| Budget Amount *help |
¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1998: ¥500,000 (Direct Cost: ¥500,000)
Fiscal Year 1997: ¥700,000 (Direct Cost: ¥700,000)
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| Research Abstract |
The research project has been pursued by two authors mainly on number theory of automorphic forms, especially on the infinite products of modular functions and on lifting theory of various automorphic forms.
1. Concerning the modular function with real coefficients of the q-expansion, it is known that the sequences of signs of the coefficients are very often periodic. The cases of Thompson series are observed by McKay- Strauss, and some special cases including 1/j are treated by the first author and Kaneko-Ninomiya.
On many cases of the infinite products type or so, the sign patterns of McKay-Strauss' type can be explained by the circle method of Hardy-Ramanajan. In fact it can be shown the signs of coefficients coincide with the signs of Kloosterman sums and so they are periodic.
The first author treated the more sign patterns of many other infinite products of general type, and in particular it was found that it sometimes happen the corresponding coefficients are all zero periodically, which relates to the vanishing problem of a certain general Kloosterman sum.
2. It is known there exists a lifting from GL(2)-automorphic forms to GL(3)-forms. In fact, Gelbart & Jaquet already gave the first construction by using the functional equations method of L-functions.
The second author has succeeded in reconstruction of the lifting by theta correspondence method, where he uses new integral expression of Eisensten series. This remarkable success of quite new method will shed new light on many other liftings.
On another direction, the second author also executed a big calculation of Fourier coefficients of some Sigel modular forms of degree 3, which supports strongly a conjecture of Miyawaki lifting.
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