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Theorem 5.1. Let \( y \) be an element of \( K \) . Let \( t \in K \) be a local parameter at the point \( P \), and let \( z \) be the element of \( K \) which is such that \( {dy} = {zdt} \) . If \( {dy}/{dt} \) is the derivative of \( y \) with respect to \( t \) taken formally from the power series expansion of \( ... | Proof. The statement of our theorem depends on the fact that every differential form of \( K \) can be written \( {zdt} \) for some \( z \), by \( §3 \) . We know that \( K \) is separable algebraic over \( k\left( t\right) \), and the irreducible polynomial equation \( f\left( {t, y}\right) = 0 \) of \( y \) over \( k... | Yes |
Proposition 5.3. The notation being as above, let \( \\operatorname{Tr} \) be the trace from \( K \) to \( F \) . Then for any \( y \in K \), we have\n\n\[ \operatorname{Tr}\left( y\right) = \\mathop{\\sum }\\limits_{{i = 1}}^{r}{\\operatorname{Tr}}_{i}\left( y\right) \] | Proof. Suppose \( y \) is a generator of \( K \) over \( F \) . If \( \\left\\lbrack {K : F}\\right\\rbrack = n \), then \( \\operatorname{Tr}\left( y\\right) \) is the coefficient of \( {Y}^{n - 1} \) in the irreducible polynomial \( g\\left( Y\\right) \) as above. A similar remark applies to the local traces, and our... | Yes |
Corollary 2. For any divisior \( \mathfrak{a} > 0 \) we have\n\n\[ \delta \left( {-a}\right) = \deg a - 1 + g \]\n\nand\n\n\[ \dim \operatorname{Diff}\left( \alpha \right) /\mathrm{{dfk}} = \deg \alpha - 1. \] | Proof. Since \( l\left( {-\alpha }\right) = 0 \) because a function which has no poles and at least one zero is identically 0 , the formulas are special cases of the Riemann-Roch theorem. | Yes |
Theorem 6.1. Let \( k \) be algebraically closed, and let \( K \) be a function field with \( k \) as constant field. Let \( E \) be a finite separable extension of \( K \) of degree n. Let \( {g}_{E} \) and \( {g}_{K} \) be the genera of \( E \) and \( K \) respectively. For each point \( P \) of \( K \), and each poi... | Proof. If \( \omega \) is any non-zero differential form of \( K \), then we know that its degree is \( 2{g}_{K} - 2 \) . Such a form can be written as \( {ydx} \), with \( x, y \in K \) . We can also view \( x, y \) as elements of \( E \) ; we can compute the degree in \( E \), and compare it with that in \( K \) to g... | Yes |
Theorem 8.1. Assume that \( K \) has characteristic 0 . Then\n\n\[ \n\dim \mathrm{{dsk}}/\text{exact} = {2g}\text{.} \n\] | Proof. We define \( \operatorname{dsk}\left( \alpha \right) \) and \( \operatorname{dtk}\left( \alpha \right) \) just as we defined \( \operatorname{Diff}\left( \alpha \right) \), that is, forms of the prescribed kind whose divisor is \( \geqq - \alpha \) .\n\nLet \( {P}_{1},\ldots ,{P}_{r} \) be distinct points, and l... | Yes |
Theorem 8.2. Let \( {P}_{1},\ldots ,{P}_{r} \) be distinct points, and let \( N \) be a positive integer such that \( \left( {N - 1}\right) r > {2g} - 2 \) . Then\n\n\[ \mathrm{{dsk}}/\text{exact} = \mathrm{{dsk}}\left( {{N\sum }{P}_{i}}\right) /{dL}\left( {\left( {N - 1}\right) \sum {P}_{i}}\right) \text{.} \] | Note that\n\n\[ \dim {dL}\left( {\left( {N - 1}\right) \sum {P}_{i}}\right) = l\left( {\left( {N - 1}\right) \sum {P}_{i}}\right) - 1, \]\n\nbecause the only functions \( z \) such that \( {dz} = 0 \) are the constants. On the other hand, also note that\n\n\[ \mathrm{{dfk}} \cap \text{exact} = 0\text{,} \]\n\nbecause a... | Yes |
Theorem 2.1. A basis for the differentials of first kind is given by \( {\omega }_{r, s} \) with \( 1 \leqq r, s \) such that \( r + s \leqq N - 1 \) . | Proof. Suppose that \( x\left( P\right) = \infty \) . Put \( t = 1/x \) . Then\n\n\[ \n{dx} = - \frac{1}{{t}^{2}}{dt} \n\]\n\nand \( {\operatorname{ord}}_{P}y = {\operatorname{ord}}_{P}x = 1 \) . Then\n\n\[ \n{\omega }_{r, s} = {x}^{r - 1}{y}^{s - 1}\left( \frac{-1}{{t}^{2}}\right) \frac{dt}{{y}^{N - 1}}, \n\]\n\nfrom ... | Yes |
Theorem 2.2. The forms \( {\omega }_{r, s} \) with\n\n\[ 1 \leqq r, s \leqq N - 1\text{ and }r + s ≢ 0{\;\operatorname{mod}\;N} \]\n\nconstitute a basis for \( \mathrm{{dsk}}/\mathrm{{exact}} \). | Proof. Let\n\n\[ \infty = \mathop{\sum }\limits_{{i = 1}}^{N}\left( {\infty }_{i}\right) \]\n\nbe the divisor of points above \( x = \infty \) on \( F\left( N\right) \), taken with multiplicity 1 First we note that the space\n\n\[ \mathrm{{dtk}}\left( \infty \right) /\mathrm{{dfk}} \]\n\nhas dimension \( N - 1 \) by th... | Yes |
Theorem 3.1. A basis of \( \mathrm{{dfk}} \) on \( F\left( {r, s}\right) \) is given by the forms\n\n\[ \n{\omega }_{\langle {mr}\rangle ,\langle {ms}\rangle } \n\] \n\nfor all \( \left( {r, s}\right) \) -admissible elements \( m \) . | Proof. It is clear that \( {x}^{\langle {mr}\rangle }{y}^{\langle {ms}\rangle } \) lies in the function field of \( F\left( {r, s}\right) \), and hence that the forms listed above are of the first kind on \( F\left( {r, s}\right) \) . Conversely, suppose \( \omega \) is a dfk on \( F\left( {r, s}\right) \) . Write\n\n\... | Yes |
If \( N = p \) is prime \( \geqq 3 \), then for every admissible pair \( \left( {r, s}\right) \) the curve \( F\left( {r, s}\right) \) has genus \( \left( {p - 1}\right) /2 \), and \( K\left( {r, s}\right) = K\left( {1,{s}^{ * }}\right) \) for a uniquely determined integer \( {s}^{ * } \) such that the pair \( \left( {... | The genus can either be computed directly as we did for the Fermat curve, or one can use Theorem 3.1. The number of \( m \) such that \( \left( {\langle {mr}\rangle ,\langle {ms}\rangle }\right) \) is admissible is trivially computed to be \( \left( {p - 1}\right) /2 \), using the remark preceding the theorem. The stat... | Yes |
Theorem 4.1. We have \( {f}^{ * } \circ f = p \cdot \mathrm{{id}} \) . | Proof. Let \( A \) and \( B \) be the automorphisms of \( F\left( N\right) \) induced by \( \left( {x, y}\right) \mapsto \) \( \left( {{\zeta x}, y}\right) \) and \( \left( {x,{\zeta y}}\right) \) respectively. We use the same letters for the induced automorphisms of the divisor class group. For any divisor \( \mathfra... | Yes |
Theorem 1.1. Taking as charts discs on t-planes as above, together with the maps given by local parameters, gives an analytic manifold structure to \( R \) . | Proof. If \( P \in R \), and \( t, u \) are two parameters at \( P \), then \( t \) has a power series expansion in terms of \( u \), say\n\n\[ t = \mathop{\sum }\limits_{{i = 0}}^{\infty }{a}_{i}{u}^{i} \]\n\nand since \( t \) is algebraic over \( k\left( u\right) \), simple estimates show that this power series is co... | Yes |
