Using only linear algebra over Zp, we de ne a discrete probability distribution on the set of isomorphism classes of short exact sequences of Zp-modules, and then conjecture that as E varies over elliptic curves over a xed global eld k, the distribution of 0→ E(k)⊗Qp/Zp → Selp∞ E →X[p∞]→ 0 is that one. This single conjecture would explain many of the known theorems and conjectures on ranks, Sel...

In this paper it is shown that for every prime p > 5 the dimension of the p-torsion in the Tate-Shafarevich group of E/K can be arbitrarily large, where E is an elliptic curve defined over a number field K, with [K : Q] bounded by a constant depending only on p. From this we deduce that the dimension of the p-torsion in the Tate-Shafarevich group of A/Q can be arbitrarily large, where A is an a...

Let E be an elliptic curve over Q and p be a prime of good supersingular reduction for E. Although the Iwasawa theory of E over the cyclotomic Zp-extension of Q is well known to be fundamentally different from the case of good ordinary reduction at p, we are able to combine the method of our earlier paper with the theory of Kobayashi [5] and Pollack [8], to give an explicit upper bound for the ...

Somatostatin (SST) is a neuropeptide present in neurons, endocrine cells, and a wide range of neuroendocrine tumors (NETs). 68Ga-DOTATOC, 68Ga-DOTANOC, and 68Ga-DOTATATE are current SST analogues used for PET/CT which bind to SST receptors expressed in NETs. These SST analogues have been used successfully for diagnosis of SST-expressing tumors with a more sensitive detection technique than conv...

Metastatic tumor is one of several etiologies of space-occupying masses in the orbit that accounts for 1-13% of all orbital masses. In the adult patient population, breast cancer is the most common tumor to metastasize to the orbit, followed by metastasis from the lung, prostate, and gastrointestinal tract. Carcinoid tumors are rare neuroendocrine neoplasms derived from enterochromaffin cells, ...

Let E be an optimal elliptic curve overQ of prime conductorN . We show that if for an odd prime p, the mod p representation associated to E is reducible (in particular, if p divides the order of the torsion subgroup of E(Q)), then the p-primary component of the ShafarevichTate group of E is trivial. We also state a related result for more general abelian subvarieties of J0(N) and mention what t...