On a Convolution of L-Series.
On a two-variable zeta function for number fields
This paper studies a two-variable zeta function attached to an algebraic number field , introduced by van der Geer and Schoof, which is based on an analogue of the Riemann-Roch theorem for number fields using Arakelov divisors. When this function becomes the completed Dedekind zeta function of the field . The function is a meromorphic function of two complex variables with polar divisor , and it satisfies the functional equation . We consider the special case , where for this function...
On congruent primes and class numbers of imaginary quadratic fields
We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a related criterion for primes p such that divides the class number of the imaginary quadratic field ℚ(√-p)....
On L-functions of elliptic curves and anticyclotomic towers.
On L-functions of elliptic curves and cyclotomic towers.
On -adic -functions
On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer.
On the Birch and Swinnerton-Dyer conjecture
On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields
Let be a modular elliptic curve defined over a totally real number field and let be its associated eigenform. This paper presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of over suitable quadratic imaginary extensions . In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when is even and not new at any prime.
On the conjecture of Birch and Swinnerton-Dyer for abelian varieties over function fields in characteristic p > 0.
On the conjectures of Birch and Swinnerton-Dyer and a geometric analog
On the de Rham and -adic realizations of the elliptic polylogarithm for CM elliptic curves
In this paper, we give an explicit description of the de Rham and -adic polylogarithms for elliptic curves using the Kronecker theta function. In particular, consider an elliptic curve defined over an imaginary quadratic field with complex multiplication by the full ring of integers of . Note that our condition implies that has class number one. Assume in addition that has good reduction above a prime unramified in . In this case, we prove that the specializations of the -adic elliptic...
On the ...-factor attached to motives.
On the leading terms of zeta isomorphisms and -adic -functions in non-commutative Iwasawa theory.
On the meromorphic continuation of degree two -functions.
On the p-adic height of Heegner cycles.
On the parity of ranks of Selmer groups. III.
On the values of Artin L-series at s=1 and annihilation of class groups
Let L be a finite Galois CM-extension of a totally real field K. We show that the validity of an appropriate special case of the Equivariant Tamagawa Number Conjecture leads to a natural construction for each odd prime p of explicit elements in the (non-commutative) Fitting invariants over of a certain tame ray class group, and hence also in the analogous Fitting invariants of the p-primary part of the ideal class group of L. These elements involve the values at s=1 of the Artin L-series of characters...
On the values of equivariant zeta functions of curves over finite fields.
On the vanishing of twisted -functions of elliptic curves.