. We also extend a result of Ostrowski on the hypertranscendence of ordinary Dirichlet series.

Let ${\mathcal{K}}$ be an imaginary quadratic field. Let ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$ be irreducible generic cohomological automorphic representation of $\text{GL}(n)/{\mathcal{K}}$ and $\text{GL}(n-1)/{\mathcal{K}}$ , respectively. Each of them can be given two natural rational structures over number fields. One is defined by the rational structure on topological cohomology, and the other is given in terms of the Whittaker model. The ratio between these rational structures is called a Whittaker period. An argument presented by Mahnkopf and Raghuram shows that, at least if ${\rm\Pi}$ is cuspidal and the weights of ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$ are in a standard relative position, the critical values of the Rankin–Selberg product $L(s,{\rm\Pi}\times {\rm\Pi}^{\prime })$ are essentially algebraic multiples of the product of the Whittaker periods of ${\rm\Pi}$ and ${\rm\Pi}^{\prime }$ . We show that, under certain regularity and polarization hypotheses, the Whittaker period of a cuspidal ${\rm\Pi}$ can be given a motivic interpretation, and can also be related to a critical value of the adjoint $L$ -function of related automorphic representations of unitary groups. The resulting expressions for critical values of the Rankin–Selberg and adjoint $L$ -functions are compatible with Deligne’s conjecture.

We obtain uniqueness theorems for L-functions in the extended Selberg class when the functions share values in a finite set and share values weighted by multiplicities.

We study analytic properties function $m\left( z,\,E \right)$ , which is defined on the upper half-plane as an integral from the shifted $L$ -function of an elliptic curve. We show that $m\left( z,\,E \right)$ analytically continues to a meromorphic function on the whole complex plane and satisfies certain functional equation. Moreover, we give explicit formula for $m\left( z,\,E \right)$ in the strip $\left| \Im z \right|\,<\,2\pi$ .

In this work, we s uggest a defnition for the category of mixed motives generated by the motive h1 (E) for E an elliptic curve without complex multiplication. We then compute the cohomology of this category. Modulo a strengthening of the Beilinson-Soulé conjecture, we show that the cohomology of our category agrees with the expected motivic cohomology groups. Finally for each pure motive (Symnh1 (E)) (–1) we construct families of nontrivial motives whose highest associated weight graded piece is (Symnh1 (E)) (–1).

Let q≥2 and N≥1 be integers. W. Zhang recently proved that for any fixed ε>0 and qε≤N≤q1/2−ε, where the sum is taken over all nonprincipal characters χ modulo q, L(1,χ) denotes the L-functions corresponding to χ, and αq=qo(1) is some explicit function of q. Here we improve this result and show that the same asymptotic formula holds in the essentially full range qε≤N≤q1−ε.

We establish the Langlands functoriality conjecture for the transfer from the generic spectrum of GSp(4) to GL(4) and give a criterion for the cuspidality of its image. We apply this to prove results toward the generalized Ramanujan conjecture for generic representations of GSp(4).

We construct families of hyperelliptic curves over ${\mathbb Q}$ of arbitrary genus g with (at least) g integral elements in K2. We also verify the Beilinson conjectures about K2 numerically for several curves with g = 2, 3, 4 and 5. The first few sections of the paper also provide an elementary introduction to the Beilinson conjectures for K2 of curves.

We prove that holonomic arithmetical ${\mathcal{D}}$-modules over curves have finite fibers. We also define L-functions associated with arithmetical ${\mathcal{D}}$-modules and, when the scheme is a curve, we show a cohomological formula. Furthermore, we prove that F-isocrystals over curves are holonomic.

In this work we study the moments of central values of Hecke L-functions associated with cubic characters, and establish quantitative non-vanishing result for the L-values.

We construct a finite-dimensional vector space of functions of two complex variables attached to a smooth algebraic curve C over a finite field $\mathbb{F}_q$, q odd, and a level. These functions collect the analytic information about the cohomology of the curve and its quadratic twists that is encoded in the corresponding L-functions; they are double Dirichlet series in two independent complex variables s and w. We prove that these series satisfy a finite, non-abelian group of functional equations in the two complex variables (s, w) and are rational functions in q-s and q-w with a specified denominator. The group is D6, the dihedral group of order 12.