By Nicholas M. Katz

This paintings is a finished remedy of contemporary advancements within the research of elliptic curves and their moduli areas. The mathematics research of the moduli areas all started with Jacobi's "Fundamenta Nova" in 1829, and the fashionable idea was once erected by means of Eichler-Shimura, Igusa, and Deligne-Rapoport. long ago decade mathematicians have made additional gigantic growth within the box. This ebook supplies an entire account of that growth, together with not just the paintings of the authors, but in addition that of Deligne and Drinfeld.

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68 X 10-- Applications 48. 5 X loJ Exercises 77--80: Percent Change If an amount changes from A to a new amount B, then the percent change is 100. 1 Calculate the percent change for the givenA and B. Rowul your answer to the nearest tenth of a percent when appropriate. 77. A = $8, B = $13 78. 47 79. 85 80. A = 1256, B = 1195 81. Percent Change Suppose that tuition is initially $100 per credit and increases by 6% from the first year to the second year. What is the cost of tuition the second year?

5 X 20 = 30 miles south, and so it is located 50 miles north of the initial location of car B. Thus its coordinates are (0, 50) at 1:30 PM. 5 X 40 = 60 miles cast, so its coordinates are (60, 0) at 1:30 PM. 17, where these points are plotted. M. we must find the distanced between the points (0, 50) and (60, 0 ). 1 miles Now Try Execcise 31 .... 18 M ~ A common way to make estimations is to average data values. For example, in 1980 the average cost of tuition and fees at public colleges and universities was $800, whereas in 1982 it was $1000.

1f a line segment is drawn between two data points, then its midpoint is the unique point on the line segment that is equidistant from the endpoints. 18. For example, the midpoint of -3 and 5isM=~= 1. 2 Visualizing and Graphing Data I 17 '*IlA Finding the distance between two points ,:t Find the exact distance bet\w:en (3, -4) and ( -2, 7). Then approximate this distance to the nearest hundredth. SOW110N In the distance formula, let (x~o y1) be (3, -4) and (x2, y2) be (-2, 7). V(x2 - xS1 + (y2 - y 1f = = = V(-2 - 3) 2 + (7 - (-4)f Dl&tanceformula \l(-5f + 112 v'146 Su171;ract.