Related papers: Murmurations of modular forms in the weight aspect

In this paper, we study the average of the Fourier coefficients of a holomorphic cusp form for the full modular group at primes of the form $[g(n)]$.

We describe a family of $q$-series generating the space of weight 1 modular forms coming from indefinite binary quadratic forms and study linear relations between these series.

The modularity of an elliptic curve $E/\mathbb Q$ can be expressed either as an analytic statement that the $L$-function is the Mellin transform of a modular form, or as a geometric statement that $E$ is a quotient of a modular curve…

The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular…

This paper investigates the relations between modular graph forms, which are generalizations of the modular graph functions that were introduced in earlier papers motivated by the structure of the low energy expansion of genus-one Type II…

We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the $k-1$-th…

The characters $\chi_\mu$ of nontwisted affine algebras at fixed level define in a natural way a representation $R$ of the modular group $SL_2(Z)$. The matrices in the image $R(SL_2(Z))$ are called the Kac-Peterson modular matrices, and…

A random ensemble of cusp forms for the full modular group is introduced. For a weight-$k$ cusp form, restricted to a compact subdomain of the modular surface, the true order of magnitude of its expected supremum is determined to be $\asymp…

We prove a sub-convex estimate for the sup-norm of $L^2$-normalized holomorphic modular forms of weight $k$ on the upper half plane, with respect to the unit group of a quaternion division algebra over $\mf Q$. More precisely we show that…

We utilize the structure of quasiautomorphic forms over an arbitrary Hecke triangle group to define a new vector analogue of an automorphic form. We supply a proof of the functional equations that hold for these functions modulo the group…

We produce first examples of p-local height three TAF homology theories. The corresponding one-dimensional formal groups arise as split summands of the formal groups of certain abelian three-folds, the Shimura variety of which can be…

We prove that the determinant (pseudo-representation) associated to the Hecke algebra of Katz modular forms of weight one and level prime to p is unramified at p.

Assuming the Riemann hypothesis for $L$-functions attached to primitive Dirichlet characters, modular cusp forms, and their tensor products and symmetric squares, we write down explicit finite sets of Hecke operators that span the Hecke…

We prove that the category of finitely generated graded modules over the quiver Hecke algebra of arbitrary type admits numerous stratifications in the sense of Kleshchev. A direct consequence is that the full subcategory corresponding to…

The group of extensions (as in the title), endowed with something like a connection at Archimedean infinity, is isomorphic to the ad\'ele-class group of $\Q$: which is a topological group with interesting Haar measure.}

In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This…

The coefficients of the higher-derivative terms in the low energy expansion of genus-one graviton Type II superstring scattering amplitudes are determined by integrating sums of non-holomorphic modular functions over the complex structure…

In earlier work, the first named author generalized the construction of Darmon-style $\mathcal{L}$-invariants to cuspidal automorphic representations of semisimple groups of higher rank, which are cohomological with respect to the trivial…

This is essentially a translated (and explained) version of a peper Hecke published in 1930 where he shows, for a prime q, a relation between the class number h(-q) and the representation of PSL(2, Z / pZ) on the space of holomorphic…

We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner…