Index Ap´ ery number, 199 Asymptotic formula for p(n), 86 Atkin-Lehner operator, 27, 51 Bailey pair, 167 Bernoulli numbers, 5 generalized, 14 Beukers’ supercongruence, 199 Birch and Swinnerton-Dyer Conjecture, 151 Borcherds’ isomorphism, 71 CM point, 70 Calabi-Yau threefold, 195 Central critical L-value, 150 Class equation, 130 Class number, 71, 133 Cohen and Lenstra heuristics, 134 Cohen-Eisenstein series, 15 Congruence subgroup, 1 Congruent number, 155 Crank of a partition, 89 Cusp, 2 Dedekind’s eta-function, 17 Degree of nilpotency, 34 Delta-function, 7 Denominator formula for the Monster, 25 Dimension formulas, 10, 16 Divisor function, 5 Divisor polynomial, 32 Dyson’s rank of a partition, 88 Eisenstein series, 5 Elliptic curve, 36, 150, 187, 190 congruent number, 151, 155, 157, 188 Legendre normal form, 187 quadratic twist of, 152 supersingular, 36 with complex multiplication, 188 Elliptic fixed point, 2 Eta-product, 18 Eta-quotient, 18 Euler’s generating function, 85 Filtration, 35 Fourier expansion, 4 Fricke involution, 27, 51 Fundamental domain of SL2( ), 1 Galois representation, 42, 63 Gaussian hypergeometric function, 184 Gauss sum, 201 Goldfeld’s Conjecture, 152 Gross-Koblitz formula, 201 Hecke eigenform, 22, 50 Hecke Gr¨ ossencharacter, 9 Hecke operator, 21, 49, 55 Hecke trace of singular moduli, 124 Heegner divisor, 70 Heine transformation, 166 Hurwitz class number, 71, 133 Hurwitz zeta-function, 175 Hypergeometric function, 166, 184 basic, 166 Gaussian, 184 ordinary, 183 Iwasawa λ-invariant, 139 Iwasawa µ-invariant, 139 j-function, 7 Jacobi sum, 184 Jacobi’s triple product identity, 17 K3 surface, 193 Kohnen newform, 55 Kohnen operator, 52 Kohnen plus space, 54 Koike’s Conjecture, 197 Kolyvagin’s Theorem, 151 L-function, 14 Dirichlet, 14 Hasse-Weil, 150 Kubota-Leopoldt, 134 modular, 149 Legendre normal form, 187 Lehmer’s Conjecture, 41 Modular curve, 109 215

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