arXiv subjects

Maryna Viazovska

Publications and source records attributed to Maryna Viazovska.

22 records · Page 2Linked to original sources

arXiv2013

Petersson inner products of weight one modular forms

In this paper we study regularized Petersson products between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight 1 modular form with integral Fourier coefficients. In our recent work \cite{Via CM Green} motivated by the conjecture of B. Gross and D. Zagier on the CM values of higher Green's functions we have discovered that such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. The main result of the present paper is the explicit factorization formula for the obtained algebraic number.

math.NT↗

arXiv2012

Theta lifts and local Maass forms

The first two authors and Kohnen have recently introduced a new class of modular objects called locally harmonic Maass forms, which are annihilated almost everywhere by the hyperbolic Laplacian operator. In this paper, we realize these locally harmonic Maass forms as theta lifts of harmonic weak Maass forms. Using the theory of theta lifts, we then construct examples of (non-harmonic) local Maass forms, which are instead eigenfunctions of the hyperbolic Laplacian almost everywhere.

math.NT↗

arXiv2011

CM Values of Higher Green's Functions

Higher Green functions are real-valued functions of two variables on the upper half plane which are bi-invariant under the action of a congruence subgroup, have logarithmic singularity along the diagonal, and satisfy the equation $Δf = k(1 - k)f$, where $Δ$ is a hyperbolic Laplace operator and $k$ is a positive integer. Such functions were introduced in the paper of Gross and Zagier "Heegner points and derivatives of $L$-series"(1986). Also it was conjectured in this paper that higher Green's functions have "algebraic" values at CM points. In many particular cases this conjecture was proven by A. Mellit in his Ph. D. thesis. In this note we present a proof of the conjecture for any pair of CM points lying in the same quadratic imaginary field.

math.NT↗

arXiv2011

Optimal asymptotic bounds for spherical designs

In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$ there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant depending only on $d$.

math.MG↗