Index abelian surface, 196 abelian variety, 41, 166 Abramovich, D., 39 accumulating subvariety, 9, 36, 195, 196, 199 accumulating surface, 196, 199 adelic intersection product, 33 adelic metric, 27, 31, 45, 46 adelic Picard group, 25, 27, 30, 33 adelic point, 4–8, 10, 56, 61, 119–121, 124, 137, 145, 206, 208, 231 adelic topology, 121 adelically metrized invertible sheaf, 27–31, 33 adjunction formula, 55 aﬃne cone, 54 algebraic Brauer–Manin obstruction, 5, 132 algebraic surfaces, 98, 106 classification of, 167 algorithm FFT point counting, 190, 192, 200 for computation of volume, 50 for numerical integration, 193 obvious, 189 of Tate, 159 to compute approximate value for Peyre’s constant, 192 to detect conics on quartic threefold, 198 to detect lines on cubic threefold, 196, 200 to search for solutions of Diophantine equation, 9, 169–182, 200 naive, 169 to solve system of equations over finite field, 196, 198 to test for conic through two points, 197 to test lines for irrationality, 195 to test lines for irrationality, 196 to test whether conic is contained in quartic threefold, 198 almost Fano variety, 66 AMD, 182 Amir-Moez, A. R., 83 Arakelov, S. Yu., 25 Arakelov degree, 6, 22 Archimedean valuation, 17 ARIBAS, 183 arithmetic cohomology, 83 arithmetic degree, 30 arithmetic intersection theory, 21 arithmetic Picard group, 21 arithmetic variety, 6, 21, 23–25 Artin conductor, 215, 236 Artin L-function, vii, 51, 207, 208, 216, 222, 231 assembly, 179 Auslander–Goldman, theorem of, 7, 98, 103, 105 automorphisms of PN , 20 Azumaya algebra, 83, 89–98, 104–106, 123, 124, 128 bad prime, 192 reduction, 153 Batyrev, V. V., 33, 40, 66, 78, 79 Batyrev and Manin, conjecture of, 6, 38, 40–43, 166, 196 Bernstein, D., 166, 199 Bezout’s theorem, 194 Bhowmik, G., 80 bielliptic surface, 166, 196 Bierce, A., 165 Billard, H., 41 biquadratic reciprocity low, 4 Bogomolov, F., 42 conjecture of, 42 Bombieri, E., 47 Bourbaki, N., 29 Brauer, R., 84 261

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