Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmüller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and develop from scratch a theory of twisted modular forms for Fuchsian groups with a modular embedding, proving dimension formulas, coefficient growth estimates and differential equations. In Part II we provide a modular proof for an Apéry-like integrality statement for solutions of Picard–Fuchs equations. We illustrate the theory on a worked example, giving explicit Fourier expansions of twisted modular forms and the equation of a Teichmüller curve in a Hilbert modular surface. In Part III we show that genus two Teichmüller curves are cut out in Hilbert modular surfaces by a product of theta derivatives. We rederive most of the known properties of those Teichmüller curves from this viewpoint, without using the theory of flat surfaces. As a consequence we give the modular embeddings for all genus two Teichmüller curves and prove that the Fourier developments of their twisted modular forms are algebraic up to one transcendental scaling constant. Moreover, we prove that Bainbridge’s compactification of Hilbert modular surfaces is toroidal. The strategy to compactify can be expressed using continued fractions and resembles Hirzebruch’s in form, but every detail is different.

Using a result of Warnaar, we prove a number of single- and multi-sum identities in the spirit of Ramanujan’s partial theta identities, but with partial indefinite binary theta functions in the role of partial theta functions. We also calculate the corresponding residual identities and use a result of Ji and Zhao to recast our identities in terms of indefinite ternary theta functions.

We describe an implementation for computing holomorphic and skew-holomorphic Jacobi forms of integral weight and scalar index on the full modular group. This implementation is based on formulas derived by one of the authors which express Jacobi forms in terms of modular symbols of elliptic modular forms. Since this method allows a Jacobi eigenform to be generated directly from a given modular eigensymbol without reference to the whole ambient space of Jacobi forms, it makes it possible to compute Jacobi Hecke eigenforms of large index. We illustrate our method with several examples.

We show that the image of the adelic Galois representation attached to a non-CM modular form is open in the adelic points of a suitable algebraic subgroup of GL2 (defined by F. Momose). We also show a similar result for the adelic Galois representation attached to a finite set of modular forms.

We construct algebras of endomorphisms in the derived category of the cohomology of arithmetic manifolds, which are generated by Hecke operators. We construct Galois representations with coefficients in these Hecke algebras and apply this technique to sharpen recent results of P. Scholze.

Let $F/\mathbf{Q}$ be a totally real field and $K/F$ a complex multiplication (CM) quadratic extension. Let $f$ be a cuspidal Hilbert modular new form over $F$ . Let ${\it\lambda}$ be a Hecke character over $K$ such that the Rankin–Selberg convolution $f$ with the ${\it\theta}$ -series associated with ${\it\lambda}$ is self-dual with root number 1. We consider the nonvanishing of the family of central-critical Rankin–Selberg $L$ -values $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})$ , as ${\it\chi}$ varies over the class group characters of $K$ . Our approach is geometric, relying on the Zariski density of CM points in self-products of a Hilbert modular Shimura variety. We show that the number of class group characters ${\it\chi}$ such that $L(\frac{1}{2},f\otimes {\it\lambda}{\it\chi})\neq 0$ increases with the absolute value of the discriminant of $K$ . We crucially rely on the André–Oort conjecture for arbitrary self-product of the Hilbert modular Shimura variety. In view of the recent results of Tsimerman, Yuan–Zhang and Andreatta–Goren–Howard–Pera, the results are now unconditional. We also consider a quaternionic version. Our approach is geometric, relying on the general theory of Shimura varieties and the geometric definition of nearly holomorphic modular forms. In particular, the approach avoids any use of a subconvex bound for the Rankin–Selberg $L$ -values. The Waldspurger formula plays an underlying role.

Let $G\subseteq \widetilde{G}$ be two quasisplit connected reductive groups over a local field of characteristic zero and having the same derived group. Although the existence of L-packets is still conjectural in general, it is believed that the L-packets of $G$ should be the restriction of those of $\widetilde{G}$ . Motivated by this, we hope to construct the L-packets of $\widetilde{G}$ from those of $G$ . The primary example in our mind is when $G=\text{Sp}(2n)$ , whose L-packets have been determined by Arthur [The endoscopic classification of representations: orthogonal and symplectic groups, Colloquium Publications, vol. 61 (American Mathematical Society, Providence, RI, 2013)], and $\widetilde{G}=\text{GSp}(2n)$ . As a first step, we need to consider some well-known conjectural properties of L-packets. In this paper, we show how they can be deduced from the conjectural endoscopy theory. As an application, we obtain some structural information about L-packets of $\widetilde{G}$ from those of  $G$ .

Given a commutative complete local noetherian ring $A$ with finite residue field $\boldsymbol{k}$ , we show that there is a topologically finitely generated profinite group $\unicode[STIX]{x1D6E4}$ and an absolutely irreducible continuous representation $\overline{\unicode[STIX]{x1D70C}}:\unicode[STIX]{x1D6E4}\rightarrow \text{GL}_{n}(\boldsymbol{k})$ such that $A$ is a universal deformation ring for $\unicode[STIX]{x1D6E4},\overline{\unicode[STIX]{x1D70C}}$ .

