We settle a part of the conjecture by Bandini and Valentino [‘On the structure and slopes of Drinfeld cusp forms’, Exp. Math. 31(2) (2022), 637–651] for $S_{k,l}(\Gamma _0(T))$ when $\mathrm {dim}\ S_{k,l}(\mathrm {GL}_2(A))\leq 2$. We frame and check the conjecture for primes $\mathfrak {p}$ and higher levels $\mathfrak {p}\mathfrak {m}$, and show that a part of the conjecture for level $\mathfrak {p} \mathfrak {m}$ does not hold if $\mathfrak {m}\ne A$ and $(k,l)=(2,1)$.

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain p-adic estimates of Satake parameters of certain nonregular self-dual automorphic representations of $\mathrm {GL}_n$ .

We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for $\mathit{GSp}_{4}/\mathbb{Q}$ in various aspects. A main tool is Arthur’s invariant trace formula. While Shin [Automorphic Plancherel density theorem, Israel J. Math.192(1) (2012), 83–120] and Shin–Templier [Sato–Tate theorem for families and low-lying zeros of automorphic $L$-functions, Invent. Math.203(1) (2016) 1–177] used Euler–Poincaré functions at infinity in the formula, we use a pseudo-coefficient of a holomorphic discrete series to extract holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms $A,B_{1}$ in Theorem 1.1 which have not been studied and a mysterious second term $B_{2}$ also appears in the second main term coming from the semisimple elements. Furthermore our explicit study enables us to treat more general aspects in the weight. We also give several applications including the vertical Sato–Tate theorem, the unboundedness of Hecke fields and low-lying zeros for degree 4 spinor $L$-functions and degree 5 standard $L$-functions of holomorphic Siegel cusp forms.

Let X be a Shimura curve of genus zero. In this paper, we first characterize the spaces of automorphic forms on X in terms of Schwarzian differential equations. We then devise a method to compute Hecke operators on these spaces. An interesting by-product of our analysis is the evaluation and other similar identities.

For the locally symmetric space X attached to an arithmetic subgroup of an algebraic group G of ℚ-rank r, we construct a compact manifold by gluing together 2r copies of the Borel–Serre compactification of X. We apply the classical Lefschetz fixed point formula to and get formulas for the traces of Hecke operators ℋ acting on the cohomology of X. We allow twistings of ℋ by outer automorphisms η of G. We stabilize this topological trace formula and compare it with the corresponding formula for an endoscopic group of the pair (G,η) . As an application, we deduce a weak lifting theorem for the lifting of automorphic representations from Siegel modular groups to general linear groups.

We study the action of the Hecke operators Un on the set of hypergeometric functions, as well as on formal power series. We show that the spectrum of these operators on the set of hypergeometric functions is the set {na:n∈ℕ,a∈ℤ}, and that the polylogarithms play an important role in the study of the eigenfunctions of the Hecke operators Un on the set of hypergeometric functions. As a corollary of our results on simultaneous eigenfunctions, we also obtain an apparently unrelated result regarding the behavior of completely multiplicative hypergeometric coefficients.

We study an equidistribution problem of Linnik using Hecke operators.

Let $F$ be a totally real number field and let $\text{G}{{\text{L}}_{n}}$ be the general linear group of rank $n$ over $F$. Let $\mathfrak{p}$ be a prime ideal of $F$ and ${{F}_{\mathfrak{p}}}$ the completion of $F$ with respect to the valuation induced by $\mathfrak{p}$. We will consider a finite quotient of the affine building of the group $\text{G}{{\text{L}}_{n}}$ over the field ${{F}_{\mathfrak{p}}}$. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

In this paper, we introduce a new way of studying derivatives of L-functions of cusp forms by associating to them cocycles analogous, in function, to the period polynomial. The main result gives a description of the effect of Hecke operators on these cocycles.

Hecke operators are used to investigate part of the ${{E}_{2}}$-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of $\text{Ex}{{\text{t}}^{1}}$ which combines use of classical Hecke operators and $p$-adic Hecke operators due to Serre.