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