Abstract. This is a textbook about algorithms for computing with modular forms. It is nontraditional in that the primary focus is not on underlying theory; instead, it answers the question "how do you explicitly compute spaces of modular forms?" v To my grandmother, Annette Maurer.

The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and Swinnerton-Dyer. One conjectures that this constant is 1; it is known to be an integer. After surveying what is known about the Manin constant, we establish a new sufficient condition that ensures that the Manin constant is an odd integer. Next, we generalize the notion of the Manin constant to certain abelian variety quotients of the Jacobians of modular curves; these quotients are attached to ideals of Hecke algebras. We also generalize many of the results for elliptic curves to quotients of the new part of J 0 (N ), and conjecture that the generalized Manin constant is 1 for newform quotients. Finally an appendix by John Cremona discusses computation of the Manin constant for all elliptic curves of conductor up to 130000.

Abstract. Rational points on elliptic curves are the gems of arithmetic: they are, to diophantine geometry, what units in rings of integers are to algebraic number theory, what algebraic cycles are to algebraic geometry. A rational point in just the right context, at one place in the theory, can inhibit and control-thanks to ideas of Kolyvagin-the existence of rational points and other mathematical structures elsewhere. Despite all that we know about these objects, the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today.We have a network of heuristics and conjectures regarding rational points, and we have massive data accumulated to exhibit instances of the phenomena. Generally, we would expect that our data support our conjectures, and if not, we lose faith in our conjectures. But here there is a somewhat more surprising interrelation between data and conjecture: they are not exactly in open conflict one with the other, but they are no great comfort to each other either. We discuss various aspects of this story, including recent heuristics and data that attempt to resolve this mystery. We shall try to convince the reader that, despite seeming discrepancy, data and conjecture are, in fact, in harmony.

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Abstract. This paper provides evidence for the Birch and Swinnerton-Dyer conjecture for analytic rank 0 abelian varieties A f that are optimal quotients of J 0 (N ) attached to newforms. We prove theorems about the ratio L(A f , 1)/Ω A f , develop tools for computing with A f , and gather data about certain arithmetic invariants of the nearly 20, 000 abelian varieties A f of level ≤ 2333. Over half of these A f have analytic rank 0, and for these we compute upper and lower bounds on the conjectural order of (A f ). We find that there are at least 168 such A f for which the Birch and Swinnerton-Dyer conjecture implies that (A f ) is divisible by an odd prime, and we prove for 37 of these that the odd part of the conjectural order of (A f ) really divides # (A f ) by constructing nontrivial elements of (A f ) using visibility theory. We also give other evidence for the conjecture. The appendix, by Cremona and Mazur, fills in some gaps in the theoretical discussion in their paper on visibility of Shafarevich-Tate groups of elliptic curves.