By Fred Diamond
This ebook introduces the speculation of modular varieties, from which all rational elliptic curves come up, with a watch towards the Modularity Theorem. dialogue covers elliptic curves as advanced tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner concept; Hecke eigenforms and their mathematics houses; the Jacobians of modular curves and the Abelian forms linked to Hecke eigenforms. because it offers those principles, the publication states the Modularity Theorem in a number of types, concerning them to one another and concerning their functions to quantity conception. The authors suppose no heritage in algebraic quantity idea and algebraic geometry. workouts are included.
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Additional resources for A First Course in Modular Forms (Graduate Texts in Mathematics, Vol. 228)
2. five Modular curves and Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . forty five forty five forty eight fifty two fifty seven sixty three three size formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sixty five three. 1 The genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sixty five three. 2 Automorphic types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seventy one three. three Meromorphic differentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . seventy seven three. four Divisors and the Riemann–Roch Theorem . . . . . . . . . . . . . . . . . . eighty three three. five measurement formulation for even ok . . . . . . . . . . . . . . . . . . . . . . . . . . . . eighty five three. 6 measurement formulation for ordinary okay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 three. 7 extra on elliptic issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety two three. eight extra on cusps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ninety eight three. nine extra measurement formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 four Eisenstein sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 four. 1 Eisenstein sequence for SL2 (Z) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 four. 2 Eisenstein sequence for Γ (N ) while ok ≥ three . . . . . . . . . . . . . . . . . . . . . . 111 four. three Dirichlet characters, Gauss sums, and eigenspaces . . . . . . . . . . . 116 xiv Contents four. four four. five four. 6 four. 7 four. eight four. nine four. 10 four. eleven Gamma, zeta, and L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . one hundred twenty Eisenstein sequence for the eigenspaces while ok ≥ three . . . . . . . . . . . . . 126 Eisenstein sequence of weight 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . one hundred thirty Bernoulli numbers and the Hurwitz zeta functionality . . . . . . . . . . . . 133 Eisenstein sequence of weight 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 The Fourier remodel and the Mellin remodel . . . . . . . . . . . . . 143 Nonholomorphic Eisenstein sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Modular types through theta features . . . . . . . . . . . . . . . . . . . . . . . . . a hundred and fifty five five Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 five. 1 The double coset operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 five. 2 The d and Tp operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 five. three The n and Tn operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 five. four The Petersson internal product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 five. five Adjoints of the Hecke Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 five. 6 Oldforms and Newforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 five. 7 the most Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 five. eight Eigenforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 five. nine the relationship with L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2 hundred five. 10 sensible equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 five. eleven Eisenstein sequence back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 6 Jacobians and Abelian forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6. 1 Integration, homology, the Jacobian, and Modularity . . . . . . . . . 212 6. 2 Maps among Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6. three Modular Jacobians and Hecke operators . . . . . . . . . . . . . . . . . . . . 226 6. four Algebraic numbers and algebraic integers . . . . . . . . . . . . . . . . . . . 230 6. five Algebraic eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 6. 6 Eigenforms, Abelian forms, and Modularity . . . . . . . . . . . . . . 240 7 Modular Curves as Algebraic Curves . . . . . . . . . . . . . . . . . . . . . . 249 7. 1 Elliptic curves as algebraic curves . . . . . . . . . . . . . . . . . . . . . . . . . 250 7. 2 Algebraic curves and their functionality fields . .