Rational points on Shimura curves and Galois representations Carlos de Vera Piquero

... models of the affine quartic curve 2y 2 = 1 − 17x4 , due to Lind and Reichardt, and of Selmer’s cubic 3x3 +4y 3 = 5. These curves have rational points locally everywhere, but fail to have global Qrational points, thus they are counterexamples to the Hasse principle (see [Lin40, Rei42, Sel51]). Genus ...

... models of the affine quartic curve 2y 2 = 1 − 17x4 , due to Lind and Reichardt, and of Selmer’s cubic 3x3 +4y 3 = 5. These curves have rational points locally everywhere, but fail to have global Qrational points, thus they are counterexamples to the Hasse principle (see [Lin40, Rei42, Sel51]). Genus ...

1314Summer14.pdf

... Note that the values of ln(x) are all irrational numbers (other than when x = 1) while the values of log2(x) are rational numbers when the argument (in this case, the input value) is a power of two. Since ln(x) takes irrational values for integer inputs, a quick means for approximating ln(x) would b ...

... Note that the values of ln(x) are all irrational numbers (other than when x = 1) while the values of log2(x) are rational numbers when the argument (in this case, the input value) is a power of two. Since ln(x) takes irrational values for integer inputs, a quick means for approximating ln(x) would b ...

presentation

... The free product of two groups G and H is denoted as G ∗ H and is defined as the set of formal strings whose letters are elements from G and elements from H modulo the product relations in G and in H. The collection of isomorphism classes of all groups forms a commutative monoid with respect to the ...

... The free product of two groups G and H is denoted as G ∗ H and is defined as the set of formal strings whose letters are elements from G and elements from H modulo the product relations in G and in H. The collection of isomorphism classes of all groups forms a commutative monoid with respect to the ...

Some structure theorems for algebraic groups

... (3) Gk̄ is reduced at e. Proof. Clearly, (1)⇒(2)⇒(3). We now show that (3)⇒(1). For this, we may replace G with Gk̄ and hence assume that k is algebraically closed. Observe that for any g ∈ G(k), the local ring OG,g is isomorphic to OG,e as the left multiplication by g in G is an automorphism that s ...

... (3) Gk̄ is reduced at e. Proof. Clearly, (1)⇒(2)⇒(3). We now show that (3)⇒(1). For this, we may replace G with Gk̄ and hence assume that k is algebraically closed. Observe that for any g ∈ G(k), the local ring OG,g is isomorphic to OG,e as the left multiplication by g in G is an automorphism that s ...