We construct the $p$-adic standard $L$-functions for ordinary families of Hecke eigensystems of the symplectic group $\operatorname{Sp}(2n)_{/\mathbb{Q}}$ using the doubling method. We explain a clear and simple strategy of choosing the local sections for the Siegel Eisenstein series on the doubling group $\operatorname{Sp}(4n)_{/\mathbb{Q}}$, which guarantees the nonvanishing of local zeta integrals and allows us to $p$-adically interpolate the restrictions of the Siegel Eisenstein series to $\operatorname{Sp}(2n)_{/\mathbb{Q}}\times \operatorname{Sp}(2n)_{/\mathbb{Q}}$.

We describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

A theorem of Gekeler compares the number of non-isomorphic automorphic representations associated with the space of cusp forms of weight $k$ on $\unicode[STIX]{x0393}_{0}(N)$ to a simpler function of $k$ and $N$, showing that the two are equal whenever $N$ is squarefree. We prove the converse of this theorem (with one small exception), thus providing a characterization of squarefree integers. We also establish a similar characterization of prime numbers in terms of the number of Hecke newforms of weight $k$ on $\unicode[STIX]{x0393}_{0}(N)$.

It follows that a hypothetical fast algorithm for computing the number of such automorphic representations for even a single weight $k$ would yield a fast test for whether $N$ is squarefree. We also show how to obtain bounds on the possible square divisors of a number $N$ that has been found not to be squarefree via this test, and we show how to probabilistically obtain the complete factorization of the squarefull part of $N$ from the number of such automorphic representations for two different weights. If in addition we have the number of such Hecke newforms for even a single weight $k$, then we show how to probabilistically factor $N$ entirely. All of these computations could be performed quickly in practice, given the number(s) of automorphic representations and modular forms as input.

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).

In this article we explore the interplay between two generalizations of the Whittaker model, namely the Klyachko models and the degenerate Whittaker models, and two functorial constructions, namely base change and automorphic induction, for the class of unitarizable and ladder representations of the general linear groups.

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.

Darmon, Lauder, and Rotger conjectured that the relative tangent space of an eigencurve at a classical, ordinary, irregular weight one point is of dimension two. This space can be identified with the space of normalized overconvergent generalized eigenforms, whose Fourier coefficients can be conjecturally described explicitly in terms of $p$-adic logarithms of algebraic numbers. This article presents the proof of this conjecture in the case where the weight one point is the intersection of two Hida families of Hecke theta series.

We answer a challenge posed in Booker [$L$-functions as distributions. Math. Ann. 363(1–2) (2015), 423–454, §1.3] by proving a version of Weil’s converse theorem [Über die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 168 (1967), 149–156] that assumes a functional equation for character twists but allows their root numbers to vary arbitrarily.

We generalize a method of Conrey and Ghosh [Simple zeros of the Ramanujan $\unicode[STIX]{x1D70F}$-Dirichlet series. Invent. Math. 94(2) (1988), 403–419] to prove quantitative estimates for simple zeros of modular form $L$-functions of arbitrary conductor.

We generalize the work of Sarnak and Tsimerman to twisted sums of Kloosterman sums and thus give evidence towards the twisted Linnik–Selberg conjecture.

In this paper various analytic techniques are combined in order to study the average of a product of a Hecke $L$-function and a symmetric square $L$-function at the central point in the weight aspect. The evaluation of the second main term relies on the theory of Maaß forms of half-integral weight and the Rankin–Selberg method. The error terms are bounded using the Liouville–Green approximation.

In this article we obtain an explicit formula for certain Rankin–Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler–Zagier–Ibukiyama correspondence. The integral weight version of the main theorem was obtained by Katsurada and Kawamura. The result of the integral weight case is a product of an $L$-function and Riemann zeta functions, while the half-integral weight case is an infinite summation over negative fundamental discriminants with certain infinite products. To calculate an explicit formula for such Rankin–Selberg type Dirichlet series, we use a generalized Maass relation and adjoint maps of index-shift maps of Jacobi forms.

We record $\binom{42}{2}+\binom{23}{2}+\binom{13}{2}=1192$ functional identities that, apart from being amazingly amusing in themselves, find application in the derivation of Ramanujan-type formulas for $1/\unicode[STIX]{x1D70B}$ and in the computation of mathematical constants.

The standard twist $F(s,\unicode[STIX]{x1D6FC})$ of $L$-functions $F(s)$ in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for example the functional equation. Here we deal with a special case, where $F(s)$ satisfies a functional equation with the same $\unicode[STIX]{x1D6E4}$-factor of the $L$-functions associated with the cusp forms of half-integral weight; for simplicity we present our results directly for such $L$-functions. We show that the standard twist $F(s,\unicode[STIX]{x1D6FC})$ satisfies a functional equation reflecting $s$ to $1-s$, whose shape is not far from a Riemann-type functional equation of degree 2 and may be regarded as a degree 2 analog of the Hurwitz–Lerch functional equation. We also deduce some results on the growth on vertical strips and on the distribution of zeros of $F(s,\unicode[STIX]{x1D6FC})$.

