ON FUNDAMENTAL FOURIER COEFFICIENTS OF SIEGEL CUSP FORMS OF DEGREE 2

Let F be a Siegel cusp form of degree $2$ , even weight $k \ge 2$ , and odd square-free level N. We undertake a detailed study of the analytic properties of Fourier coefficients $a(F,S)$ of F at fundamental matrices S (i.e., with $-4\det (S)$ equal to a fundamental discriminant). We prove that as S varies along the equivalence classes of fundamental matrices with $\det (S) \asymp X$ , the sequence $a(F,S)$ has at least $X^{1-\varepsilon }$ sign changes and takes at least $X^{1-\varepsilon }$ ‘large values’. Furthermore, assuming the generalized Riemann hypothesis as well as the refined Gan–Gross–Prasad conjecture, we prove the bound $\lvert a(F,S)\rvert \ll _{F, \varepsilon } \frac {\det (S)^{\frac {k}2 - \frac {1}{2}}}{ \left (\log \lvert \det (S)\rvert \right )^{\frac 18 - \varepsilon }}$ for fundamental matrices S.

Sign changes of Fourier coefficients of Siegel cusp forms of degree two on Hecke congruence subgroups

In this paper, we give a lower bound on the number of sign changes of Fourier coefficients of a non-zero degree two Siegel cusp form of even integral weight on a Hecke congruence subgroup. We also provide an explicit upper bound for the first sign change of Fourier coefficients of such Siegel cusp forms. Explicit upper bound on the first sign change of Fourier coefficients of a non-zero Siegel cusp form of even integral weight on the Siegel modular group for arbitrary genus was dealt in an earlier work of Choie, the first author and Kohnen.

On ratios of Petersson norms for Yoshida lifts

We prove an algebraicity property for a certain ratio of Petersson norms associated to a Siegel cusp form of degree 2 (and arbitrary level) whose adelization generates a weak endoscopic lift. As a preparation for this, we explicate various features of the correspondence between scalar valued Siegel cusp forms of degree

SIGN CHANGES IN SHORT INTERVALS OF COEFFICIENTS OF SPINOR ZETA FUNCTION OF A SIEGEL CUSP FORM OF GENUS 2

In this paper, we establish a Voronoi formula for the spinor zeta function of a Siegel cusp form of genus 2. We deduce from this formula quantitative results on the number of its positive (respectively, negative) coefficients in some short intervals.

ON THE NATURAL DENSITIES OF EIGENVALUES OF A SIEGEL CUSP FORM OF DEGREE 2

We prove explicit lower bounds for the density of the sets of primes p such that eigenvalues λp of a Siegel cusp form of degree 2 satisfy c2 > λp > c1, for c1, c2 real. A similar result is also proved for the set of primes such that ∣λp∣ > c.

A non-solvable extension of ℚ unramified outside 7

We consider a mod 7 Galois representation attached to a genus 2 Siegel cusp form of level 1 and weight 28 and using some of its Fourier coefficients and eigenvalues computed by N. Skoruppa and the classification of maximal subgroups of PGSp(4,p) we show that its image is as large as possible. This gives a realization of PGSp(4,7) as a Galois group over ℚ and the corresponding number field provides a non-solvable extension of ℚ which ramifies only at 7.

On LevelpSiegel Cusp Forms of Degree Two

We give a simple formula for the Fourier coefficients of some degree-two Siegel cusp form with levelp.

PRIME DENSITY RESULTS FOR HECKE EIGENVALUES OF A SIEGEL CUSP FORM

Let F ∈ Sk( Sp (2g, ℤ)) be a cuspidal Siegel eigenform of genus g with normalized Hecke eigenvalues μF(n). Suppose that the associated automorphic representation πF is locally tempered everywhere. For each c > 0, we consider the set of primes p for which |μF(p)| ≥ c and we provide an explicit upper bound on the density of this set. In the case g = 2, we also provide an explicit upper bound on the density of the set of primes p for which μF(p) ≥ c.

SYMMETRIC SQUARE L-FUNCTIONS AND SHAFAREVICH–TATE GROUPS, II

We re-examine some critical values of symmetric square L-functions for cusp forms of level one. We construct some more of the elements of large prime order in Shafarevich–Tate groups, demanded by the Bloch–Kato conjecture. For this, we use the Galois interpretation of Kurokawa-style congruences between vector-valued Siegel modular forms of genus two (cusp forms and Klingen–Eisenstein series), making further use of a construction due to Urban. We must assume that certain 4-dimensional Galois representations are symplectic. Our calculations with Fourier expansions use the Eholzer–Ibukiyama generalization of the Rankin–Cohen brackets. We also construct some elements of global torsion which should, according to the Bloch–Kato conjecture, contribute a factor to the denominator of the rightmost critical value of the standard L-function of the Siegel cusp form. Then we prove, under certain conditions, that the factor does occur.