This note continues our study of the period-index problem in the Weil-Châtelet group of an elliptic curve, begun in [6]. In this paper, we conjectured that for any elliptic curve E/K defined over a number field and any prime number p, there exists an infinite subgroup G of H(K,E), every nonzero element of which has period p and index p. This conjecture was proved when E has full p-torsion defin...

We prove a realization of the local Langlands correspondence for two-dimensional representations Weil group conductor three in cohomology Lubin–Tate curves by purely geometric method.

The present paper deals with the classical results of A. Weil [11] on regularization of pre-groups and pre-transformation spaces (see Definitions 3.1 and 4.1). As pointed out in [4], those purely algebraic results appear to be very useful in the following complex analytic setting. Let D ⊂ C be a bounded domain and Aut(D) the group of all holomorphic automorphisms of D . By a theorem of H. Carta...

results; out of 634 patients, 61 subjects (9.62%) had significant complications including: acute renal failure, alveolar hemorrhage, ards, clotting disorders, pancreatitis, gastrointestinal bleeding and intracranial bleeding; age, jaundice, renal and pulmonary involvement (oliguria and anuria), hypotension, leukocytosis, thrombocytopenia, hyponatremia, elevated bilirubin, alanine aminotransfera...

Recall that a Q-Fano variety is by definition a normal projective variety X such that the anticanonical divisor class −K = −K X is Q-Cartier and ample. For such X we define the (Weil) index i = i(X) to be the largest integer such that K X /i exists as a Weil divisor (see [R] for a discussion of Weil divisors and reflexive sheaves, and also Lemma 2 below; NB i differs from the (industry-standard...

We show that the special case of Serre duality involved in the Borel-Bott-Weil theorem can be formulated and proved in the context of equi-variant Kasparov theory by combining the Atiyah-Singer index theorem and the framework of equivariant correspondences developed in another paper by the first author and Ralf Meyer. The twisted Dolbeault cohomology groups of a flag variety that figure in the ...

We formulate and prove the special case of Serre duality involved in the Borel-Bott-Weil theorem in the language of equivariant Kasparov theory. The method is to combine the Atiyah-Singer index theorem and the framework of equivariant correspondences developed in another paper by the first author and Ralf Meyer. The twisted Dolbeault cohomology groups of the flag variety figuring in the Borel-B...

We study an infinite family of Mordell curves (i.e. the elliptic curves in the form y = x + n, n ∈ Z) over Q with three explicit integral points. We show that the points are independent in certain cases. We describe how to compute bounds of the canonical heights of the points. Using the result we show that any pair in the three points can always be a part of a basis of the free part of the Mord...

Three lectures on elliptic surfaces and curves of high rank Noam D. Elkies Over the past two years we have improved several of the (Mordell–Weil) rank records for elliptic curves over Q and nonconstant elliptic curves over Q(t). For example, we found the first example of a curve E/Q with 28 independent points P i ∈ E(Q) (the previous record was 24, by R. Martin and W. McMillen 2000), and the fi...