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AuthorStevens, Glenn. author
TitleArithmetic on Modular Curves [electronic resource] / by Glenn Stevens
ImprintBoston, MA : Birkhรคuser Boston, 1988
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Descript XVIII, 218 p. online resource


One of the most intriguing problems of modern number theory is to relate the arithmetic of abelian varieties to the special values of associated L-functions. A very precise conjecture has been formulated for elliptic curves by Birc̃ and Swinnerton-Dyer and generalized to abelian varieties by Tate. The numerical evidence is quite encouraging. A weakened form of the conjectures has been verified for CM elliptic curves by Coates and Wiles, and recently strengthened by K. Rubin. But a general proof of the conjectures seems still to be a long way off. A few years ago, B. Mazur [26] proved a weak analog of these c- jectures. Let N be prime, and be a weight two newform for r 0 (N) . For a primitive Dirichlet character X of conductor prime to N, let i\ f (X) denote the algebraic part of L (f , X, 1) (see below). Mazur showed in [ 26] that the residue class of Af (X) modulo the "Eisenstein" ideal gives information about the arithmetic of Xo (N). There are two aspects to his work: congruence formulae for the values Af(X) , and a descent argument. Mazur's congruence formulae were extended to r 1 (N), N prime, by S. Kamienny and the author [17], and in a paper which will appear shortly, Kamienny has generalized the descent argument to this case


1. Background -- 1.1. Modular Curves -- 1.2. Hecke Operators -- 1.3. The Cusps -- 1.4. $$ % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf % gDOjdaryqr1ngBPrginfgDObcv39gaiuqacqWFtcpvaaa!41F4! \mathbb{T} $$-modules and Periods of Cusp Forms -- 1.5. Congruences -- 1.6. The Universal Special Values -- 1.7. Points of finite order in Pic0(X(?)) -- 1.8. Eisenstein Series and the Cuspidal Group -- 2. Periods of Modular Forms -- 2.1. L-functions -- 2.2. A Calculus of Special Values -- 2.3. The Cocycle ?f and Periods of Modular Forms -- 2.4. Eisenstein Series -- 2.5. Periods of Eisenstein Series -- 3. The Special Values Associated to Cuspidal Groups -- 3.1. Special Values Associated to the Cuspidal Group -- 3.2. Hecke Operators and Galois Modules -- 3.3. An Aside on Dirichlet L-functions -- 3.4. Eigenfunctions in the Space of Eisenstein Series -- 3.5. Nonvanishing Theorems -- 3.6. The Group of Periods -- 4. Congruences -- 4.1. Eisenstein Ideals -- 4.2. Congruences Satisfied by Values of L-functions -- 4.3. Two Examples: X1(13), X0(7,7) -- 5. P-adic L-functions and Congruences -- 5.1. Distributions, Measures and p-adic L-functions -- 5.2. Construction of Distributions -- 5.3. Universal measures and measures associated to cusp forms -- 5.4. Measures associated to Eisenstein Series -- 5.5. The Modular Symbol associated to E -- 5.6. Congruences Between p-adic L-functions -- 6. Tables of Special Values -- 6.1. X0(N), N prime ? 43 -- 6.2. Genus One Curves, X0(N) -- 6.3. X1(13), Odd quadratic characters

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