1 Algebraic Groups 7 1.1 Group Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Restriction of Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Algebraic Groups over a Nonalgebraically Closed Field K . . . . . . . . . . . 13 1.4 Structure of Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Topologizing G(R...

Let Y C Pm be a subvariety of codimension d defined by an ideal / in charp > 0 with //'(Y, 0 (-1)) = 0. If t is an integer greater than log (d) and Hi( Y, f/l"+i) = 0 for n » 0 and i =1,2, then Pic(Y) is an extension of a finite p-primary group of exponent at most pt by Z[ 0 (1 )1 and Br'(Y)(p) is a group of exponent at most pl'. If Y is also smooth and defined over a finite field with dim Y < ...

Let A be a quotient of J0(N) associated to a newform f such that the special L-value of A (at s = 1) is non-zero. We give a formula for the ratio of the special L-value to the real period of A that expresses this ratio as a rational number. We extract an integer factor from the numerator of this formula; this factor is non-trivial in general and is related to certain congruences of f with eigen...

Let E be an elliptic curve over Q attached to a newform f of weight 2 on 00(N ), and let K be a real quadratic field in which all the primes dividing N are split. This paper relates the canonical R/Z-valued “circle pairing” on E(K ) defined by Mazur and Tate [MT1] to a period integral I ( f, K ) defined in terms of f and K . The resulting conjecture can be viewed as an analogue of the classical...

Modular curves and their close relatives, Shimura curves attached to multiplicative subgroups of quaternion algebras, are equipped with a distinguished collection of points defined over class fields of imaginary quadratic fields and arising from the theory of complex multiplication: the so-called Heegner points. It is customary to use the same term to describe the images of degree zero divisors...

For N and k positive integers, let M(N, k)C denote the C-vector space of cuspidal modular forms of level N and weight k. This vector space is equipped with the usual Hecke operators Tn, n ≥ 1. If we need to consider several levels or weights at the same time, we will denote this Tn by T N n , or T N,k n . If p is a prime number dividing N , our Tp is also known under the name Up. One of our mai...

Heegner points play an outstanding role in the study of the Birch and Swinnerton-Dyer conjecture, providing canonical Mordell–Weil generators whose heights encode first derivatives of the associated Hasse–Weil L-series. Yet the fruitful connection between Heegner points and L-series also accounts for their main limitation, namely that they are torsion in (analytic) rank> 1. This partly exposito...