(1)レベルと重さに条件を付け，半整数の重さのHilbert保型形式の志村liftができることを示し，フーリエ係数の対応を記述した．またケータ級数の3乗もlift可能であり，応用としてルート2を含む2次体Kで整数が3つの平方数の和に書ける条件を求め，その表現数とKの総虚2次拡大体の相対類数の関係を求めた．
(2)保型形式f,gのn番目のフーリエ係数をそれぞれa(n), b(n)とし，a(n), そしてb(n)の複素共役を積をn番目の係数とするディリクレ級数を考える. f,gの重さや指標が異なっていてもそのL関数が全複素平面への解析接続を示し，またその関数等式を具体的に求めた．二次形式等への応用も示した．
(1)It is shown that Hilbert modular forms of half integral weight have Shimura lifting, provided that the level is divisible by 16 and the weight is at least 5/2. The lifting map is described explicitly in terms of Fourier coefficients. Though the condition is not satisfied, the third power of theta series has lift. As its application, the condition that algebraic integers in the quadratic field K containing square root of 2 are the sums of three squares, is obtained. Also the class numbers of the imaginary quadratic extensions of K is ordained.
(2)Let f,g be elliptic modular forms of level N, and let a(n), b(n) be their n-th Fourier coefficients respectively. Let L(s;f,g) be the Dirichlet series whose n-th coefficient is a product of a(n) and the complex conjugate of b(n). Then it is shown L(s;f,g) extends meromorphically to the whole a plane, and that it has a functional equation. Also the applications to quadratic forms are shown.
Hilbert保型形式のShimura対応とその応用
2020RP0008
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