Given $k\geqslant 2$ , we show that there are at most finitely many rational numbers $x$ and $y\neq 0$ and integers $\ell \geqslant 2$ (with $(k,\ell )\neq (2,2)$ ) for which

$$\begin{eqnarray}x(x+1)\cdots (x+k-1)=y^{\ell }.\end{eqnarray}$$

$\ell$

$(x,y,\ell )$

$y=0$

$\ell <\exp (3^{k})$

We present a ready to compute trace formula for Hecke operators on vector-valuedmodular forms of integral weight for SL2(ℤ) transforming under the Weil representation. As a corollary, we obtain a ready to compute dimension formula for the corresponding space of vector-valued cusp forms, which is more general than the dimension formulae previously published in the vector-valued setting.

We prove a general result on Bailey pairs and show that two Bailey pairs of Bringmann and Kane are special cases. We also show how to use a change of base formula to pass from the pairs of Bringmann and Kane to pairs used by Andrews in his study of Ramanujan's seventh order mock theta functions. We derive several more Bailey pairs of a similar type and use these to construct a number of new q-hypergeometric double sums which are mock theta functions. Finally, we prove identities between some of these mock theta double sums and classical mock theta functions.

We consider the category of smooth $W(k)[\text{GL}_{n}(F)]$ -modules, where $F$ is a $p$ -adic field and $k$ is an algebraically closed field of characteristic $\ell$ different from  $p$ . We describe a factorization of this category into blocks, and show that the center of each such block is a reduced, $\ell$ -torsion free, finite type $W(k)$ -algebra. Moreover, the $k$ -points of the center of a such a block are in bijection with the possible ‘supercuspidal supports’ of the smooth $k[\text{GL}_{n}(F)]$ -modules that lie in the block. Finally, we describe a large explicit subalgebra of the center of each block and give a description of the action of this algebra on the simple objects of the block, in terms of the description of the classical ‘characteristic zero’ Bernstein center of Bernstein and Deligne [Le ‘centre’ de Bernstein, in Representations des groups redutifs sur un corps local, Traveaux en cours (ed. P. Deligne) (Hermann, Paris), 1–32].

The aim of this paper is to carry out an explicit construction of CAP representations of $\text{GL}(2)$ over a division quaternion algebra with discriminant two. We first construct cusp forms on such a group explicitly by lifting from Maass cusp forms for the congruence subgroup ${\rm\Gamma}_{0}(2)$. We show that this lifting is nonzero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then show that our cuspidal representations provide examples of CAP (cuspidal representation associated to a parabolic subgroup) representations, and, in fact, counterexamples to the Ramanujan conjecture.

We prove a level raising mod $\ell =2$ theorem for elliptic curves over $\mathbb{Q}$ . It generalizes theorems of Ribet and Diamond–Taylor and also explains different sign phenomena compared to odd $\ell$ . We use it to study the 2-Selmer groups of modular abelian varieties with common mod 2 Galois representation. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families.

We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $\text{GL}_{n}$ over a totally real field $F$ .

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and let $\bar{\unicode[STIX]{x1D70C}}$ be a continuous, absolutely irreducible representation of its absolute Galois group with values in a finite field of characteristic $p$ . We prove that the Galois representations that become crystalline of a fixed regular weight after an abelian extension are Zariski-dense in the generic fiber of the universal deformation ring of $\bar{\unicode[STIX]{x1D70C}}$ . In fact we deduce this from a similar density result for the space of trianguline representations. This uses an embedding of eigenvarieties for unitary groups into the spaces of trianguline representations as well as the corresponding density claim for eigenvarieties as a global input.

We investigate two kinds of Fricke families, those consisting of Fricke functions and those consisting of Siegel functions. In terms of their special values we then generate ray class fields of imaginary quadratic fields over the Hilbert class fields, which are related to the Lang–Schertz conjecture.

In this paper we perform an extensive study of the spaces of automorphic forms for $\text{GL}_{2}$ of weight $2$ and level $\mathfrak{n}$ , for $\mathfrak{n}$ an ideal in the ring of integers of the quartic CM field $\mathbb{Q}({\it\zeta}_{12})$ of twelfth roots of unity. This study is conducted through the computation of the Hecke module $H^{\ast }({\rm\Gamma}_{0}(\mathfrak{n}),\mathbb{C})$ , and the corresponding Hecke action. Combining this Hecke data with the Faltings–Serre method for proving equivalence of Galois representations, we are able to provide the first known examples of modular elliptic curves over this field.

Supplementary materials are available with this article.

In this article, we construct examples of $L$ -indistinguishable overconvergent eigenforms for an inner form of $\text{SL}_{2}$ .

We show that every modular form on Γ0(2n) (n ⩾ 2) can be expressed as a sum of eta-quotients, which is a partial answer to Ono's problem. Furthermore, we construct a primitive generator of the ring class field of the order of conductor 4N (N ⩾ 1) in an imaginary quadratic field in terms of the special value of a certain eta-quotient.