Bounds for Siegel Modular Forms of genus 2 modulo ppp (original) (raw)
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manuscripta mathematica, 2021
A. We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms mod ℓ, 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 mod ℓ. 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 mod ℓ of a Siegel modular form with integral algebraic Fourier coefficients provided ℓ is large enough. We also make some efforts to make this "largeness" of ℓ effective.
A note on Fourier-Jacobi coefficients of Siegel modular forms
Archiv der Mathematik, 2013
Let F be a Siegel cusp form of weight k and genus n > 1 with Fourier-Jacobi coefficients f m. In this article, we estimate the growth of the Petersson norms of f m , where m runs over an arithmetic progression. This result sharpens a recent result of Kohnen in [5].
On Fundamental Fourier Coefficients of Siegel Modular Forms
Journal of the Institute of Mathematics of Jussieu, 2021
We prove that if F is a nonzero (possibly noncuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many nonzero Fourier coefficients which are indexed by half-integral matrices having odd, square-free (and thus fundamental) discriminant. The proof uses an induction argument in the setting of vector-valued modular forms. Further, as an application of a variant of our result and complementing the work of A. Pollack, we show how to obtain an unconditional proof of the functional equation of the spinor L-function of a holomorphic cuspidal Siegel eigenform of degree 333 and level 111 .
Bounds on Fourier coefficients and global sup-norms for Siegel cusp forms of degree 2
arXiv (Cornell University), 2023
Let F be an L 2-normalized Siegel cusp form for Sp 4 (Z) of weight k that is a Hecke eigenform and not a Saito-Kurokawa lift. Assuming the Generalized Riemann Hypothesis, we prove that its Fourier coefficients satisfy the bound |a(F, S)| ≪ǫ k 1/4+ǫ (4π) k Γ(k) c(S) − 1 2 det(S) k−1 2 +ǫ where c(S) denotes the gcd of the entries of S, and that its global sup-norm satisfies the bound (det Y) k 2 F ∞ ≪ǫ k 5 4 +ǫ. The former result depends on new bounds that we establish for the relevant local integrals appearing in the refined global Gan-Gross-Prasad conjecture (which is now a theorem due to Furusawa and Morimoto) for Bessel periods.
Fourier Coefficients of Half-Integral Weight Modular Forms Modulo ℓ
The Annals of Mathematics, 1998
S. Chowla conjectured that for a given prime p there are infinitely many imaginary quadratic fields whose class number is not a multiple of p. For p = 2 this conjecture is a consequence of Gauss's genus theory, and forp = 3 it follows from the work of Davenport and Heilbronn [DH] (who ...
On mod ppp singular modular forms
We show that an elliptic modular form with integral Fourier coefficients in a number field KKK, for which all but finitely many coefficients are divisible by a prime ideal frakp\frak{p}frakp of KKK, is a constant modulo frakp\frak{p}frakp. A similar property also holds for Siegel modular forms. Moreover, we define the notion of mod frakp\frak{p}frakp singular modular forms and discuss some relations between their weights and the corresponding prime ppp. We discuss some examples of mod frakp\frak{p}frakp singular modular forms arising from Eisenstein series and from theta series attached to lattices with automorphisms. Finally, we apply our results to properties mod frakp\frak{p}frakp of Klingen-Eisenstein series.