We prove that if $F$ is a non-zero (possibly non-cuspidal) vector-valued Siegel modular form of any degree, then it has infinitely many non-zero 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. In an Appendix, as an application of a variant of our result and building upon 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 $3$.

On fundamental fourier coefficients of siegel modular forms

We prove an $\Omega $-result for the Fourier coefficients of a half-integral weight cusp form of arbitrary level, nebentypus and weights. In particular, this implies that the analogue of the Ramanujan-Petersson conjecture for such forms is essentially the best possible. As applications, we show similar $\Omega $-results for Fourier coefficients of Siegel cusp forms of any degree and on Hecke congruence subgroups.

Omega Results for Fourier Coefficients of Half-Integral Weight and Siegel Modular Forms

We prove that if $f$ is a non zero cusp form of weight $k$ on $\Gamma_0(N)$ with character $\chi$ such that $N/(\text{conductor }\chi)$ square-free, then there exists a square-free $n\ll_{\epsilon} k^{3+\epsilon}N^{7/2+\epsilon}$ such that $a(f,n)\neq 0$. This significantly improves the already known existential and quantitative result from previous works.

Hecke-Siegel type threshold for square-free Fourier coefficients: an improvement

We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms $$\bmod \, \ell $$
 , partially answering a question of Bellaiche-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.

On Fourier coefficients of elliptic modular forms $$\bmod \, \ell $$ mod ℓ with applications to Siegel modular forms

We prove that given any $$\epsilon >0$$
 , a non-zero adelic Hilbert cusp form $${\mathbf {f}}$$
 of weight $$k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n$$
 and square-free level $$\mathfrak {n}$$
 with Fourier coefficients $$C_{{\mathbf {f}}}(\mathfrak {m})$$
 , there exists a square-free integral ideal $$\mathfrak {m}$$
 with $$N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }$$
 such that $$C_{{\mathbf {f}}}(\mathfrak {m})\ne 0$$
 . The implied constant depends on $$\epsilon , F$$
 .

On the Fourier coefficients of certain Hilbert modular forms

We prove that Hermitian cusp forms of weight k for the Hermitian modular group of degree 2 are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which are also treated in this paper.

/pdf/distinguishing-hermitian-cusp-forms-of-degree-2-by-a-certain-1nu2gvvu5h.pdf

Distinguishing Hermitian cusp forms of degree 2 by a certain subset of all Fourier coefficients

In this short article, we show the existence of an analogue of the classical Sturm’s bound in the context of the square-free Fourier coefficients for cusp forms of square-free levels. This number is a cut-off to determine a cusp form from its initial few square-free Fourier coefficients. We also mention some questions in this regard.

Sturm-Like Bound for Square-Free Fourier Coefficients

Bounds for the Petersson norms of the pullbacks of Saito–Kurokawa lifts

We prove a result on the distribution of Hecke eigenvalues, $\mu_F(p^r)$ (for $r=1,2$ or $3$) of a non Saito--Kurokawa lift $F$ of degree $2$. As a consequence, we obtain an Omega result for the Hecke eigenvalues for such an $F$, which is the best possible in terms of orders of magnitude.

Large Hecke eigenvalues and an Omega result for non Saito--Kurokawa lifts

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