Chapter 1. Introduction ........................................ 1
1.1 Congruence relations ....................................... 2
1.2 Selmer groups for elliptic curves .......................... 6
1.3 Behavior of Iwasawa invariants ............................. 9
1.4 Selmer atoms .............................................. 10
1.5 Parity questions .......................................... 12
1.6 Other situations .......................................... 14
1.7 Organization and acknowledgements ......................... 16
Chapter 2. Projective and quasi-projective modules ............ 17
2.1 Criteria for projectivity and quasi-projectivity .......... 17
2.2 Nonzero μ-invariant ....................................... 24
2.3 The structure of ΛG/ΛGθ ................................... 26
2.4 Projective dimension ...................................... 27
Chapter 3. Projectivity or quasi-projectivity of ХЕΣ0 (К∞) ..... 31
3.1 The proof of Theorem 1 .................................... 31
3.2 Quasi-projectivity ........................................ 37
3.3 Partial converses ......................................... 39
3.4 More general situations ................................... 41
3.5 Δ ⋊ Г-extensions ......................................... 44
Chapter 4. Selmer atoms ....................................... 47
4.1 Various cohomology groups. Coranks. Criteria for
vanishing ................................................. 47
4.2 Selmer groups for E|p|  α ................................ 56
4.3 Justification of (1.4.b) and (1.4.c) ...................... 59
4.4 Justification of (1.4.d) and the proof of Theorem 2 ....... 62
4.5 Finiteness of Selmer atoms ................................ 63
Chapter 5. The structure of Hυ(К∞, E) ......................... 69
5.1 Determination of δE,υ(σ) .................................. 70
5.2 Determination of (ρE,υ, χ) ................................ 72
5.3 Projectivity and Herbrand quotients ....................... 74
Chapter 6. The case where Δ is a p-group ...................... 77
Chapter 7. Other specific groups .............................. 81
7.1 The groups A4, S4, and S5 ................................. 81
7.2 The group PGL2(Fp) ........................................ 84
7.3 The groups PGL2(Z/pr+1Z) for r ≥ 1 ........................ 90
7.4 Extensions of (Z/pZ)× by a p-group ........................ 96
Chapter 8. Some arithmetic illustrations ..................... 105
8.1 An illustration where Σ0 is empty ........................ 105
8.2 An illustration where Σ0 is non-empty .................... 109
8.3 An illustration where the σss's have abelian image ....... 114
8.4 False Tate extensions of Q ............................... 127
Chapter 9. Self-dual representations ......................... 131
9.1 Various classes of groups ................................ 131
9.2 ΠΩ groups ................................................ 133
9.3 Some parity results concerning multiplicities ............ 137
9.4 Self-dual representations and the decomposition map ...... 139
Chapter 10. A duality theorem ................................. 141
10.1 The main result .......................................... 142
10.2 Consequences concerning the parity of sE(ρ) .............. 145
Chapter 11. p-modular functions ............................... 151
11.1 Basic examples of p-modular functions .................... 151
11.2 Some p-modular functions involving multiplicities ........ 152
Chapter 12. Parity ............................................ 159
12.1 The proof of Theorem 3 ................................... 160
12.2 Consequences concerning WDel(E,ρ) and WSelρ(E,ρ) .......... 167
Chapter 13. More arithmetic illustrations .................... 169
13.1 An illustration where ΨE ∩ ΦK/F is empty ................. 170
13.2 An illustration where К  Q(E[p∞]) ...................... 172
13.3 An illustration where Gal(K/Q) is isomorphic to Bn or
Hn ....................................................... 174
Bibliography .................................................. 183
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