Related papers: A B\"ocherer-Type Conjecture for Paramodular Forms

We prove the Sato-Tate conjecture for Hilbert modular forms. More precisely, we prove the natural generalisation of the Sato-Tate conjecture for regular algebraic cuspidal automorphic representations of $\GL_2(\A_F)$, $F$ a totally real…

Let $E/L$ be a real quadratic extension of number fields. We construct an explicit map from an irreducible cuspidal automorphic representation of $\mathrm{GL}(2,E)$ which contains a Hilbert modular form with $\Gamma_0$ level to an…

For a given cusp form $\phi$ of even integral weight satisfying certain hypotheses, Waldspurger's Theorem relates the critical value of the $\mathrm{L}$-function of the $n$-th quadratic twist of $\phi$ to the $n$-th coefficient of a certain…

We consider a Dirichlet series $D(F,G;s)$ attached to two automorphic forms $F$ and $G$ of an orthogonal group of real signature $(2,4)$, involving their Fourier--Jacobi coefficients. When $F$ is a Hecke eigenform and $G$ a lift of a…

We restrict a Siegel modular cusp form of degree 2 and square free level that is a Yoshida lifting (a lifting from the orthogonal group of a definite quaternion algebra) to the embedded product of two half planes and compute the Petersson…

In this paper we use automorph class theory formalism to construct a lifting of similitudes of quadratic Z-modules of arbitrary ternary nondegenerate quadratic forms to morphisms between certain subrings of associated Clifford algebras. The…

In this note we give a detailed construction of a $\Lambda$-adic $\mathfrak{d}$-th Shintani lifting. We derive a $\Lambda$-adic version of Kohnen's formula relating Fourier coefficients of half-integral weight modular forms and special…

The aim of this paper is to prove inequalities towards instances of the Bloch-Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the $L$-function at the central point is zero or one. We achieve…

We develop $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of cubic Hecke $L$-functions of prime moduli over the Eisenstein field using multiple Dirichlet series under the…

In the present paper we study the central values of additive twists of Maa{\ss} forms $L$-series. In the case of the modular group, we show that the additive twists (when averaged over denominators) are asymptotically normally distributed.…

We explain how the work of Johnson-Leung and Roberts on lifting Hilbert modular forms for real quadratic fields to Siegel modular forms can be adapted to imaginary quadratic fields. For this we use archimedean results from Harris, Soudry,…

We extend the results of [CGLS22] to higher weight modular forms and prove a rank $0$ Tamagawa number formula (also known as the Bloch-Kato conjecture) for modular forms at good Eisenstein primes, under some technical assumption on periods.…

The Gromov-Lawson-Rosenberg-conjecture for a group G states that a closed spin manifold M^n (n>4) with fundamental group G admits a metric with positive scalar curvature if and only if its C^*-index A(M) in KO_n(C^*_r(G)) vanishes. We prove…

We study sums of additively twisted Fourier coefficients of a holomorphic cusp form, a Maass cusp form, and the symmetric-square lift of a holomorphic cusp form. We obtain bounds that are uniform with respect to both the form and the terms…

We prove a conjectural formula relating the Bessel period of certain automorphic forms on $\mathrm{GSp}_4$ to a central $L$-value. This formula is proposed by Liu \cite{liu} as the refined Gan-Gross-Prasad conjecture for the groups…

In 2006, Kaneko and Koike defined extremal quasimodular forms and proved their existence in depth $1$ and $2$. After normalizing and restricting to the case of depth at most $4$, they conjectured a certain bound on the Fourier coefficients…

We propose a formulation of the Equivariant Tamagawa Number Conjecture for modular motives with coefficients in universal deformation rings and Hecke algebras; something which seems to have been heretofore missing because the complexes of…

We develop vanishing and cuspidality criteria for quaternionic modular forms on $G=\mathrm{Spin}(4,4)$ using a theory of scalar Fourier coefficients. By analyzing a Fourier-Jacobi expansion for these forms, we prove that a level one…

In this paper, we prove a converse theorem for half-integral weight modular forms assuming functional equations for $L$-series with additive twists. This result is an extension of Booker, Farmer, and Lee's result in [BFL22] to the…

We construct special cycles on the moduli stack of unitary shtukas. We prove an identity between (1) the r-th central derivative of non-singular Fourier coefficients of a normalized Siegel--Eisenstein series, and (2) the degree of special…