The goal of this paper is to explain how a simple but apparently new fact of linear algebra together with the cohomological interpretation of L-functions allows one to produce many examples of L-functions over function fields vanishing to high order at the center point of their functional equation. Conjectures of Birch and Swinnerton-Dyer, Bloch, and Beilinson relate the orders of vanishing of ...

In this talk I shall attempt to introduce some of the main features of the Birch and Swinnerton-Dyer conjecture, (BSD). The congruent number problem, deciding whether an integer D is the area of a right angle triangle with rational sides, is not easy. It turns out that the problem is equivalent to finding out if a certain elliptic curve has an infinite number of rational points. In 1983 Tunnell...

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...

A polynomial relation f(x, y) = 0 in two variables defines a curve C. If the coefficients of the polynomial are rational numbers then one can ask for solutions of the equation f(x, y) = 0 with x, y ∈ Q, in other words for rational points on the curve. If we consider a non-singular projective model C of the curve then over C it is classified by its genus. Mordell conjectured, and in 1983 Falting...

This essay starts by first explaining, for elliptic curves defined over Q, the statement of the conjecture of Birch and Swinnerton-Dyer. Alongside, it contains a discussion of some results that have been proved in the direction of the conjecture, such as the theorem of Kolyvagin-Gross-Zagier and the weak parity theorem of Tim and Vladimir Dokchitser. The second, third and fourth part of the ess...

We provide two proofs that the conjecture of Artin-Tate for a fibered surface is equivalent to Birch-Swinnerton-Dyer Jacobian generic fibre. As byproduct, we obtain new proof theorem Geisser relating orders Brauer group and Tate-Shafarevich group.

Abstract This article proves a case of the p -adic Birch and Swinnerton–Dyer conjecture for Garrett L -functions (Bertolini et al. in On analogues -functions, 2021), imaginary dihedral exceptional zero setting extended analytic rank 4.