arXiv subjects
Publications and source records attributed to Siegfried Böcherer.
For a paramodular group of any degree and square free level we study the Hecke algebra and the boundary components. We define paramodular theta series and show that for square free level and large enough weight they generate the space of cusp forms (basis problem), using the doubling and pullback of Eisenstein series method. For this we give a new geometric proof of Garrett's double coset decomposition which works in our more general situation.
We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms $\bmod \ell$, partially answering a question of Bellaïche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto $x$ which do not vanish $\bmod \ell$. We also propose a precise conjecture as a possible answer to this question. Further, we prove several results related to the nonvanishing of arithmetically interesting (e.g., primitive or fundamental) Fourier coefficients $\bmod \ell $ of a Siegel modular form with integral algebraic Fourier coefficients provided $\ell$ is large enough. We also make some efforts to make this "largeness" of $\ell$ effective.
We show that an elliptic modular form with integral Fourier coefficients in a number field $K$, for which all but finitely many coefficients are divisible by a prime ideal $\frak{p}$ of $K$, is a constant modulo $\frak{p}$. A similar property also holds for Siegel modular forms. Moreover, we define the notion of mod $\frak{p}$ singular modular forms and discuss some relations between their weights and the corresponding prime $p$. We discuss some examples of mod $\frak{p}$ singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms. Finally, we apply our results to properties mod $\frak{p}$ of Klingen-Eisenstein series.
Let $f$ and $g$, of weights $k'>k\geq 2$, be normalised newforms for $Γ_0(N)$, for square-free $N>1$, such that, for each Atkin-Lehner involution, the eigenvalues of $f$ and $g$ are equal. Let $λ\mid\ell$ be a large prime divisor of the algebraic part of the near-central critical value $L(f\otimes g,\frac{k+k'-2}{2})$. Under certain hypotheses, we prove that $λ$ is the modulus of a congruence between the Hecke eigenvalues of a genus-two Yoshida lift of (Jacquet-Langlands correspondents of) $f$ and $g$ (vector-valued in general), and a non-endoscopic genus-two cusp form. In pursuit of this we also give a precise pullback formula for a genus-four Eisenstein series, and a general formula for the Petersson norm of a Yoshida lift. Given such a congruence, using the 4-dimensional $λ$-adic Galois representation attached to a genus-two cusp form, we produce, in an appropriate Selmer group, an element of order $λ$, as required by the Bloch-Kato conjecture on values of $L$-functions.
We restrict a Siegel modular cusp form of degree 2 and square free level that is a Yoshida lifting (a lifting from the orthogonal group of a definite quaternion algebra) to the embedded product of two half planes and compute the Petersson product against the product of two elliptic cuspidal Hecke eigenforms. The square of this integral can be explicitly expressed in terms of the central critical value of an L-function attached to the situation. The result is related to a conjecture of Gross and Prasad about restrictions of automorphic representations of special orthogonal groups.
We continue our study of Yoshida's lifting, which associates to a pair of automorphic forms on the adelic multiplicative group of a quaternion algebra a Siegel modular form of degree 2. We consider here the case that the automorphic forms on the quaternion algebra correspond to modular forms of arbitrary even weights and square free levels; in particular we obtain a construction of Siegel modular forms of weight 3 attached to a pair of elliptic modular forms of weights 2 and 4.
We compute the central critical value of the triple product $L$-function associated to three cusp forms $f_1,f_2,f_3$ with trivial character for groups $Γ_0(N_i)$ with square free levels $N_i$ not all of which are $1$ and weights $k_i$ satisfying $k_1\ge k_2\ge k_3$ and $k_1<k_2+k_3$. This generalizes work of Gross and Kudla and gives an alternative classical proof of their results in the case $N_1=N_2=N_3$ with $k_1=k_2=k_3=2$.