Theorem 1.2. Every meromorphic function on \( R \) is in \( K \) . | Proof. Let \( L \) be the field of meromorphic functions on \( R \) . If \( L \neq K \), then the degree \( {\left( L : K\right) }_{\mathrm{C}} \) of the factor space of \( L{\;\operatorname{mod}\;K} \) over the complex is infinite. We have:\n\n\[{\left( L : K\right) }_{\mathrm{C}} = {\left( L + \Lambda \left( 0\right)... | Yes |
Theorem 1.3. The Riemann surface is connected. | Proof. Let \( S \) be a connected component. Let \( P \in S \) and let \( z \in K \) be a function having a pole only at \( P \) (such a \( z \) exists by the Riemann-Roch theorem). Then \( z \) is holomorphic on any other component, without pole, hence constant, equal to \( c \) on such a component. But \( z - c \) ha... | No |
Theorem 1.4. Let \( z \in K \) be a non-constant function. The points of \( K \) induce points of \( \mathbf{C}\left( z\right) \), and thereby induce a mapping of \( R \) onto the \( z \) -sphere \( {S}_{z} \), which is a ramified topological covering. The algebraic ramification index \( {e}_{P} \) at a point \( P \) o... | Proof. Let \( P \) be a point of \( R \) and \( t \) a parameter at \( P \) . Then \( P \) induces a point \( z = a \) on the \( z \) -sphere, and \( {t}^{e} \) is equal to \( z - a \) times a unit in the power series ring \( \mathbf{C}\left\lbrack \left\lbrack t\right\rbrack \right\rbrack \) . Since one can extract \(... | Yes |
Theorem 1.6. Let \( g \) be the topological genus of \( R \) . Then\n\n\[ \n{2g} - 2 = - {2n} + \mathop{\sum }\limits_{P}\left( {{e}_{P} - 1}\right) .\n\] | Proof. We have \( {\chi }_{z} = {B}_{0} - {B}_{1} + {B}_{2} \) where \( {B}_{i} \) is the \( i \) -th Betti number.\n\nWe have \( {B}_{0} = {B}_{2} = 1 \) . But also \( {\chi }_{z} = V - E + T \), so \( {\chi }_{z} = 2 \).\n\nNow \( {\chi }_{R} = {V}^{\prime } - {E}^{\prime } + {T}^{\prime } \) and by our previous resu... | Yes |
Theorem 2.1. The kernels of this pairing on both sides are 0 . In other words, if \( \omega \) is a differential of first kind whose integral along every cycle is 0, then \( \omega = 0 \) . Conversely, if \( \gamma \) is a cycle such that the integral along \( \gamma \) of every \( \mathrm{{dfk}} \) is 0, then \( \gamm... | Proof. As to the first statement, fix a point \( O \) on \( R \) . Under the hypothesis that \( \omega \) is orthogonal to every cycle, it follows that the association\n\n\[ P \mapsto {\int }_{O}^{P}\omega \]\n\n is a holomorphic function on \( R \), whence constant, and therefore that \( \omega = 0 \) . The converse i... | No |
Theorem 1.1. Let \( \omega \) be a differential holomorphic on \( \mathcal{P} \) . Then\n\n\[{\int }_{\mathcal{P}}{f\omega } = - \mathop{\sum }\limits_{{i = 1}}^{{2g}}{w}_{i}{\widetilde{\alpha }}_{i} = {2\pi }\sqrt{-1}\sum {\operatorname{res}}_{P}\left( {f\omega }\right)\]\n\nwhere the sum over \( P \) is taken over po... | Proof. Our function \( f \) is uniquely defined inside the polygon. To get the\n\nintegral over the polygon itself, where \( f \) has been defined separately for the two representations of the same side, we approximate the polygon \( \mathcal{P} \) by a polygon \( {\mathcal{P}}^{\prime } \) as indicated (Fig. 3)\n\n![1... | Yes |
Theorem 1.2. If \( \omega \) is holomorphic on \( \mathcal{P} \), then\n\n\[ - \mathop{\sum }\limits_{{i = 1}}^{{2g}}{w}_{i}{\widetilde{A}}_{i} = {2\pi }\sqrt{-1}\mathop{\sum }\limits_{P}{\operatorname{res}}_{P}\left( {F\omega }\right) ,\]\nthe residue being the vector of residues, and the sum over \( P \) is taken ove... | We can always find a polygon representation of \( R \) such that \( z \) has no pole on \( \mathcal{P} \), and we let \( \omega = {dz}/z \) in Theorem 1.2. Then \( {\operatorname{res}}_{P}\left( {F\omega }\right) = {n}_{P}F\left( P\right) \) . Indeed, for any holomorphic function \( f \) consider a local parameter \( t... | Yes |
Lemma 2.1. Let \( {x}_{i}\left( {i = 1,\ldots ,{2g}}\right) \) be complex numbers. Then\n\n\[ \sum {x}_{i}{\widetilde{A}}_{i} = 0\;\text{ if and only if }\;{x}_{i} = B \cdot {A}_{i} \]\n\nfor some complex vector \( B = \left( {{b}_{1},\ldots ,{b}_{g}}\right) \) . The vectors \( {A}_{1},\ldots ,{A}_{2g} \) span a g-dime... | Proof. We prove first that the relation \( X \cdot {A}_{i} = 0 \) for all \( i \) implies that \( X = 0 \) (for a complex vector \( X \) ). Let \( \omega = X \cdot \Phi \) . Then\n\n\[ {\int }_{{a}_{i}}\omega = {\int }_{{a}_{i}}X \cdot \Phi = X \cdot {\int }_{{a}_{i}}\Phi = X \cdot {A}_{i} = 0 \]\n\nby hypothesis. Henc... | Yes |
Lemma 2.2. Let \( \mathfrak{a} = \sum {n}_{P}P \) be a divisor of degree 0 . Then there exists a differential of third kind \( \omega \) such that \( {\operatorname{res}}_{P}\omega = {n}_{P} \) for all \( P \) . | Proof. It suffices to prove our assertion for the case where all coefficients are zero except for two points. Indeed, by induction, suppose it proved for points \( {P}_{1},\ldots ,{P}_{r} \) . Let \( Q \) be a new point unequal to the given points. Let \( {\omega }_{j}\left( {j = 1,\ldots, r}\right) \) have poles only ... | Yes |
Lemma 3.1. There exists \( g \) distinct points \( {M}_{1},\ldots ,{M}_{g} \) such that the divisor \( {M}_{1} + \cdots + {M}_{g} \) is non-special. | Proof. Let \( {\omega }_{1} \neq 0 \) be a dfk, and \( {M}_{1} \) a point which is not zero of \( {\omega }_{1} \) . The space of dfk having a zero at \( {M}_{1} \) has dimension \( g - 1 \) by Riemann-Roch. Let \( {\omega }_{2} \neq 0 \) be in it and let \( {M}_{2} \) be a point which is not a zero of \( {\omega }_{2}... | No |
Theorem 3.2. Let \( {M}_{1},\ldots ,{M}_{g} \) be \( g \) distinct points such that the divisor \( {M}_{1} + \cdots + {M}_{g} \) is non-special. Then the map\n\n\[ \n\left( {{P}_{1},\ldots ,{P}_{g}}\right) \mapsto \sum {\int }_{{M}_{i}}^{{P}_{i}}\Phi \n\]\n\ngives an analytic isomorphism of a product of small discs \( ... | We now prove Theorem 3.2. Let \( {t}_{i} \) be a local parameter at \( {M}_{i} \) . We contend that the determinant\n\n\[ \n\left| \begin{matrix} \frac{{\varphi }_{1}}{d{t}_{1}}\left( {M}_{1}\right) & \cdots & \frac{{\varphi }_{1}}{d{t}_{g}}\left( {M}_{g}\right) \\ & \cdots & \\ \frac{{\varphi }_{g}}{d{t}_{1}}\left( {M... | Yes |
Theorem 3.3. The periods \( {A}_{1},\ldots ,{A}_{2g} \) are linearly independent over the reals. Hence the factor space of \( {\mathbf{C}}^{g} \) by the periods is a 2g-real-dimensional torus. | Proof. It will suffice to prove that any complex vector \( X \) is congruent to a vector of bounded length modulo periods. By the Riemann-Roch theorem, any divisor of degree 0 is linearly equivalent to a divisor of type\n\n\[\n\left( {{P}_{1} + \cdots + {P}_{g}}\right) - g \cdot O,\n\]\n\nfor suitable points \( {P}_{1}... | Yes |
