arXiv subjects
Publications and source records attributed to Siegfried Boecherer.
We generalize the notion of mod $p^m$ singular Siegel modular forms of $p$-rank $r$ to the vector-valued case and we show that also in this case a congruence mod $(p-1)p^{m-1}$ between the scalar weight and the $p$-rank must hold. In some sense our proof is even simpler than the one we gave previously in the scaler valued case.
We construct a Siegel--Eisenstein series of level $p$ with a quadratic character mod $p$ which is a $U(p)$-eigenfunction with eigenvalue $1$, and calculate its Fourier coefficients explicitly. We show that this Siegel--Eisenstein series is a $p$-adic Siegel--Eisenstein series, i.e., it is a $p$-adic limit of a sequence of Siegel--Eisenstein series of level $1$. We prove also that the Siegel--Eisenstein series with a nonquadratic character mod $p$ constructed by Takemori is also a $p$-adic Siegel--Eisenstein series.
In a previous paper, the authors showed that two kinds of $p$-adic Siegel--Eisenstein series of degree $n$ coincide with classical modular forms of weight $k$ for $\Gamma _0(p)$, under the assumption that $p$ is a regular prime. The purpose of this paper is to show that this condition on $p$ can be removed if the degree $n$ is low compared with $k$, namely, $n\le 2k+1$.
We prove that weights of two Siegel modular forms of nonquadratic nebentypus should satisfy some congruence relations if these modular forms are congruent to each other. Applying this result, we prove that there are no mod $p$ singular forms of nonquadratic nebentypus. Here we consider the case where the Fourier coefficients of the modular forms are algebraic integers, and we emphasize that $p$ is a rational prime. Moreover, we construct some examples of mod $\frak{p}$ singular forms of nonquadratic nebentypus using the Eisenstein series studied by Takemori.
We show that the $p$-adic Siegel Eisenstein series of general degree attached to two kind of number sequences are both linear combinations of genus theta series of level $p$, by applying the theory of mod $p$-power singular forms. As special cases of this result, we derive the result of Nagaoka and Katsurada--Nagaoka.
We prove that all mod $p^m$ singular forms of level $N$, degree $n+r$, and $p$-rank $r$ with $n\ge r$ are congruent mod $p^m$ to linear combinations of theta series of degree $r$ attached to quadratic forms of some level. Moreover, we prove that, the levels of theta series are of the form ``$p\mbox{-power}\times N$''. Additionally, in some cases of mod $p$ singular forms with smallest possible weight, we prove that the levels of theta series should be $p$.
We construct many examples of level one Siegel modular forms in the kernel of theta operators mod $p$ by using theta series attached to positive definite quadratic forms.
We give some relations between the weights and the prime $p$ of elements of the mod $p$ kernel of the generalized theta operator $Θ^{[j]}$. In order to construct examples of the mod $p$ kernel of $Θ^{[j]}$ from any modular form, we introduce new operators $A^{(j)}(M)$ and show the modularity of $F|A^{(j)}(M)$ when $F$ is a modular form. Finally, we give some examples of the mod $p$ kernel of $Θ^{[j]}$ and the filtrations of some of them.
We show that Siegel modular forms of level Γ_0(p^m) are p-adic modular forms. Moreover we show that derivatives of such Siegel modular forms are p-adic. Parts of our results are also valid for vector-valued modular forms. In our approach to p-adic Siegel modular forms we follow Serre closely; his proofs however do not generalize to the Siegel case or need some modifications.
The space spanned by theta series of adjoints of maximal even lattices of exact level $N$ and determinant $N^2$ has the Weierstrass property and hence allows to define extremality for arbitrary squarefree level $N$. We find examples of such dual extremal lattices.
Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the Laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of Hecke operators which act on these spherical harmonics. We formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moments tend to moments of the Gaussian as the eigenvalue increases. For such orthonormal bases we show that these conjectures are related to the analytic properties of degree eight arithmetic L-functions associated to triples of eigenfunctions. Moreover we establish the conjecture for the third moments and give a conditional (on standard analytic conjectures about these arithmetic L-functions) proof of the equdistribution of mass conjecture.