In 1991, Andrews and Hickerson established a new Bailey pair and combined it with the constant term method to prove some results related to sixth-order mock theta functions. In this paper, we study how this pair gives rise to new mock theta functions in terms of Appell–Lerch sums. Furthermore, we establish some relations between these new mock theta functions and some second-order mock theta functions. Meanwhile, we obtain an identity between a second-order and a sixth-order mock theta functions. In addition, we provide the mock theta conjectures for these new mock theta functions. Finally, we discuss the dual nature between the new mock theta functions and partial theta functions.

The generalized Gauss circle problem concerns the lattice point discrepancy of large spheres. We study the Dirichlet series associated to $P_{k}(n)^{2}$, where $P_{k}(n)$ is the discrepancy between the volume of the $k$-dimensional sphere of radius $\sqrt{n}$ and the number of integer lattice points contained in that sphere. We prove asymptotics with improved power-saving error terms for smoothed sums, including $\sum P_{k}(n)^{2}e^{-n/X}$ and the Laplace transform $\int _{0}^{\infty }P_{k}(t)^{2}e^{-t/X}\,dt$, in dimensions $k\geqslant 3$. We also obtain main terms and power-saving error terms for the sharp sums $\sum _{n\leqslant X}P_{k}(n)^{2}$, along with similar results for the sharp integral $\int _{0}^{X}P_{3}(t)^{2}\,dt$. This includes producing the first power-saving error term in mean square for the dimension-3 Gauss circle problem.

We construct an Euler system—a compatible family of global cohomology classes—for the Galois representations appearing in the geometry of Hilbert modular surfaces. If a conjecture of Bloch and Kato on injectivity of regulator maps holds, this Euler system is nontrivial, and we deduce bounds towards the Iwasawa main conjecture for these Galois representations.

We prove, under some assumptions, a Greenberg type equality relating the characteristic power series of the Selmer groups over $\mathbb{Q}$ of higher symmetric powers of the Galois representation associated to a Hida family and congruence ideals associated to (different) higher symmetric powers of that Hida family. We use $R=T$ theorems and a sort of induction based on branching laws for adjoint representations. This method also applies to other Langlands transfers, like the transfer from $\text{GSp}(4)$ to $U(4)$. In that case we obtain a corollary for abelian surfaces.

We finalize the analysis of the trace formula initiated in S. A. Altuğ [Beyond endoscopy via the trace formula-I: Poisson summation and isolation of special representations, Compos. Math.151(10) (2015), 1791–1820] and developed in S. A. Altuğ [Beyond endoscopy via the trace formula-II: asymptotic expansions of Fourier transforms and bounds toward the Ramanujan conjecture. Submitted, preprint, 2015, Available at:arXiv:1506.08911.pdf], and calculate the asymptotic expansion of the beyond endoscopic averages for the standard $L$-functions attached to weight $k\geqslant 3$ cusp forms on $\mathit{GL}(2)$ (cf. Theorem 1.1). This, in particular, constitutes the first example of beyond endoscopy executed via the Arthur–Selberg trace formula, as originally proposed in R. P. Langlands [Beyond endoscopy, in Contributions to Automorphic Forms, Geometry, and Number Theory, pp. 611–698 (The Johns Hopkins University Press, Baltimore, MD, 2004), chapter 22]. As an application we also give a new proof of the analytic continuation of the $L$-function attached to Ramanujan’s $\unicode[STIX]{x1D6E5}$-function.

Let $p$ and $\ell$ be distinct primes, and let $\overline{\unicode[STIX]{x1D70C}}$ be an orthogonal or symplectic representation of the absolute Galois group of an $\ell$-adic field over a finite field of characteristic $p$. We define and study a liftable deformation condition of lifts of $\overline{\unicode[STIX]{x1D70C}}$ ‘ramified no worse than $\overline{\unicode[STIX]{x1D70C}}$’, generalizing the minimally ramified deformation condition for $\operatorname{GL}_{n}$ studied in Clozel et al. [Automorphy for some$l$-adic lifts of automorphic mod$l$Galois representations, Publ. Math. Inst. Hautes Études Sci. 108 (2008), 1–181; MR 2470687 (2010j:11082)]. The key insight is to restrict to deformations where an associated unipotent element does not change type when deforming. This requires an understanding of nilpotent orbits and centralizers of nilpotent elements in the relative situation, not just over fields.