Tohoku Mathematical Journal
2026
September
SECOND SERIES VOL. 78, NO. 3

Tohoku Math. J.
78 (2026), 311-366

Title ZETA FUNCTIONS AND NONVANISHING THEOREMS FOR TORIC PERIODS ON $\mathop{\mathrm{GL}}_2$

Author Miyu Suzuki and Satoshi Wakatsuki

(Received June 12, 2024, revised January 6, 2025)
Abstract. Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $\pi$ of $D_\mathbb{A}^\times$ with trivial central character. We study the zeta functions with period integrals on $\pi$ for the prehomogeneous vector space $(D^\times \times D^\times \times \mathop{\mathrm{GL}}_2, D\oplus D)$. We show their meromorphic continuation and functional equation, determine the location and orders of possible poles and compute the residue. Arguing along the theory of Saito and computing unramified local factors, the explicit formula of the zeta functions is obtained. Counting the order of possible poles of this explicit formula, we show that if $L(1/2, \pi)\neq 0$, there are infinitely many quadratic extensions $E$ of $F$ which embeds in $D$, such that $\pi$ has a nonvanishing toric period with respect to $E$.

Mathematics Subject Classification. Primary 11F70; Secondary 11F67, 11S90.
Key words and phrases. Automorphic representations, Toric periods, Prehomogeneous zeta functions.

To the top of this page

Back to the Contents