We employ a modular method to establish the new result that two types of Eisenstein series to the tredecic base may be parametrised in terms of the eta quotients ${\it\eta}^{13}({\it\tau})/{\it\eta}(13{\it\tau})$ and ${\it\eta}^{2}(13{\it\tau})/{\it\eta}^{2}({\it\tau})$. The method can also be used to give short and simple proofs for the analogous cubic, quintic and septic theories.

We prove modularity of some two-dimensional, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$-adic Galois representations over a totally real field that are nearly ordinary at all places above $2$ and that are residually dihedral. We do this by employing the strategy of Skinner and Wiles, using Hida families, together with the $2$-adic patching method of Khare and Wintenberger. As an application we deduce modularity of some elliptic curves over totally real fields that have good ordinary or multiplicative reduction at places above $2$.

We prove a simple level-raising result for regular algebraic, conjugate self-dual automorphic forms on $\mathrm{GL}_n$ . This gives a systematic way to construct irreducible Galois representations whose residual representation is reducible.

We address the problem of evaluating an $L$ -function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that it is possible to evaluate the $L$ -function more precisely than one would expect from the standard approach. The method, however, requires considerably more computational effort to achieve a given accuracy than would be needed if more Dirichlet coefficients were available.

We give a new proof of a recent result of Kohnen–Martin on a characterization of degree 2 Siegel cusp forms by the growth of their Fourier coefficients. Our main tools are Koecher–Maass series, Imai’s converse theorem and the theory of singular modular forms.

Let $F$ be a totally real field, and $v$ a place of $F$ dividing an odd prime $p$. We study the weight part of Serre’s conjecture for continuous totally odd representations $\overline{{\it\rho}}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbb{F}}_{p})$ that are reducible locally at $v$. Let $W$ be the set of predicted Serre weights for the semisimplification of $\overline{{\it\rho}}|_{G_{F_{v}}}$. We prove that, when $\overline{{\it\rho}}|_{G_{F_{v}}}$ is generic, the Serre weights in $W$ for which $\overline{{\it\rho}}$ is modular are exactly the ones that are predicted (assuming that $\overline{{\it\rho}}$ is modular). We also determine precisely which subsets of $W$ arise as predicted weights when $\overline{{\it\rho}}|_{G_{F_{v}}}$ varies with fixed generic semisimplification.

We study the automorphic Green function $\mathop{\rm gr}\nolimits _\Gamma $ on quotients of the hyperbolic plane by cofinite Fuchsian groups $\Gamma $, and the canonical Green function $\mathop{\rm gr}\nolimits ^{\rm can}_X$ on the standard compactification $X$ of such a quotient. We use a limiting procedure, starting from the resolvent kernel, and lattice point estimates for the action of $\Gamma $ on the hyperbolic plane to prove an “approximate spectral representation” for $\mathop{\rm gr}\nolimits _\Gamma $. Combining this with bounds on Maaß forms and Eisenstein series for $\Gamma $, we prove explicit bounds on $\mathop{\rm gr}\nolimits _\Gamma $. From these results on $\mathop{\rm gr}\nolimits _\Gamma $ and new explicit bounds on the canonical $(1,1)$-form of $X$, we deduce explicit bounds on $\mathop{\rm gr}\nolimits ^{\rm can}_X$.

We study the limiting behavior of the discrete spectra associated to the principal congruence subgroups of a reductive group over a number field. While this problem is well understood in the cocompact case (i.e., when the group is anisotropic modulo the center), we treat groups of unbounded rank. For the groups $\text{GL}(n)$ and $\text{SL}(n)$ we show that the suitably normalized spectra converge to the Plancherel measure (the limit multiplicity property). For general reductive groups we obtain a substantial reduction of the problem. Our main tool is the recent refinement of the spectral side of Arthur’s trace formula obtained in [Finis, Lapid, and Müller, Ann. of Math. (2) 174(1) (2011), 173–195; Finis and Lapid, Ann. of Math. (2) 174(1) (2011), 197–223], which allows us to show that for $\text{GL}(n)$ and $\text{SL}(n)$ the contribution of the continuous spectrum is negligible in the limit.

We apply the endoscopic classification of automorphic forms on $U(3)$ to study the growth of the first Betti number of congruence covers of a Picard modular surface. As a consequence, we establish a case of a conjecture of Sarnak and Xue on cohomology growth.

Let $\nu _{f}(n)$ be the $n\mathrm{th}$ normalized Fourier coefficient of a Hecke–Maass cusp form $f$ for ${\rm SL }(2,\mathbb{Z})$ and let $\alpha $ be a real number. We prove strong oscillations of the argument of $\nu _{f}(n)\mu (n) \exp (2\pi i n \alpha )$ as $n$ takes consecutive integral values.

