Related papers: Explicit images for the Shimura Correspondence

In a recent preprint, we constructed a sesquiharmonic Maass form $\mathcal{G}$ of weight $\frac{1}{2}$ and level $4N$ with $N$ odd and squarefree. Extending seminal work by Duke, Imamo\={g}lu, and T\'{o}th, $\mathcal{G}$ maps to Zagier's…

The space of Siegel cuspforms of degree $2$ of weight $3$ with respect to the congruence subgroup $\G_2(2,4,8)$ was studied by van Geemen and van Straten in Math. computation. {\bf 61} (1993). They showed the space is generated by six-tuple…

Using recent results, we prove strong multiplicity one theorems for Goss zeta functions and their Teichmuller lifts.

In \cite{Shimura}, Shimura introduced modular forms of half-integral weight, their Hecke algebras and their relation to integral weight modular forms via the Shimura correspondence. For modular forms of integral weight, Sturm's bounds give…

The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. In this paper, we establish for O(p,2) an adjointness result between Borcherds' singular theta lift and the Kudla-Millson…

We give an explicit formula for dimensions of spaces of rational-weight modular forms whose multiplier systems are induced by eta-quotients of fractional exponents. As the first application, we give series expressions of Fourier…

Let $(M, \omega)$ be a compact connected K\"ahler manifold of complex dimension four and let $[\chi] \in H^{1,1}(M; \mathbb{R})$. We confirmed the conjecture by Collins--Jacob--Yau [arXiv:1508.01934] of the solvability of the deformed…

We define a regularized lift from harmonic weak Maass forms of weight $2-N$ to differential forms of degree $N-1$ on the symmetric space $\SL_N(\R)/\SO(N)$, that are smooth outside of certain modular symbols. We show that this lift is…

We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak…

Given a rational elliptic curve E, a suitable imaginary quadratic field K and a quaternionic Hecke eigenform g of weight 2 obtained from E by level raising such that the sign in the functional equation for L_K(E,s) (respectively, L_K(g,1))…

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 paper, we study the Hecke eigenvalues of Ikeda lifts. Using the spherical map for the Hecke algebra of the symplectic group, we obtain an explicit formula for the eigenvalues $\lambda_F(p^r)$. From this formula, we show that…

Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus subspace of weight k-n/2+1/2 and level 4, let I(h) be the Duke-Imamoglu-Ikeda lift of h in the space of cusp forms of weight k for Sp(n,Z), and f the…

We consider two families of arithmetic divisors defined on integral models of Shimura curves. The first was studied by Kudla, Rapoport and Yang, who proved that if one assembles these divisors in a formal generating series, one obtains the…

Let $\ell \geq 5$ be a prime, and let $\nu_\eta$ denote the Dedekind eta multiplier. For an odd integer $r$, and a real Dirichlet character $\psi$, recent work of Ahlgren, Andersen, and the author showed that quadratic congruences modulo…

We look at genera of even unimodular lattices of rank $12$ over the ring of integers of $\mathbb{Q}(\sqrt{5})$ and of rank $8$ over the ring of integers of $\mathbb{Q}(\sqrt{3})$, using Kneser neighbours to diagonalise spaces of…

In this paper we establish a new Fundamental Lemma for Hecke correspondences. Let $F$ be a local field containing the cube roots of unity. We exhibit an algebra isomorphism of the spherical Hecke algebra of $PGL_3(F)$ and the spherical…

In the 1980s B\"ocherer formulated a conjecture relating the central value of the quadratic twists of the spinor L-function attached to a Siegel modular form F to the coefficients of F . He proved the conjecture when F is a Saito-Kurokawa…

We prove that for $s=\sigma+it$ with $\sigma\ge0$ and $0<t\le x$, we have \[\zeta(s)=\sum_{n\le x}n^{-s}+\frac{x^{1-s}}{(s-1)}+\Theta\frac{29}{14} x^{-\sigma},\qquad \frac{29}{14}=2.07142\dots\] where $\Theta$ is a complex number with…

In this paper, we introduce higher level versions of the theta group $\Gamma_{\theta}.$ In particular, we treat level 3 and 4 versions of the theta group, $\Gamma_{\theta,3}$ and $\Gamma_{\theta,4}$ and prove that $\displaystyle…