Lemma 5.1. Numbering the sides of the polygon as at the beginning, \( {a}_{1},\ldots ,{a}_{2g} \), and given an index \( i \), there exists \( a\mathrm{{dfk}}\varphi \) with periods satisfying\n\n\[ \operatorname{Re}{\int }_{{a}_{i}}\varphi = 1,\;\operatorname{Re}{\int }_{{a}_{j}}\varphi = 0\;\text{ for }\;j \neq i. \] | Proof. Consider the period matrix:\n\n\n\nwhere the \( u, v \) are the real and imaginary parts of the periods, and are, therefore, real. Then the column vectors of the above matrix are our periods \( {A}_{1},\ldots ,{... | Yes |
Lemma 5.2. A \( d\mathrm{{fk}} \) cannot have all its periods pure imaginary. | Proof. The preceding system of linear equations cannot have a solution when made homogeneous. | No |
Theorem 5.3. Given a divisor \( \alpha = \sum {n}_{P}P \) of degree 0, there exists a unique \( \mathrm{{dtk}}{\omega }_{a} \) such that\n\n(1) \( {\operatorname{res}}_{P}{\omega }_{\alpha } = {n}_{P} \) for all \( P \) .\n\n(2) The periods of \( {\omega }_{a} \) are all pure imaginary. | Proof. Let \( \omega \) be a dtk with \( {\operatorname{res}}_{P}\omega = {n}_{P} \) . It suffices to find a dfk \( \varphi \) having the same real parts for the canonical periods (integrals around \( {a}_{i} \) ). This is\nimmediate from Lemma 5.1. The uniqueness is clear, because the difference between two differenti... | No |
Theorem 5.4. If \( \sigma \) is a cycle such that \( {\int }_{\sigma }\varphi = 0 \) for all \( \mathrm{{dfk}} \) then \( \sigma \sim 0 \) . | Proof. Let \( \sigma \sim {n}_{1}{a}_{1} + \cdots + {n}_{2g}{a}_{2g} \) . Find a dfk \( \varphi \) having periods such that\n\n\[ \operatorname{Re}{\int }_{{a}_{1}}\varphi = 1,\;\text{ and }\;\operatorname{Re}{\int }_{{a}_{j}}\varphi = 0,\;j \neq 1. \]\n\nThen\n\n\[ 0 = {\int }_{\sigma }\varphi = {n}_{1} + \text{ pure ... | Yes |
Theorem 5.6. The bilinear map\n\n\\[ \n{H}_{1}\\left( {R,\\mathbf{Z}}\\right) \\times {\\Omega }_{1}\\left( R\\right) \\rightarrow \\mathbf{R} \n\\]\n\ngiven by\n\n\\[ \n\\left( {\\gamma ,\\omega }\\right) \\mapsto \\operatorname{Re}{\\int }_{\\gamma }\\omega \n\\]\n\ninduces a duality of real vector spaces, making \\(... | Proof. By Lemma 5.2, the kernel of the pairing on the right, i.e. in the space of dfk, is equal to 0 . As a real vector space, the dfk have dimension \\( {2g} \\), by Lemma 5.1. Since \\( {H}_{1}\\left( {R,\\mathbf{Z}}\\right) \\) has real dimension \\( \\leqq {2g} \\), it follows that its image in the dual space of \\... | Yes |
Lemma 1.1. The symbol \( L\left( {f,\sigma }\right) \) is bimultiplicative in \( f \) and \( \sigma \) . | Proof. The formulas\n\n\[ L\left( {{fg},\sigma }\right) = L\left( {f,\sigma }\right) + L\left( {g,\sigma }\right) \text{ and } L\left( {f,\sigma {\sigma }^{\prime }}\right) = L\left( {f,\sigma }\right) + L\left( {f,{\sigma }^{\prime }}\right) \]\n\nare trivially proved from the definitions, and the fact that the integr... | Yes |
Theorem 4.1 (Rohrlich). The period lattice of \( F\left( {r, s}\right) \) is generated by the vectors\n\n\[ \n\\text{(. . .,}{\\zeta }^{{rmj} + {smk}}\\left( {1 - {\\zeta }^{rm}}\\right) \\left( {1 - {\\zeta }^{sm}}\\right) \\frac{1}{N}B\\left( {\\frac{\\langle {rm}\\rangle }{N},\\frac{\\langle {sm}\\rangle }{N}}\\righ... | Proof. The integral of \( {\\omega }_{\\langle {rm}\\rangle ,\\langle {sm}\\rangle } \) over \( {l}_{A}^{s}{l}_{B}^{-r} \) is the only new one, not already considered in determining the period lattice on the Fermat curve itself. The integral over \( {l}_{A}^{s} \) is obtained by integrating over the path shown, letting... | Yes |
Theorem 1.1. Let\n\n\[ E\left( {x, y}\right) = L\left( {x, y}\right) - L\left( {y, x}\right) . \]\n\nThen \( E \) is \( \mathbf{R} \) -bilinear, alternating, and real valued on \( V \times V \) . Furthermore, \( E \) takes on integral values on \( D \times D \) . | Proof. The last statement follows from (2). Since \( L \) is \( \mathbf{R} \) -bilinear, it follows that \( E \) is real valued, so our theorem is proved. | No |
Theorem 1.2. Let\n\n\[ S\left( {x, y}\right) = E\left( {{ix}, y}\right) . \]\n\nThen \( S \) is symmetric. Also the form\n\n\[ H\left( {x, y}\right) = E\left( {{ix}, y}\right) + {iE}\left( {x, y}\right) \]\n\nis hermitian, so \( S \) is the real part of \( H \) . | Proof. We expand the value for \( S\left( {x, y}\right) - S\left( {y, x}\right) \) in terms of \( L \) . We find at once that\n\n\[ S\left( {x, y}\right) - S\left( {y, x}\right) = i\left\lbrack {E\left( {x, y}\right) - E\left( {{ix},{iy}}\right) }\right\rbrack . \]\n\nSince the left-hand side is real and the right-hand... | Yes |
Theorem 1.3. In any equivalence class of theta functions, there exists a normalized theta function, unique up to a non-zero constant factor. | Proof. We have shown existence. As to uniqueness, we have already noted the uniqueness of the quadratic factor. For the linear factor, if \( \operatorname{Im}K = 0 \) then the \( \mathbf{C} \) -linear function \( \lambda \) in the previous discussion is 0, which shows the uniqueness. | No |
Theorem 1.5. Suppose that \( F \) is a normalized entire theta function. Then :\n\n(i) There exists a number \( C > 0 \) such that\n\n\[ \left| {F\left( z\right) }\right| \leqq C{e}^{\left( {\pi /2}\right) H\left( {z, z}\right) }\;\text{ for all }z \in V. \]\n\n(ii) The associated hermitian form \( H \) is positive (no... | Proof. As to the first inequality, let\n\n\[ g\left( z\right) = F\left( z\right) e\left( {-\left( {\pi /2}\right) H\left( {z, z}\right) }\right) ,\]\n\nwhere \( e\left( w\right) = {e}^{w} \) . Then for \( u \in D \) ,\n\n\[ g\left( {z + u}\right) = g\left( z\right) {e}^{{i\pi }\left\lbrack {E\left( {z, u}\right) + K\le... | Yes |
Theorem 2.1. Let \( F \) be a normalized entire theta function on the complex space \( V \), with respect to the lattice \( D \), and let \( H \) be its associated hermitian form. Let \( {V}_{H} \) be the kernel of \( H \) . Then:\n\n(i) The image of the lattice \( D \) in \( V/{V}_{H} \) is discrete.\n\n(ii) The value... | Proof. Let \( {z}_{0} \in {V}_{H} \) and let \( x \in V \) . By assumption, for any complex \( z \), we have\n\n\[ H\left( {x + z{z}_{0}, x + z{z}_{0}}\right) = H\left( {x, x}\right) . \]\n\nThe estimate in Theorem 1.5 shows that\n\n\[ \left| {F\left( {x + z{z}_{0}}\right) }\right| \leqq {Ce}\left( {\frac{\pi }{2}H\lef... | Yes |
Lemma 1. Let \( E \) be an alternating non-degenerate bilinear form on a free Z-module \( D \), having values in \( \mathbf{Z} \) . Then \( D \) is an E-orthogonal direct sum\n\n\[ D = \left\lbrack {{e}_{1},{v}_{1}}\right\rbrack \oplus \cdots \oplus \left\lbrack {{e}_{n},{v}_{n}}\right\rbrack \]\n\nof 2-dimensional sub... | Proof. The lemma is proved by induction. Among all values \( E\left( {u, v}\right) \) with \( u, v \in D \) we select a least positive one, say \( {d}_{1} \), and we take a pair \( {e}_{1},{v}_{1} \) such that \( E\left( {{e}_{1},{v}_{1}}\right) = {d}_{1} \) . We let \( \left\lbrack {{e}_{1},{v}_{1}}\right\rbrack = {D}... | Yes |
