By Nicholas M. Katz

ISBN-10: 0691083525

ISBN-13: 9780691083520

This paintings is a finished remedy of modern advancements within the research of elliptic curves and their moduli areas. The mathematics examine of the moduli areas started with Jacobi's "Fundamenta Nova" in 1829, and the fashionable thought used to be erected by way of Eichler-Shimura, Igusa, and Deligne-Rapoport. some time past decade mathematicians have made additional huge development within the box. This publication offers an entire account of that development, together with not just the paintings of the authors, but in addition that of Deligne and Drinfeld.

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**Extra resources for Arithmetic Moduli of Elliptic Curves.**

**Example text**

Remember that division by zero is undeﬁned! Dividing Integers 1. The quotient of two positive integers or two negative integers is the quotient of their absolute values. 2. The quotient of a positive integer and a negative integer (or a negative and a positive) is the opposite of the quotient of their absolute values. 3. The quotient of zero and any nonzero integer (zero divided by any nonzero integer) is zero. The next example illustrates this description of division. EXAMPLE 4 Find the quotient of the integers.

EXAMPLE 4 ■ Find the greatest common factor of 24 and 35. Solution 24 ϭ 2 # 2 # 2 # 3 ϭ 23 # 3 35 ϭ 5 # 7 Because there are no common prime factors, the greatest common factor is 1. ▼ PRACTICE YOUR SKILL Find the greatest common factor of 49 and 80. ■ The concept of “greatest common factor” can be extended to more than two numbers, as the next example demonstrates. EXAMPLE 5 Find the greatest common factor of 24, 28, and 120. Solution 24 ϭ 4 # 6 ϭ 2 # 2 # 2 # 3 ϭ 23 # 3 28 ϭ 4 # 7 ϭ 2 # 2 # 7 ϭ 22 # 7 120 ϭ 10 # 12 ϭ 2 # 5 # 2 # 6 ϭ 2 # 5 # 2 # 2 # 3 ϭ 23 # 3 # 5 The only factor that is common to all three numbers is 2.

3 3 3 3 3 3 3 a ba b a ba b 4 2 4 2 4 4 2 3 3 9 ϭ ± ≤ ± ≤ ϭ ϭ a ba b ϭ ϭ 2 2 3 1 4 2 8 3 2 a ba b 3 3 2 3 2 Notice that this is a form of 1 and 3 2 is the reciprocal of 2 3 3 3 2 3 divided by is equivalent to times . 5 Division of Rational Numbers If b, c, and d are nonzero integers, and a is any integer, then c a a Ϭ ϭ b d b # d c c a c a d by , we multiply times the reciprocal of , which is . The c b d b d following example demonstrates the important steps of a division problem. Note that to divide EXAMPLE 8 Find the quotient.

### Arithmetic Moduli of Elliptic Curves. by Nicholas M. Katz

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