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We combine the Duke-Imamoglu-Ikeda lifting with the theta lifting to produce new CAP representations of metaplectic, symplectic and orthogonal groups. These constructions partially generalize the theories of Waldspurger on the Shimura…

The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let's mention : (1) the control of the image of the Galois representation modulo $p$, (2) Hida's…

It is well known that the normaized characters of integrable highest weight modules of given level over an affine Lie algebra $\hat{\frak{g}}$ span an $SL_2(\mathbf{Z})$-invariant space. This result extends to admissible…

Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times…

We establish special value results of convolutions of Goss and Pellarin $L$-series attached to Drinfeld modules that take values in Tate algebras. Applying the class module formula of Demeslay to certain rigid analytic twists of one…

In this paper, we investigate cycle integrals of weakly holomorphic modular forms. We show that these integrals coincide with the cycle integrals of classical cusp forms. We use these results to define a Shintani lift from integral weight…

In this article we study the Iwasawa theory for Hecke characters associated with CM abelian varieties and Hilbert modular forms at ordinary primes. We formulate and prove a result concerning the anticyclotomic Iwasawa main conjecture for CM…

We obtain a first moment formula for Rankin-Selberg convolution $L$-series of holomorphic modular forms or Maass forms of arbitrary level on $GL(2)$, with an orthonormal basis of Maass forms. One consequence is the best result to date,…

Bruinier, Funke, and Imamoglu have proved a formula for what can philosophically be called the "central $L$-value" of the modular $j$-invariant. Previously, this had been heuristically suggested by Zagier. Here, we interpret this…

Let $F$ be a totally real number field. We prove that a character of the spherical Hecke algebra appearing in the completed cohomology of Hilbert modular varieties is modular if the associated Galois representation is absolutely…

Let $f$ be a cuspidal eigenform of weight two and level $N$, let $p\nmid N$ be a prime at which $f$ is congruent to an Eisenstein series and let $V_f$ denote the $p$-adic Tate module of $f$. Beilinson constructed a class $\kappa_f\in…

Let r : G_Q -> GL_2(Fpbar) be a p-ordinary and p-distinguished irreducible residual modular Galois representation. We show that the vanishing of the algebraic or analytic Iwasawa mu-invariant of a single modular form lifting r implies the…

After a brief review of recent rigorous results concerning the representation theory of rational chiral conformal field theories (RCQFTs) we focus on pairs (A,F) of conformal field theories, where F has a finite group G of global symmetries…

We give a characterisation of the field into which quotients of values of L-functions associated to a cusp form belong. The construction involves shifted convolution series of divisor sums and to establish it we combine parts of F. Brown's…

Let $(\mathbb{G},\mathbb{H})$ be a symmetric pair of reductive groups over a $p$-adic field with $p\neq 2$, attached to the involution $\theta$. Under the assumption that there exists a maximally $\theta$-split torus in $\mathbb{G}$, which…

Following Rankin's method, D. Zagier computed the $n$-th Rankin-Cohen bracket of a modular form $g$ of weight $k_1$ with the Eisenstein series of weight $k_2$ and then computed the inner product of this Rankin-Cohen bracket with a cusp form…

We show that modular forms of fractional weights on principal congruence subgroups of odd levels, which are found by T. Ibukiyama, naturally appear as characters being multiplied $\eta^{c_{\text{eff}}}$ of the so-called minimal models of…

Given a holomorphic newform $f$ of weight $k$ and with rational coefficients, a question of Mazur and van Straten asks if there is an associated Calabi-Yau variety $X$ over ${\mathbb Q}$ of dimension $k-1$ such that the $\ell$-adic Galois…

We prove in some cases a formula for the Greenberg-Benois $\mathcal{L}$-invariant of the spin, standard and adjoint Galois representations associated with Siegel-Hilbert modular forms. In order to simplify the calculation, we give a new…

The purpose of this paper is to build on results in {\it{higher codimension Iwasawa theory}}. The setting of our results involves Galois representations arising as cyclotomic twist deformations associated to (i) the tensor product of two…