Lemma 2. Let \( \left\lbrack {{e}_{1},{v}_{1}}\right\rbrack \oplus \cdots \oplus \left\lbrack {{e}_{n},{v}_{n}}\right\rbrack \) be a Frobenius decomposition of \( D \) with respect to a non-degenerate Riemann form. Then \( \left\{ {{e}_{1},\ldots ,{e}_{n}}\right\} \) is a C-basis for \( V \) . | Proof. Let \( {V}^{\prime } \) be the \( \mathbf{R} \) -space generated by \( {e}_{1},\ldots ,{e}_{n} \), and let \( {V}^{\prime \prime } = i{V}^{\prime } \) . Suppose that we have a relation\n\n\[ x + {iy} = 0,\;x, y \in {V}^{\prime }.\]\n\nThen iy is in \( {V}^{\prime } \) (because equal to \( - x \) ), hence is \( E... | Yes |
Theorem 3.1 (Frobenius). Let \( \\left( {L, J}\\right) \) be a non-degenerate type with respect to \( \\left( {V, D}\\right) \). Then the entire theta functions on \( V \) with respect to \( D \) having this type, together with 0, form a complex vector space of dimension equal to the pfaffian of \( E \) with respect to... | Proof. We first observe that \( L \) is symmetric on \( \\left\\lbrack {{e}_{1},\\ldots ,{e}_{n}}\\right\\rbrack \) (notation as above), and hence on the space \( {V}^{\\prime } \) generated by \( {e}_{1},\\ldots ,{e}_{n} \) over \( \\mathbf{R} \). Consequently, there is a symmetric C-bilinear form \( B \) on \( V \) s... | Yes |
Lemma 3.3. If \( L \) is such that \( L\left( {V,{e}_{i}}\right) = 0 \) for all \( i \), that is, \( L \) satisfies \( \left( *\right) \) , then the imaginary part of \( L \) is negative definite on the \( \mathbf{R} \) -space generated by \( {v}_{1},\ldots ,{v}_{n} \) . | Proof. Let \( z = x + {iy} \), where \( x, y \) lie in the space \( {V}^{\prime } \) above, that is are real linear combinations of \( {e}_{1},\ldots ,{e}_{n} \) . Then for \( y \neq 0 \) we get\n\n\[ 0 < E\left( {{iy}, y}\right) = E\left( {x + {iy}, y}\right) = E\left( {z, y}\right) = L\left( {z, y}\right) - L\left( {... | Yes |
Theorem 3.4. The dimension of \( \operatorname{Th}\left( {L, J}\right) \) is equal to the reduced pfaffian of the associated alternating form \( E \) . | Proof. This follows from the theorem, which gives us the dimension in the non-degenerate case. | No |
Theorem 3.5. Let \( \left( {L, J}\right) \) be a non-degenerate type with respect to \( \left( {V, D}\right) \). Then all theta functions of this type except possibly those lying in a finite union of subspaces of dimension \( < \operatorname{pf}\left( E\right) \) are not theta functions with respect to a lattice strict... | Proof. If \( F \) is a theta function of type \( \left( {L,{J}^{\prime }}\right) \) with respect to \( \left( {V,{D}^{\prime }}\right) \), then the pfaffian of \( E \) with respect to \( {D}^{\prime } \) is equal to\n\n\[ \frac{d}{\left( {D}^{\prime } : D\right) } < d \]\n\nwhere \( d = {d}_{1}\cdots {d}_{n} \) is its ... | Yes |
Theorem 4.1 (Riemann-Roch). Let \( {X}_{0},\ldots ,{X}_{m} \) be positive divisors such that \( {X}_{0} \) is non-degenerate. There exists a polynomial \( P \) in \( m + 1 \) variables such that\n\n\[ l\left( {{r}_{0}{X}_{0} + \cdots + {r}_{m}{X}_{m}}\right) = P\left( {{r}_{0},\ldots ,{r}_{m}}\right) \]\n\nfor any inte... | Proof. This is obvious from Theorem 3.1 because the desired dimension is given as the pfaffian of \( {r}_{0}{E}_{0} + \cdots + {r}_{m}{E}_{m} \) . | No |
Lemma 5.1. (i) If \( \theta \) is of type \( \left( {L, J}\right) \) then \( {\theta }_{a} \) is of type \( \left( {L, J - {L}_{a}}\right) \), where \( {L}_{a}\left( u\right) = L\left( {a, u}\right) . | Proof. We have\n\n\[\n\frac{\theta \left( {x - a + u}\right) }{\theta \left( {x - a}\right) } = \exp \{ {2\pi i}\left\lbrack {L\left( {x - a, u}\right) + J\left( u\right) }\right\rbrack \}\n\]\n\nfrom which our first assertion is obvious. Note that the bilinear form \( L \) is the same for any translation of \( \theta ... | No |
Theorem 5.3. Let \( \theta \) be an entire non-degenerate theta function with Riemann form \( E \) . Then the kernel of \( {\varphi }_{\theta } \) is finite in \( V/D \), and is represented by those elements \( a \in V \) such that \( E\left( {a, u}\right) \in \mathbf{Z} \) for all \( u \in D \) . We have \[ \text{orde... | Proof. We have seen in Lemma 5.1 that if \( \left( {L, J}\right) \) is the type of \( \theta \), which we may assume to be normalized, then the type of the normalized theta function in the equivalence class of \( {\theta }_{a} \) is \( \left( {L, J - {E}_{a}}\right) \) . Suppose that \( a \) is in the kernel of \( {\va... | Yes |
Theorem 5.4. Let \( \theta \) be an entire theta function, normalized, with associated hermitian form \( H \) . Let \( {V}_{H} \) be the null space of \( H \) . Then \( {V}_{H} + D \) is of finite index in the kernel of \( {\varphi }_{\theta } \), in \( V \) . | Proof. This results at once from the non-degenerate case. Indeed, we know from \( §2 \) that \( \theta \) induces a theta function on \( \bar{V} = V/{V}_{H} \), with respect to the image \( \bar{D} \) of \( D \) in \( V/{V}_{H} \) . If \( {\theta }_{a}^{\prime } \sim \theta \), and \( {\theta }_{a}^{\prime } \) is the ... | Yes |
Theorem 1.1. The matrix\n\n\[ \n\left( \frac{U}{U}\right) \n\]\n\nis non-singular (where the bar denotes complex conjugate). | Proof. Let \( M \) be the above matrix. Suppose \( M \) is singular. Then there\nexists a complex vector \( Z \) ( \( {2n} \) -tuple) such that \( {MZ} = 0 \) (we view \( Z \) as a column vector). This implies that \( {UZ} = 0 \) and \( \bar{U}Z = 0 \), so that also \( U\bar{Z} = 0 \).\n\nHence\n\n\[ \nU\left( {Z + \ba... | Yes |
Theorem 1.2. The rational representation \( \lambda \mapsto Q\left( \lambda \right) \) is equivalent to the direct sum of the complex representation and its conjugate. | Proof. By Theorem 1.1, the matrix\n\n\[ \left( \frac{U}{U}\right) \]\n\nis invertible, and by the above relation, we also get the complex conjugate relation, namely\n\n\[ \overline{C\left( \lambda \right) U} = \bar{U}Q\left( \lambda \right) \]\n\nTherefore\n\n\[ \left( \begin{matrix} C\left( \lambda \right) & 0 \\ 0 & ... | Yes |
Theorem 2.1. The p-adic representation of \( \operatorname{End}\left( A\right) \) is equivalent to the rational representation. | Proof. Obvious from the boxed isomorphism (2) above. | No |
Theorem 4.1. Let \( {A}^{\prime } = {V}^{\prime }/{D}^{\prime } \) be a subtorus of \( A = V/D \), and assume that \( \left( {V, D}\right) \) has a non-degenerate Riemann form. Then there exists a subtorus \( {A}^{\prime \prime } = {V}^{\prime \prime }/{D}^{\prime \prime } \) such that\n\n\[ A = {A}^{\prime } + {A}^{\p... | Proof. Let \( {V}^{\prime \prime } \) be the orthogonal complement of \( {V}^{\prime } \) with respect to the positive definite hermitian form \( H \) associated with the Riemann form on \( \left( {V, D}\right) \) . By the Gram-Schmidt orthogonalization process already used in Chapter VI, \( §3 \), we can see easily th... | Yes |
