Related papers: An explicit trace formula of Jacquet-Zagier type f…

Let $A$ be a simple C*-algebra of stable rank one and let $p$ and $q$ be two $\sigma$-compact open projections. It is proved that there is a continuous path of unitaries in ${\tilde A}$ which connects open sub-projections of $p$ which is…

This thesis studies modular forms from a classical and adelic viewpoint. We use this interplay to obtain results about the arithmetic of the Fourier coefficients of modular forms and their generalisations. In Chapter 2, we compute lower…

In this article, for $n\geq 2$, we compute asymptotic, qualitative, and quantitative estimates of the Bergman kernel of Picard modular cusp forms associated to torsion-free, cocompact subgroups of $\mathrm{SU}\big((n,1),\mathbb{C}\big)$.…

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier…

We refine known dimension formulas for spaces of cusp forms of squarefree level, determining the dimension of subspaces generated by newforms both with prescribed global root numbers and with prescribed local signs of Atkin-Lehner…

Let $f$ be a holomorphic cusp form of weight $k$ with respect to $SL_2(\mathbb{Z})$ which is a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper, we shall give the relation of the number of zeros…

We construct holomorphic elliptic modular forms of weight 2 and weight 1, by special values of Weierstrass p-functions, and by differences of special values of Weierstrass zeta-functions, respectively. Also we calculated the values of these…

This article constructs the Satake parameter for any irreducible smooth $J$-spherical representation of a $p$-adic group, where $J$ is any parahoric subgroup. This parametrizes such representations when $J$ is a special maximal parahoric…

In 2016, Ahlgren and Samart used the theory of holomorphic modular forms to obtain lower bounds on $p$-adic valuations related to the Fourier coefficients of three cusp forms. In particular, their work strengthened a previous result of…

We give an explicit dimension formula for paramodular forms of degree two of prime level with plus or minus sign of the Atkin--Lehner involution of weight $\det^k\operatorname{Sym}(j)$ with $k\geq 3$, as well as a dimension formula for…

We study degenerate Whittaker vectors in scalar type holomorphic discrete series representations of tube type Hermitian Lie groups and their analytic continuation. In four different realizations, the bounded domain picture, the tube domain…

We consider the Rankin-Selberg L-functions associated with a fixed modular form of full level and holomorphic cuspidal newforms of large even weight, fixed level and fixed primitive nebentypus. We compute the second moment of this family in…

We provide a simple and new induction based treatment of the problem of distinguishing cusp forms from the growth of the Fourier coefficients of modular forms. Our approach gives the best possible ranges of the weights for this problem, and…

We study $\ell$-regular partitions by defining a sequence of modular forms of level $\ell$ and quadratic character which encode their $\ell$-adic behavior. We show that this sequence is congruent modulo increasing powers of $\ell$ to level…

We give a new construction of $p$-adic overconvergent Hilbert modular forms by using Scholze's perfectoid Shimura varieties at infinite level and the Hodge--Tate period map. The definition is analytic, closely resembling that of complex…

We show that the finite part of the adjoint $L$ function (including contributions from all nonarchimedean places, including ramified places) is holomorphic in $\Re(s) \ge 1/2$ for a cuspidal automorphic representation of $GL_3$ over a…

We investigate the asymptotic distribution of integrals of the $j$-function that are associated to ideal classes in a real quadratic field. To estimate the error term in our asymptotic formula, we prove a bound for sums of Kloosterman sums…

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two. Moreover, we determine the arithmetic…

Let $S_k$ be the space of holomorphic cusp forms of weight $k$ with respect to $SL_2(\mathbb{Z})$. Let $f \in S_k$ be a normalized Hecke eigenform, $L_f(s)$ the $L$-function attached to the form $f$. In this paper we consider the…

In this paper we formulate a conjecture which partially generalizes the Gross-Kohnen-Zagier theorem to higher weight modular forms. For f in S_k(N) satisfying certain conditions, we construct a map from the Heegner points of level N to a…