## Number Theory Seminar

Seminar information archive ～02/27｜Next seminar｜Future seminars 02/28～

Date, time & place | Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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Organizer(s) | Naoki Imai, Shane Kelly |

### 2023/06/07

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

On the dimension of spaces of p-ordinary half-integral weight Siegel modular forms

of degree 2 (日本語)

**Hirofumi Yamamoto**(The University of Tokyo)On the dimension of spaces of p-ordinary half-integral weight Siegel modular forms

of degree 2 (日本語)

[ Abstract ]

Hecke eigenforms whose eigenvalues at $p$ are $p$-adic units are called $p$-ordinary. The dimension of the spaces spanned by $p$-ordinary Siegel modular eigenforms is known to be bounded regardless of the power of $p$ of levels and weights. Furthermore, Hida showed that the degree is bounded in the case of half-integral weight modular forms, using the Shimura correspondence. In this talk, I will explain that a similar result holds for half-integral weight $p$-ordinary Siegel modular forms of degree 2. Let $F$ be a $p$-ordinary Hecke eigen cuspform, and $\pi_F$ be the corresponding cuspidal representation of $Mp_4(\mathbb{A}_\mathbb{Q})$. Then the weight of $F$ determines the Hecke eigenvalue of $\pi_F$ at $p$. Therefore, we can relate $F$ to $p$-ordinary integral weight Siegel modular forms or elliptic modular forms by using the local theta correspondence and the method of Ishimoto(Ibukiyama conjecture).

Hecke eigenforms whose eigenvalues at $p$ are $p$-adic units are called $p$-ordinary. The dimension of the spaces spanned by $p$-ordinary Siegel modular eigenforms is known to be bounded regardless of the power of $p$ of levels and weights. Furthermore, Hida showed that the degree is bounded in the case of half-integral weight modular forms, using the Shimura correspondence. In this talk, I will explain that a similar result holds for half-integral weight $p$-ordinary Siegel modular forms of degree 2. Let $F$ be a $p$-ordinary Hecke eigen cuspform, and $\pi_F$ be the corresponding cuspidal representation of $Mp_4(\mathbb{A}_\mathbb{Q})$. Then the weight of $F$ determines the Hecke eigenvalue of $\pi_F$ at $p$. Therefore, we can relate $F$ to $p$-ordinary integral weight Siegel modular forms or elliptic modular forms by using the local theta correspondence and the method of Ishimoto(Ibukiyama conjecture).