Lemma 5.1. Let \( {\mathbf{C}}_{1} \) be the group of complex numbers of absolute value 1. For each \( \xi \in {V}^{ * } \), let \( {\chi }_{\xi } \) be the element of \( \operatorname{Hom}\left( {D,{\mathbf{C}}_{1}}\right) \) defined by \[ {\chi }_{\xi }\left( u\right) = {e}^{{2\pi i}\operatorname{Im}\langle \xi, u\ra... | Proof. Our map \( \xi \mapsto {\chi }_{\xi } \) is clearly a homomorphism. Suppose that \[ {\chi }_{\xi }\left( u\right) = 1\;\text{ for all }\;u \in D. \] This means that \( \xi \in {D}^{ * } \), and conversely. Hence our map is injective on the factor group \( {V}^{ * }/{D}^{ * } \) . To see surjectivity, suppose giv... | Yes |
Theorem 5.3. Let \( E \) be a non-degenerate Riemann form on \( \left( {V, D}\right) \), with associated hermitian form \( H \) . Let \( {\lambda }^{\prime } = {\lambda }_{E}^{\prime } \) . Then \( {\lambda }^{\prime } \) is the adjoint of \( \lambda \) with respect to \( H \), that is\n\n\[ H\left( {{\lambda x}, y}\ri... | Proof. Trivial computation as follows.\n\n\[ H\left( {{\lambda }^{\prime }x, y}\right) = H\left( {{\varphi }_{E}^{-1}{}^{t}\lambda {\varphi }_{E}\left( x\right), y}\right) = \left\langle {{}^{t}\lambda {\varphi }_{E}\left( x\right), y}\right\rangle \]\n\n\[ = \left\langle {{\varphi }_{E}\left( x\right) ,{\lambda y}}\ri... | Yes |
Theorem 6.1. Let \( \theta \) be an entire theta function with non-degenerate Riemann form E. Then we have a diagram which commutes with factor \( - 1 \) : | Proof. The theorem is obvious by putting together what we know. Start with an element \( x \in V \) . Its image \( {\varphi }_{E}\left( x\right) \) is characterized by the condition\n\n\[ H\left( {x, u}\right) = \left\langle {{\varphi }_{E}\left( x\right), u}\right\rangle \]\n\nThe corresponding character is given by e... | Yes |
Theorem 6.2. The following diagram is commutative: | Proof. From the equalities\n\n\[ \n{F}_{\xi } \circ \lambda \left( {{z}_{1} + {u}_{1}}\right) = {F}_{\xi }\left( {\lambda {z}_{1}}\right) {\psi }_{{F}_{{\xi }^{ \circ }}\lambda }\left( {u}_{1}\right) \]\n\n\[ \n= {F}_{\xi }\left( {\lambda {z}_{1} + \lambda {u}_{1}}\right) = {F}_{\xi }\left( {\lambda {z}_{1}}\right) {\p... | Yes |
Theorem 6.3. Let \( E \) be the Riemann form associated with \( \theta \) . Then the two involutions \( {\lambda }_{E}^{\prime } \) and \( {\lambda }_{\theta }^{\prime } \) are equal. | Proof. This is an immediate consequence of the preceding two theorems, and of the diagram:\n\n\n\nThe central square is commutative, and the two end triangles have sign -1 , which cancels. | Yes |
Theorem 8.1. The map \( {\rho }_{V} \) is a \( \mathbf{C} \) -linear isomorphism, inducing an isomorphism\n\n\[{\rho }_{A} : V/D \rightarrow {\widehat{\Omega }}_{1}/\widehat{D} = {\widehat{\Omega }}_{1}/{H}_{1}\left( {A,\mathbf{Z}}\right) . | Proof. For each \( j = 1,\ldots, n \) we have\n\n\[{\int }_{0}^{z}d{z}_{j} = {z}_{j}\]\n\nThus it is clear that the map is C-linear, and we have already seen above that it induces an injective map on \( D \) . This proves the theorem, since \( D \) generates \( V \) over \( \mathbf{R} \), and its image generates \( {\w... | No |
Theorem 8.2. The map\n\n\[ \nP \mapsto {\int }_{0}^{P}\left( {{\omega }_{1},\ldots ,{\omega }_{n}}\right) \]\n\nestablishes a complex analytic isomorphism of \( A = V/D \) with \( {\mathbf{C}}^{n}/\Lambda \), sending \( D \) on \( \Lambda \) . | Proof. The map is well defined, from \( V \) into \( {\mathbf{C}}^{n}/\Lambda \) . The above duality immediately implies that the kernel is \( D \) . The map is locally surjective in a neighborhood of the origin because \( d{z}_{1},\ldots, d{z}_{n} \) are local analytic coordinates, and it is surjective by additivity, ... | Yes |
Theorem 8.3. Let\n\n\\[ \nf : V/D \rightarrow {V}^{\prime }/{D}^{\prime }\n\\]\n\nbe a complex analytic map such that \\( f\\left( 0\\right) = 0 \\) . Then \\( f \\) is a homomorphism. | Proof. Let \\( {\\Omega }_{1}\\left( f\\right) : {\\Omega }_{1}\\left( {{V}^{\prime }/{D}^{\prime }}\\right) \n\rightarrow {\\Omega }_{1}\\left( {V/D}\\right) \\) be the induced linear map on holomorphic differential forms, obtained by pull back. We further get the dual map (linear)\n\n\\[ \n{\\widehat{\\Omega }}_{1}\\... | Yes |
Lemma 1.1. The symmetry of \( {}^{t}{CP} \) is equivalent with the condition\n\n\[ W{P}^{-{1t}}W = 0. \] | Proof. This symmetry is equivalent with the condition\n\n\[ - \left( \frac{W}{W}\right) C{P}^{-1}\left( {{}^{t}W,{}^{t}\bar{W}}\right) = \left( \frac{W}{W}\right) {P}^{-1}{}^{t}C\left( {{}^{t}W,{}^{t}\bar{W}}\right) . \]\n\nUsing once more the definition \( {iW} = {WC} \) and performing the matrix multiplications, yiel... | Yes |
Lemma 1.2. The matrix associated with \( H \) is\n\n\[ M = {2i}{\left( \bar{W}{P}^{-{1t}}W\right) }^{-1} > 0, \]\n\nand so for \( u, v \in {\mathbf{C}}^{n} \) we have\n\n\[ H\left( {u, v}\right) = {}^{t}{uM}\bar{v} \] | Proof. Let \( M \) be the matrix as indicated above. It is clear that \( M \) is hermitian. Thus it will suffice to show that\n\n\[ H\left( {u, u}\right) = {}^{t}{uM}\bar{u} \]\n\nLet \( {i}_{n} = i{1}_{n} \) be \( i \) times the unit \( n \times n \) matrix. Then\n\n\[ \left( \begin{matrix} {i}_{n} & 0 \\ 0 & - {i}_{n... | Yes |
Lemma 1.3. Let \( \left( {{\omega }_{1},{\omega }_{2}}\right) \in \mathcal{R} \) .\n\n(i) If \( g \in {\mathrm{{GL}}}_{n}\left( \mathbf{C}\right) \) then \( g\left( {{\omega }_{1},{\omega }_{2}}\right) \in \mathcal{R} \) .\n\n(ii) If \( M \in {\operatorname{Sp}}_{2n}\left( \mathbf{R}\right) \) then \( \left( {{\omega }... | Proof. The first two assertions are immediate from the definitions of \( \mathcal{R} \) , \( {\mathrm{{Sp}}}_{2n} \), and the Riemann relations. As for (iii), suppose there exists a vector \( v \in {\mathbf{C}}^{n} \) such that \( {}^{t}v{\omega }_{1} = 0 \) . Then\n\n\[ \n{}^{t}v\left( {{\omega }_{1}{}^{t}{\bar{\omega... | Yes |
Every isomorphism class of principally polarized abelian manifold contains a representative\n\n\\[ \n\\left( {{\\mathbf{C}}^{n},\\left( {z,{1}_{n}}\\right), J}\\right) \\;\\text{ with }\\;z \\in {\\mathfrak{H}}_{n},\n\\]\n\nfor which the columns of \\( \\left( {z,{1}_{n}}\\right) \\) form a Frobenius basis. | Proof. Let \\( W = \\left( {{\\omega }_{1},{\\omega }_{2}}\\right) \\) be a matrix satisfying the Riemann relations, and whose columns form a basis for the period lattice of the abelian manifold. We note:\n\nIf \\( {\\omega }_{2} = {1}_{n} \\), then the Riemann relations are equivalent with the property\n\nthat \\( {\\... | No |
