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In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is one of the surfaces obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group. Hilbert modular surfaces were first described by using some unpublished notes written by Hilbert about 10 years before. ==Definitions== If ''R'' is the ring of integers of a real quadratic field, then the Hilbert modular group SL2(''R'') acts on the product ''H''×''H'' of two copies of the upper half plane ''H''. There are several birationally equivalent surfaces related to this action, any of which may be called Hilbert modular surfaces: *The surface ''X'' is the quotient of ''H''×''H'' by SL2(''R''); it is not compact and usually has quotient singularities coming from points with non-trivial isotropy groups. *The surface ''X'' * is obtained from ''X'' by adding a finite number of points corresponding to the cusps of the action. It is compact, and has not only the quotient singularities of ''X'', but also singularities at its cusps. *The surface ''Y'' is obtained from ''X'' * by resolving the singularities in a minimal way. It is a compact smooth algebraic surface, but is not in general minimal. *The surface ''Y''0 is obtained from ''Y'' by blowing down certain exceptional −1-curves. It is smooth and compact, and is often (but not always) minimal. There are several variations of this construction: *The Hilbert modular group may be replaced by some subgroup of finite index, such as a congruence subgroup. *One can extend the Hilbert modular group by a group of order 2, acting on the Hilbert modular group via the Galois action, and exchanging the two copies of the upper half plane. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hilbert modular surface」の詳細全文を読む スポンサード リンク
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