Abstract. Extending the approach of Iwaniec and Duke, we present strong uniform bounds for Fourier coefficients of half-integral weight cusp forms of level N . As an application, we consider a Waring-type problem with sums of mixed powers.
We prove uniform bounds for the Petersson norm of the cuspidal part of the theta series. This gives an improved asymptotic formula for the number of representations by a quadratic form. As an application, we show that every integer n ‰ 0, 4, 7 pmod 8q is represented as n3 for integers x 1 , x 2 , x 3 such that the product x 1 x 2 x 3 has at most 72 prime divisors.
We compute the second moment of spinor L-functions at central points of Siegel modular forms on congruence subgroups of large prime level N and give applications to non-vanishing.
We prove uniform bounds for the Petersson norm of the cuspidal part of the theta series. This gives an improved asymptotic formula for the number of representations by a quadratic form. As an application, we show that every integer
$n \neq 0,4,7 \,(\textrm{mod}\ 8)$
is represented as
$n= x_1^2 + x_2^2 + x_3^3$
for integers
$x_1,x_2,x_3$
such that the product
$x_1x_2x_3$
has at most 72 prime divisors.
First and foremost, I am very grateful to Prof. Dr. Valentin Blomer for raising my interest in modular forms, his continuous support during my master and PhD thesis and his valuable feedback. Furthermore, I sincerely thank Prof. Dr. Brüdern for his counsel, constant support and his open-door policy. Moreover, I thank my girlfriend Bri, my office mates Burkhard and Rebecca as well as Max for advice, emotional support and consistent encouragement. Last but not least I would like to thank my parents for their counsel and support.
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