Theorem 3.1. Let \( L = \left\lbrack {z,\delta }\right\rbrack \), where \( \delta = \operatorname{diag}\left( {{d}_{1},\ldots ,{d}_{n}}\right) \) with\n\n\[ 0 < {d}_{1}\left| {d}_{2}\right| \ldots \mid {d}_{n} \]\n\nLet j range over a complete system of representatives for \( {\delta }^{-1}{\mathbf{Z}}^{n}/{\mathbf{Z}}... | Proof. After multiplying by the inverse of a character, and a trivial theta function, we are reduced to proving the equivalent statement that the functions\n\n\[ \theta \left( {u, z, r + j, s}\right) \]\n\nform a basis for the space \( \operatorname{Th}\left( {z,\delta }\right) \) of entire functions such that\n\n\[ \f... | Yes |
Theorem 1.1. The map \( \alpha \mapsto {\alpha }^{\prime } \) is an involution of \( Q \), that is an anti-automorphism of order 2. | Proof. The map is obviously linear. It suffices to prove the property\n\n\[ \n{\left( \alpha \beta \right) }^{\prime } = {\beta }^{\prime }{\alpha }^{\prime } \n\]\n\nwhen \( \alpha ,\beta \) are \( 2 \times 2 \) matrices, and the trace, norm are the ordinary trace and determinant. By specialization, it suffices to pro... | Yes |
Proposition 1.2. Every inner automorphism of \( Q \) commutes with the involution. | Proof. Immediate from the fact that \( \alpha {\alpha }^{\prime } \) is an element of \( k \), and so commutes with all elements of \( Q \) . | Yes |
Theorem 1.3. Let \( F \) be a separable quadratic extension of \( k \) contained in \( Q \) . Let \( \varphi : F \mapsto Q \) be a \( k \) -linear embedding which is not the identity. Then there exists an inner automorphism of \( Q \) which induces \( \varphi \) on \( F \) . In particular, every automorphism of \( Q \)... | Proof. We shall need the remark that there is an isomorphism\n\n\[ Q \otimes Q\overset{ \approx }{ \rightarrow }{\operatorname{End}}_{k}\left( {Q}_{\mathrm{{vs}}}\right) \]\n\nwhere \( {Q}_{\mathrm{{vs}}} \) denotes \( Q \) viewed as a vector space over \( k \) . The isomorphism is given as follows. An element \( \sum ... | Yes |
Theorem 1.4. The map \( \alpha \mapsto {\alpha }^{ * } \) is an involution if and only if \( {\gamma }^{2} \in k \) . Every involution is of this type, for some \( \gamma \) . | Proof. Since \( {\alpha }^{* * } = {\gamma }^{-2}\alpha {\gamma }^{2} \), and \( {\alpha }^{* * } = \alpha \) for all \( \alpha \in Q \) if and only if \( {\gamma }^{2} \in k \) (because \( k \) is the center), the first assertion is clear. Conversely, let \( \alpha \mapsto {\alpha }^{ * } \) be an involution. Then \( ... | Yes |
Theorem 2.1. Let \( F \) be a subfield of degree 2 over \( k, F = k\left( \beta \right) \) , \( {\beta }^{2} = b \in k \) . Then:\n\n(i) There is a basis \( 1,\beta ,\gamma ,{\beta \gamma } \) of \( Q \) over \( k \) such that \( \gamma \) is invertible, and\n\n\[{\gamma }^{2} = c \in k,\;{\beta \gamma } = - {\gamma \b... | Proof. The second statement is immediate in view of the commutation rules between \( \beta \) and \( \gamma \), and the fact that the canonical involution induces the non-trivial automorphisms of \( k\left( \beta \right) \) and \( k\left( \gamma \right) \) respectively, so \( {\beta }^{\prime } = - \beta \) and \( {\ga... | No |
Theorem 2.2. The algebra \( Q \) is indefinite if and only if \( Q \) contains a real quadratic subfield. If \( Q = \left( {b, c}\right) \), this is the case if and only if \( b > 0 \) or \( c > 0 \) . | Proof. If \( Q \) contains a real quadratic subfield \( F \), then \( F \otimes Q \approx {\mathbf{M}}_{2}\left( F\right) \) so the splitting is clear. Conversely, suppose \( {Q}_{\mathbf{R}} \approx {\mathbf{M}}_{2}\left( \mathbf{R}\right) \) . Suppose \( b < 0 \) and \( c < 0 \) . Then \( b = - {b}_{1}^{2} \) and \( ... | Yes |
Theorem 2.3. Assume that \( k = \mathbf{Q} \) and that \( Q \) is indefinite. Let \( * \) be the involution defined by an element \( \gamma \) . Then \( \operatorname{tr}\left( {\alpha {\alpha }^{ * }}\right) > 0 \) for all \( \alpha \in Q \) , \( \alpha \neq 0 \) if and only if \( {\gamma }^{2} < 0 \) . | Proof. Let \( Q = \left( {b, c}\right) \) with \( {\gamma }^{2} = c,{\beta }^{2} = b \) . Any element \( \alpha \) can be written in the form\n\n\[ \alpha = x + {y\beta }\text{with}x, y \in k\left( \gamma \right) \text{.} \]\n\nA trivial computation using the commutation rule between \( \beta \) and \( \gamma \) shows ... | Yes |
Lemma 4.1. Let\n\n\\[ \nw = \\left( \\begin{array}{l} {w}_{1} \\ {w}_{2} \\end{array}\\right) \\in {\\mathbf{C}}^{2} \\]\n\nLet \\( {\\alpha }_{1},\\ldots ,{\\alpha }_{4} \\in {\\mathbf{M}}_{2}\\left( \\mathbf{R}\\right) \\) be linearly independent over \\( \\mathbf{R} \\) . Let \\( L \\) be the \\( \\mathbf{Z} \\) -mo... | Proof. If \\( {w}_{1} \\) or \\( {w}_{2} = 0 \\), or if \\( {w}_{1},{w}_{2} \\) are real multiples of each other, then there is some real linear combination\n\n\\[ \n\\alpha = \\mathop{\\sum }\\limits_{{i = 1}}^{4}{c}_{i}{\\alpha }_{i},\\;{c}_{i} \\in \\mathbf{R},\\text{ not all }{c}_{i} = 0\n\\]\n\nsuch that \\( {\\al... | Yes |
Theorem 4.2. Let \( A = {\mathbf{C}}^{2}/\Lambda \) be a complex torus of dimension 2, such that \( \left( {A,\iota }\right) \) is of type \( \left( {Q,\rho }\right) \). Let \( u \in \Lambda, u \neq 0 \). Then \( u \) is non-degenerate. There exists a lattice \( \mathfrak{a} \) in \( Q \) such that \( \Lambda = \rho \l... | The converse statement in the theorem is obvious. If we want to give all the data in the notation, we shall say that \( \left( {A,\iota }\right) \) is of type \( \left( {Q,\rho ,\alpha, u}\right) \) with respect to \( \Theta \). We often omit \( \Theta \) from the notation, and identify \( A \) with \( {\mathbf{C}}^{2}... | No |
Theorem 4.3. Let \( {\mathbf{C}}^{2}/\Lambda \) be of type \( \left( {Q,\rho ,\alpha, u}\right) \) . Let \( E \) be a \( \rho \) -admissible Riemann form on \( {\mathbf{C}}^{2}/\Lambda \) . Then there exists a rational number \( c \) such that\n\n\[ E\left( {\rho \left( \alpha \right) u,\rho \left( \beta \right) u}\rig... | Proof. The map \( \alpha \mapsto E\left( {\rho \left( 1\right) u,\rho \left( \alpha \right) u}\right) \) is a \( \mathbf{Q} \) -linear functional on \( Q \), so there exists \( \xi \in Q \) such that\n\n\[ E\left( {\rho \left( 1\right) u,\rho \left( \alpha \right) u}\right) = \operatorname{tr}\left( {\xi \alpha }\right... | Yes |
Theorem 5.1. Assume that \( \mathfrak{a} = \mathfrak{o} \) . Then \( \left( {A\left( {\tau }_{1}\right) ,\rho }\right) \) and \( \left( {A\left( {\tau }_{2}\right) ,\rho }\right) \) are isomorphic if and only if there exists a unit \( \epsilon \) in \( \mathfrak{o} \) with \( \operatorname{nr}\left( \epsilon \right) = ... | Proof. In the discussion preceding the theorem, \( h \) is an isomorphism if and only if \( {M\Lambda }\left( {\tau }_{1}\right) = \Lambda \left( {\tau }_{2}\right) \), or equivalently\n\n\[ \rho \left( \mathfrak{o}\right) \rho \left( \lambda \right) = \rho \left( \mathfrak{o}\right) \]\n\nThis is equivalent with \( \m... | Yes |
