The Birch and Swinnerton--Dyer conjecture famously predicts that the rank of an elliptic curve can be computed from its $L$-function. In this article we consider a weaker version called parity prove following. Let $E_1$ $E_2$ two curves defined over number field $K$ whose 2-torsion groups are isomorphic as Galois modules. Assuming finiteness Shafarevich-Tate $E_2$, show Swinnerton-Dyer correctl...

The paper begins in §1 with a foundational discussion of a new notion, that of a semistandard filtration in a highest weight category. The main result is Theorem 1, which says that “multiplicities” of standard modules in such filtration are well-defined. In §2, we specialize to the case of semistandard filtrations of maximal submodules of standard modules. The main result is Theorem 2, which un...

Steinhaus graphs on n vertices are certain simple graphs in bijective correspondence with binary {0,1}-sequences of length n−1. A conjecture of Dymacek in 1979 states that the only nontrivial regular Steinhaus graphs are those corresponding to the periodic binary sequences 110...110 of any length n − 1 = 3m. By an exhaustive search the conjecture was known to hold up to 25 vertices. We report h...

Let E be a one-parameter family of elliptic curves over a number field K. It is natural to expect the average root number of the curves in the family to be zero. All known counterexamples to this folk conjecture occur for families obeying a certain degeneracy condition. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Further...

This conjecture can be traced to Chowla ([2], p. 96); it is closely related to the Bunyakovsky/Schinzel conjecture on primes represented by irreducible polynomials. The one-variable analogue of (1.2) is classical for deg f = 1 and quite hopeless for deg f > 1. We know (1.2) itself when deg f ≤ 2. (The main ideas of the proof go back to de la Vallée-Poussin ([5], [6]); see [10], §3.3, for an exp...

Abstract We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non‐Galois extensions whose Galois closure has a group permutation‐isomorphic to prescribed G (in short, “ ‐extensions”). In particular, for alternating groups (an infinite family of) projective linear , we show that most curves (for example) infinitely many ‐extensions, conditional only on pa...