Modular Forms of Degree n and Representation by Quadratic Forms

Abstract

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Modular forms of degree n and representation by quadratic forms II

“…But his method is rather rude for our aim. It was effective for T close to scalar matrices [9,13]. Hence we improve it although it is technical.…”

Cusp forms of degree 2 and weight 3

“…and proceeding exactly as in [9], It can happen that, for some S, the associated theta series vanish identically for a given k and every spherical harmonic ~3 of 2 columns! Let S be integral, m-rowed and positive-definite and ~3, a complex matrix of m rows and 2 columns with rank ~3=2 such that S[~B]=0.…”

On the Theory of θ-Functions, the Moduli of Abelian Varieties, and the Moduli of Curves

“…Hence f, fg, and g have Fourier expansions of the form (28) f (Z) = jT c1(T)e(TZ) (29) f (Z)g(Z) = ET c2(T)e(TZ) (30) g(Z) = ST C3(T)e(TZ), where the summations are over symmetric, positive semi-definite n x n matrices T such that t T is half integral, and c1(T), c2(T) e k for all T. We define a partial ordering >-on real, positive semi-definite matrices T as follows: Let T = (tij) and T' = (tV,) be two such matrices. Since F if of finite index in F,, there exists an integer t > 0 such that ify= ( ) SeF, then yt = ( tS e ]) .…”