Theorem 1.1. Let \( \bar{X} \) be a positive divisor on \( {\mathbf{C}}^{n}/D \), and let \( X \) be its inverse image on \( {\mathbf{C}}^{n} \) . Then there exists an entire theta function \( F \) representing this divisor on \( {\mathbf{C}}^{n} \) . | Proof. The proof will be carried out by juggling with differential forms, and reproving ad hoc some results valid on Kähler manifolds. Everything becomes much simpler because we work on the torus and \( {\mathbf{C}}^{n} \) . | No |
Lemma 1. Let \( M \) be a \( {C}^{\infty } \) manifold, and \( \left\{ {U}_{i}\right\} \) a locally finite open covering. For each pair \( \left( {i, j}\right) \) such that \( {U}_{i} \cap {U}_{j} \) is not empty, suppose given a differential form \( {\omega }_{ij} \) of degree \( p \), satisfying\n\n\[ \n{\omega }_{ij... | Proof. Let \( \left\{ {g}_{i}\right\} \) be a partition of unity subordinated to the given covering. We let\n\n\[ \n{\omega }_{i} = \mathop{\sum }\limits_{j}{g}_{j}{\omega }_{ij} \n\] \n\nwith the obvious convention that the expression on the right is equal to 0 wherever it is not defined. Using the cocycle equation, a... | Yes |
Lemma 2. Let \( \left( {a}_{ij}\right) \) be a real symmetric positive definite matrix. Let\n\n\[ \Delta = \sum {a}_{ij}\frac{{\partial }^{2}}{\partial {x}_{i}\partial {x}_{j}} \]\n\nLet \( \omega \) be a p-form on the torus. There exists a p-form \( \psi \) on the torus such that \( {\Delta \psi } = \omega \) if and o... | Proof. Since our operators \( I \) and \( \Delta \) actually operate on functions, we can just deal with functions \( f \) on the torus, viewed as periodic functions on \( {\mathbf{R}}^{m} \) . Since these functions are assumed to be \( {C}^{\infty } \), they have Fourier expansions which converge rapidly to 0 , as one... | Yes |
A subset \( C \) of \( {\mathbb{R}}^{d} \) is convex if and only if any convex combination of points from \( C \) is again in \( C \) . | Proof. If any convex combination of points from \( C \) is again in \( C \), then, in particular, any convex combination of two points from \( C \) is in \( C \) . Therefore, \( C \) is convex.\n\nConversely, assume that \( C \) is convex. We shall prove by induction on \( n \) that any point from \( {\mathbb{R}}^{d} \... | Yes |
For any subset \( M \) of \( {\mathbb{R}}^{d} \), the convex hull conv \( M \) is the set of all convex combinations of points from \( M \) . | Let \( C \) denote the set of all convex combinations of points from \( M \) . Since \( M \subset \operatorname{conv}M \), each \( x \in C \) is also a convex combination of points from the convex set conv \( M \) ; the \ | No |
For any subset \( M \) of \( {\mathbb{R}}^{d} \), the convex hull conv \( M \) is the set of all convex combinations\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{n}{\lambda }_{i}{x}_{i} \]\n\nsuch that \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) is an affinely independent family of points from \( M \) . | Proof. We shall prove that if a point \( x \) is a convex combination of \( n \) points \( {x}_{1},\ldots ,{x}_{n} \) such that \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) is affinely dependent, then \( x \) is already a convex combination of \( n - 1 \) of the points \( {x}_{1},\ldots ,{x}_{n} \) . Repeating this ar... | Yes |
Corollary 2.4. For any subset \( M \) of \( {\mathbb{R}}^{d} \) with \( \dim \left( {\operatorname{aff}M}\right) = n \), the convex hull conv \( M \) is the set of all convex combinations of at most \( n + 1 \) points from \( M \) . | Proof. For any affinely independent \( m \) -family \( \left( {{x}_{1},\ldots ,{x}_{m}}\right) \) of points from \( M \) , we have \( m \leq n + 1 \) by the assumption. Therefore, the set of all convex combinations of \( n + 1 \) or fewer points from \( M \) contains conv \( M \) by Theorem 2.3. On the other hand, it i... | Yes |
For any subset \( M \) of \( {\mathbb{R}}^{d} \) with \( \dim \left( {\operatorname{aff}M}\right) = n \), the convex hull conv \( M \) is the set of all convex combinations of precisely \( n + 1 \) points from \( M \) . | Proof. In a convex combination one may always add terms of the form \( {0x} \) . Therefore, the statement follows from Corollary 2.4. | No |
Theorem 2.6. Let \( M = \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) be a finite set of \( n \) points from \( {\mathbb{R}}^{d} \) such that the \( n \) -family \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) is affinely dependent. Then there are subsets \( {M}_{1} \) and \( {M}_{2} \) of \( M \) with \( {M}_{1} \cap {M}... | Proof. By the affine dependence there are reals \( {\lambda }_{1},\ldots ,{\lambda }_{n} \), not all 0, such that\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{n}{\lambda }_{i}{x}_{i} = o,\;\mathop{\sum }\limits_{{i = 1}}^{n}{\lambda }_{i} = 0. \]\n\n(4)\n\nDenoting the set \( \{ 1,\ldots, n\} \) by \( I \), we let\n\n\[ {I}_... | Yes |
Corollary 2.7. Let \( M = \left\{ {{x}_{1},\ldots ,{x}_{n}}\right\} \) be a finite set of \( n \) points from \( {\mathbb{R}}^{d} \) such that \( n \geq d + 2 \) . Then there are subsets \( {M}_{1} \) and \( {M}_{2} \) of \( M \) with \( {M}_{1} \cap {M}_{2} = \varnothing \) and \( {M}_{1} \cup {M}_{2} = M \) such that... | Proof. The maximum number of members in an affinely independent family of points from \( {\mathbb{R}}^{d} \) is \( d + 1 \) . Therefore, \( \left( {{x}_{1},\ldots ,{x}_{n}}\right) \) must be affinely dependent, whence Theorem 2.6 applies. | No |
Corollary 2.9. Any convex polytope \( P \) in \( {\mathbb{R}}^{d} \) is a compact set. | One should observe, however, that a direct proof of Corollary 2.9 does not require Carathéodory’s Theorem. In fact, if \( M \) is the finite set \( \left\{ {{x}_{1},\ldots ,{x}_{m}}\right\} \) , then each \( {y}_{v} \) (in the notation of the proof above) has a representation\n\n\[ \n{y}_{v} = \mathop{\sum }\limits_{{i... | Yes |
Theorem 3.1. Let \( C \) be any non-empty convex set in \( {\mathbb{R}}^{d} \) . Then \( \operatorname{ri}C \neq \varnothing \) . | We first prove Theorem 3.1 for simplices:\n\nLemma 3.2. Let \( S \) | No |
Lemma 3.2. Let \( S \) be a simplex in \( {\mathbb{R}}^{d} \). Then \( \operatorname{ri}S \neq \varnothing \). | Proof. When \( \dim S = k \), there is a \( \left( {k + 1}\right) \)-family \( \left( {{x}_{1},\ldots ,{x}_{k + 1}}\right) \), affinely independent, such that\n\n\[ S = \operatorname{conv}\left\{ {{x}_{1},\ldots ,{x}_{k + 1}}\right\} \]\n\nThen \( \left( {{x}_{1},\ldots ,{x}_{k + 1}}\right) \) is an affine basis of aff... | Yes |
