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We determine all possible values of the integer group determinant of C 2 4 , where C 2 is the cyclic group of order 2. [ABSTRACT FROM AUTHOR]

Let λ π (1 , ... , 1 , n) be the normalized Fourier coefficients of an even Hecke–Maass form π for S L (m , Z) with m ≥ 3 , and r 3 (n) = # { (n 1 , n 2 , n 3) ∈ Z 3 : n = n 1 2 + n 2 2 + n 3 2 } . In this paper, we introduce a refined version of the circle method to derive a sharp bound for the shifted convolution sum of GL(m) Fourier coefficients λ π (1 , ... , 1 , n) and r 3 (n) , which improves previous results (even under the generalized Ramanujan conjecture). [ABSTRACT FROM AUTHOR]

Let d and n be positive integers and let K be a totally real number field of discriminant d and degree n. We construct a theta series θ K ∈ M d , n , where M d , n is a space of modular forms defined in terms of n and d. Moreover, if d is square free and n is at most 4 then θ K is a complete invariant for K. We also investigate whether or not the collection of θ -series, associated to the set of isomorphism classes of quartic number fields of a fixed squarefree discriminant d, is a linearly independent subset of M d , 4 . This is known to be true if the degree of the number field is less than or equal to 3. We give computational and heuristic evidence suggesting that in degree 4 these theta series should be independent as well. [ABSTRACT FROM AUTHOR]