Non-injectivity of Nonzero Discriminant Polynomials and Applications to Packing Polynomials

Borcherds 乘积的逆定理与 Kudla-Millson theta 提升的单射性

A converse theorem for Borcherds products and the injectivity of the Kudla-Millson theta lift

本文证明了无平方因子级且判别群各向异性的格的乘法 Borcherds 提升逆定理，并利用 Kudla-Millson theta 提升的单射性得到满射性，推广了 Bruinier 和 BF1 的结果，同时改进了 Scheithauer 关于反射自守乘积不存在性的定理。

On representation of integers by binary quadratic forms

Given a negative $D>-(\log X)^{\log 2-δ}$, we give a new upper bound on the number of square free integers $<X$ which are represented by some but not all forms of the genus of a primitive positive definite binary quadratic form $f$ of discriminant $D$. We also give an analogous upper bound for square free integers of the form $q+a<X$ where $q$ is prime and $a\in\mathbb Z$ is fixed. Combined with the 1/2-dimensional sieve of Iwaniec, this yields a lower bound on the number of such integers $q+a<X$ represented by a binary quadratic form of discriminant $D$, where $D$ is allowed to grow with $X$ …