Theorem 3.3. Let \( C \) be a convex set in \( {\mathbb{R}}^{d} \). Then for any \( {x}_{0} \in \mathrm{{ri}}C \) and any \( {x}_{1} \in \operatorname{cl}C \) with \( {x}_{0} \neq {x}_{1} \) we have \( \left\lbrack {{x}_{0},{x}_{1}\lbrack \subset }\right. \) ri \( C \) . | Proof. It is easy to prove the statement in the particular case where we have \( {x}_{0} \in \operatorname{int}C \) and \( {x}_{1} \in C \). For \( \lambda \in \rbrack 0,1\left\lbrack \right. \), let \( {x}_{\lambda } \mathrel{\text{:=}} \left( {1 - \lambda }\right) {x}_{0} + \lambda {x}_{1} \). From \( {x}_{0} \in \) ... | Yes |
Theorem 3.4. For any convex set \( C \) in \( {\mathbb{R}}^{d} \) one has:\n\n(a) \( \mathrm{{cl}}C \) is convex. | Proof. For \( C = \varnothing \), there is nothing to prove. So, we may assume that \( C \) is non-empty, whenever necessary.\n\n(a) Let \( {x}_{0},{x}_{1} \in \operatorname{cl}C \), and let \( \lambda \in \rbrack 0,1\lbrack \) . We shall prove that the point\n\n\[ \n{x}_{\lambda } \mathrel{\text{:=}} \left( {1 - \lamb... | Yes |
Theorem 3.5. For any convex set \( C \) in \( {\mathbb{R}}^{d} \) and any point \( x \in C \) the following three conditions are equivalent:\n\n(a) \( x \in \operatorname{ri}C \) .\n\n(b) For any line \( A \) in aff \( C \) with \( x \in A \) there are points \( {y}_{0},{y}_{1} \in A \cap C \) such that \( x \in \rbrac... | Proof. The implications (a) \( \Rightarrow \) (b) and (b) \( \Rightarrow \) (c) are obvious. Therefore, we need only prove (c) \( \Rightarrow \) (a). By Theorem 3.1 there is a point \( y \in \mathrm{{ri}}C \) . If \( y = x \), there is nothing more to prove. If \( y \neq x \), then by (c) there is a point \( z \in C \)... | Yes |
Theorem 3.6. Let \( M \) be any subset of \( {\mathbb{R}}^{d} \). Then\n\n\[ \text{clconv}M = \operatorname{cl}\left( {\operatorname{conv}M}\right) \text{,}\]\n\ni.e. the closed convex hull of \( M \) is the closure of the convex hull of \( M \). | Proof. Using Theorem 3.4(a) we see that \( \operatorname{cl}\left( {\operatorname{conv}M}\right) \) is a closed convex set containing \( M \). Since clconv \( M \) is the smallest such set, it follows that\n\n\[ \text{clconv}M \subset \operatorname{cl}\left( {\operatorname{conv}M}\right) \text{.}\]\n\nOn the other hand... | Yes |
Theorem 4.1. Let \( C \) be a non-empty convex set in \( {\mathbb{R}}^{d} \), and let \( H \) be a hyperplane in \( {\mathbb{R}}^{d} \). Then the following two conditions are equivalent:\n\n(a) \( H \cap \operatorname{ri}C = \varnothing \).\n\n(b) \( C \) is contained in one of the two closed halfspaces bounded by \( H... | Proof. Assume that (a) holds. Let \( {x}_{0} \in \mathrm{{ri}}C \), cf. Theorem 3.1. Then \( {x}_{0} \notin H \) by (a). In particular, \( C \) is not contained in \( H \). Suppose that there is a point \( {x}_{1} \in C \) such that \( {x}_{0} \) and \( {x}_{1} \) are on opposite sides of \( H \). Then there are \( y \... | Yes |
Lemma 4.4. Let \( C \) be a non-empty open convex set in \( {\mathbb{R}}^{d} \), and let \( x \) be a point of \( {\mathbb{R}}^{d} \) not in \( C \) . Then there is a hyperplane \( H \) in \( {\mathbb{R}}^{d} \) such that \( x \in H \) and \( H \cap C = \varnothing \) . | Proof. We shall use induction on \( d \) . The statement is trivially true for \( d = 0,1 \) . We also need a proof for \( d = 2 \), however. So, let \( C \) be a non-empty open convex set in \( {\mathbb{R}}^{2} \), and let \( x \in {\mathbb{R}}^{2} \smallsetminus C \) . We shall prove that there exists a line \( L \) ... | No |
Theorem 4.5. Let \( C \) be a non-empty closed convex set in \( {\mathbb{R}}^{d} \). Then \( C \) is the intersection of its supporting halfspaces. | Proof. When \( \dim C = 0 \), the theorem is clearly true. When \( C = {\mathbb{R}}^{d} \), there are no supporting halfspaces; hence, the theorem is also true in this case. So, let \( \dim C \geq 1 \), and let \( \mathrm{x} \) be a point of \( {\mathbb{R}}^{d} \) outside \( C \); we shall prove that there is a support... | Yes |
Theorem 5.1. Every face \( F \) of a closed convex set \( C \) in \( {\mathbb{R}}^{d} \) is closed. | Proof. For \( \dim F = - 1,0 \) there is nothing to prove. Assume that \( \dim F \geq 1 \) , and let \( x \) be any point in cl \( F \) . Let \( {x}_{0} \) be a point in ri \( F \), cf. Theorem 3.1. If \( x = {x}_{0} \), we have \( x \in F \) as desired. If \( x \neq {x}_{0} \), then \( \left\lbrack {{x}_{0}, x\lbrack ... | No |
Theorem 5.2. Let \( F \) be a face of a closed convex set \( C \) in \( {\mathbb{R}}^{d} \), and let \( G \) be a subset of \( F \) . Then \( G \) is a face of \( C \) if (and only if) \( G \) is a face of \( F \) . | Proof. It follows immediately from the definition that if the set \( G \) is a face of \( C \), then it is also a face of \( F \). Conversely, suppose that \( G \) is a face of \( F \), and let \( y \) and \( z \) be points of \( C \) such that \( \rbrack y, z\lbrack \) intersects \( G \). Since \( G \subset F \), the ... | Yes |
Theorem 5.3. Let \( F \) be a face of a closed convex set \( C \) in \( {\mathbb{R}}^{d} \) such that \( F \neq C \) . Then \( F \subset \mathrm{{rb}}C \) . | Proof. For \( \dim C = - 1,0 \) there is nothing to prove. So, assume that we have \( \dim C \geq 1 \) . Let \( F \) be a face of \( C \) such that some point \( x \) from \( F \) is in \( \operatorname{ri}C \) . We shall complete the proof by showing that \( F = C \) . Let \( y \) be an arbitrary point in \( C \) . If... | No |
Corollary 5.4. Let \( F \) and \( G \) be faces of a closed convex set \( C \) in \( {\mathbb{R}}^{d} \) such that \( G \subsetneq F \) . Then \( G \subset \operatorname{rb}F \) . | Proof. First note that \( G \) is a face of \( F \), cf. Theorem 5.2. The statement then follows immediately from Theorem 5.3. | No |
Corollary 5.5. Let \( F \) and \( G \) be faces of a closed convex set \( C \) in \( {\mathbb{R}}^{d} \) such that \( G \subsetneq F \) . Then \( \dim G < \dim F \) . | Proof. First note that we have aff \( G \subset \operatorname{aff}F \) since \( G \subset F \) . Suppose that aff \( G = \) aff \( F \) . Then ri \( G \subset \) ri \( F \) since \( G \subset F \) . Combining with Corollary 5.4 we obtain ri \( G = \varnothing \) . By Theorem 3.1 this implies \( G = \varnothing \), when... | Yes |
Theorem 5.6. Let \( C \) be a closed convex set in \( {\mathbb{R}}^{d} \), let \( x \) be a point in \( C \), and let \( F \) be a face of \( C \) containing \( x \) . Then \( F \) is the smallest face of \( C \) containing \( x \) if and only if \( x \in \mathrm{{ri}}F \) . | Proof. If \( x \in \mathrm{{ri}}F \), then \( F \) is the smallest face containing \( x \) by Corollary 5.4. If \( x \in \mathrm{{rb}}F \), then by Theorem 4.3 there is a face \( G \) (in fact, exposed) of \( F \) such that \( x \in G \subsetneq F \) . By Theorem 5.2, \( G \) is also a face of \( C \), and therefore \(... | Yes |
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