Let C / K C/K be a curve over...

We define three hard problems in the theory of elliptic divisibility sequences (EDS Association, EDS Residue and EDS Discrete Log), each of which is solvable in subexponential time if and only if the elliptic curve discrete logarithm problem is solvable in sub-exponential time. We also relate the problem of EDS Association to the Tate pairing and the MOV, Frey-Rück and Shipsey EDS attacks on th...

Suppose K = W (k)[1/p], where W (k) is the ring of Witt vectors with coefficients in algebraically closed field k of characteristic p 6= 2. We discuss an explicit construction of p-adic semi-stable representations of the absolute Galois group of K with Hodge-Tate weights from [0, p). This theory is applied to projective algebraic varieties over Q with good reduction outside 3 and semi-stable re...

We prove the finiteness of class numbers and Tate-Shafarevich sets for all affine group schemes of finite type over global function fields, as well as the finiteness of Tamagawa numbers and Godement’s compactness criterion (and a local analogue) for all such groups that are smooth and connected. This builds on the known cases of solvable and semisimple groups via systematic use of the recently ...

Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E∗E is flat over E∗. We wish to address the following question: given a commutative E∗-algebra A in E∗E-comodules, is there an E∞-ring spectrum X with E∗X ∼= A as comodule algebras? We will formulate this as a moduli problem, and give a way – suggested by work of Dwyer, Kan, and Stover – of dissecting the result...

For a noetherian scheme, we introduce its unbounded stable derived category. This leads to a recollement which reflects the passage from the bounded derived category of coherent sheaves to the quotient modulo the subcategory of perfect complexes. Some applications are included, for instance an analogue of maximal CohenMacaulay approximations, a construction of Tate cohomology, and an extension ...

BACKGROUND The multifunctional involvement and infiltration of myeloid cells (tumor-associated tissue eosinophils [TATE] and mast cells) can provide a unique opportunity to define relevant effectors functions that may represent novel, therapeutic options for modulation of tumor onset/growth. AIM Our study aimed to evaluate infiltration of myeloid cells (TATE and Mast cells) infiltration in di...

The discretisation problem for even quadratic twists is almost understood, with the main question now being how the arithmetic Delaunay heuristic interacts with the analytic random matrix theory prediction. The situation for odd quadratic twists is much more mysterious, as the height of a point enters the picture, which does not necessarily take integral values (as does the order of the Shafare...