In this paper, we prove that the Sato–Tate conjecture for primitive Maass forms holds on average. We also investigate the rate of convergence in the Sato–Tate conjecture and establish some estimates of the discrepancy with respect to the Sato–Tate measure on the average of primitive Maass forms.

We describe algorithms that allow the computation of fundamental domains in the Bruhat–Tits tree for the action of discrete groups arising from quaternion algebras. These algorithms are used to compute spaces of rigid modular forms of arbitrary even weight, and we explain how to evaluate such forms to high precision using overconvergent methods. Finally, these algorithms are applied to the calculation of conjectural equations for the canonical embedding of p-adically uniformizable rational Shimura curves. We conclude with an example in the case of a genus 4 Shimura curve.

As the simplest case of Langlands functoriality, one expects the existence of the symmetric power $S^n(\pi )$, where $\pi $ is an automorphic representation of ${\rm GL}(2,{\mathbb{A}})$ and ${\mathbb{A}}$ denotes the adeles of a number field $F$. This should be an automorphic representation of ${\rm GL}(N,{\mathbb{A}})$ ($N=n+1)$. This is known for $n=2,3$ and $4$. In this paper we show how to deduce the general case from a recent result of J.T. on deformation theory for ‘Schur representations’, combined with expected results on level-raising, as well as another case (a particular tensor product) of Langlands functoriality. Our methods assume $F$ totally real, and the initial representation $\pi $ of classical type.

In this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on ${\rm GL}_2$ over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.

While investigating the Doi–Naganuma lift, Zagier defined integral weight cusp forms $f_D$ which are naturally defined in terms of binary quadratic forms of discriminant $D$. It was later determined by Kohnen and Zagier that the generating function for the function $f_D$ is a half-integral weight cusp form. A natural preimage of $f_D$ under a differential operator at the heart of the theory of harmonic weak Maass forms was determined by the first two authors and Kohnen. In this paper, we consider the modularity properties of the generating function of these preimages. We prove that although the generating function is not itself modular, it can be naturally completed to obtain a half-integral weight modular object.

When the branch character has root number $- 1$ , the corresponding anticyclotomic Katz $p$ -adic $L$ -function vanishes identically. For this case, we determine the $\mu $ -invariant of the cyclotomic derivative of the Katz $p$ -adic $L$ -function. The result proves, as an application, the non-vanishing of the anticyclotomic regulator of a self-dual CM modular form with root number $- 1$ . The result also plays a crucial role in the recent work of Hsieh on the Eisenstein ideal approach to a one-sided divisibility of the CM main conjecture.

Given a prime $p\gt 2$ , an integer $h\geq 0$ , and a wide open disk $U$ in the weight space $ \mathcal{W} $ of ${\mathbf{GL} }_{2} $ , we construct a Hecke–Galois-equivariant morphism ${ \Psi }_{U}^{(h)} $ from the space of analytic families of overconvergent modular symbols over $U$ with bounded slope $\leq h$ , to the corresponding space of analytic families of overconvergent modular forms, all with ${ \mathbb{C} }_{p} $ -coefficients. We show that there is a finite subset $Z$ of $U$ for which this morphism induces a $p$ -adic analytic family of isomorphisms relating overconvergent modular symbols of weight $k$ and slope $\leq h$ to overconvergent modular forms of weight $k+ 2$ and slope  $\leq h$ .

Generalizing previous results of Deligne–Serre and Taylor, Galois representations are attached to cuspidal automorphic representations of unitary groups whose Archimedean component is a holomorphic limit of discrete series. The main ingredient is a construction of congruences, using the Hasse invariant, that is independent of $q$ -expansions.

Using the $\ell $ -invariant constructed in our previous paper we prove a Mazur–Tate–Teitelbaum-style formula for derivatives of $p$ -adic $L$ -functions of modular forms at trivial zeros. The novelty of this result is to cover the near-central point case. In the central point case our formula coincides with the Mazur–Tate–Teitelbaum conjecture proved by Greenberg and Stevens and by Kato, Kurihara and Tsuji at the end of the 1990s.

Ramanujan’s last letter to Hardy concerns the asymptotic properties of modular forms and his ‘mock theta functions’. For the mock theta function $f(q)$, Ramanujan claims that as $q$ approaches an even-order $2k$ root of unity, we have

$$\begin{eqnarray*}f(q)- (- 1)^{k} (1- q)(1- {q}^{3} )(1- {q}^{5} )\cdots (1- 2q+ 2{q}^{4} - \cdots )= O(1).\end{eqnarray*}$$

$q$

$\vartheta $