부산대학교 도서관

Let $N$ be the product of an odd number of distinct primes, and let $\Cal{O}_N$ be a maximal order in a definite quaternion algebra over $\Bbb{Q}$ ramified exactly at the primes dividing $N$. The authors study the zeros of functions $\varphi$ in the space $S(\Cal{O}_N)$ of quaternionic cuspforms, which are functions on the set of right ideal classes of $\Cal{O}_N$. Zeros of these functions are of two types: `trivial' (i.e., forced by the Atkin-Lehner signs of $\varphi$) or `nontrivial'. \par If $\roman{Sq}_r$ is the set of such $N$ with exactly $r$ prime factors, then the authors consider all eigenforms in $S(\Cal{O}_N)$ for $N \in \roman{Sq}_r$ and conjecture that, when these eigenforms are ordered by $N$, \roster \item 100\% of their zeros are trivial; and \item 100\% of them have no trivial zeros. \endroster They show that the second of these conjectures follows from a conjecture on Galois orbits of eigenforms, and give numerical evidence for the first. \par Finally, the authors link these zeros to values of $L$-functions via Waldspurger's formula. As a sample of their results, they show that if $N$ is prime, $\varphi$ is a zero-free eigenform in $S(\Cal{O}_N)$, and $K = \Bbb{Q}(\sqrt{-D})$ is a quadratic extension with class number one in which $N$ does not split, then $$ L(\tfrac{1}{2}, \varphi_K) \neq 0 $$ for $\varphi_K$ the base change of $\varphi$ to $K$.