Question

Hello everyone,

After reading RH ( Riemann's Hypothesis ) and Swinnerton-Dyer conjecture, I asked myself why can't RH hold for $L$-Functions ( Hasse-Weil L-function ). In particular the GRH imposes the location of finding zeroes of a $L$-function with $\chi$ character.

So in particular Swinnerton-Dyer's conjecture states that

The underlying Mordell-Weil group of an elliptic curve has an infinite cardinality if there is a zero at $s=1$. i.e. $$E(\mathbb{Q})=\infty \iff L(E,1)=0.$$ $$E(\mathbb{Q})<\infty \iff L(E,1)\neq 1.$$ So can we predict that Hasse-Weil L-function of an elliptic-cruve satisfies the GRH. After normalizing can we put it this way :

There are infinitely many zeroes in the critical strip of Hasse-Weil L-function $L(E,s). $ i.e. Let $\mathfrak{K} $ be the number of zeroes of the Hasse-Weil L-function. Then $\mathfrak{K}=\infty \iff s=1+it$ ? . ( Assume that $E$ has infinitely many points, other wise $L(E,s)\neq0$.

I have some more set of questions concerning the significance of zeroes . They can be stated as

1. Are there any zeroes existing in the critical strip of $L(E,1+it)$ ? .
2. We know that $\rm{Rank(E(\mathbb{Q}))}= \rm{ord}_{s=1} L(E,s).$ So what about the significance of order of vanishing for other zeroes which are located at $s=1+it$ . Do they have some interesting relation with the properties of elliptic curves ? .

Are there any interesting results that are published in this direction so far ?

Thank you.

Answer 1

In answer to question 1, there are certainly zeroes on the critical strip. A great way you can investigate this is to go to the L-Functions and Modular Forms Database, where you can view plots of the associated Hardy Z-functions associated to the Hasse-Weil L-function of any elliptic curve over $\Bbb Q$ with conductor less than 240000.

For example you can go to L-function of the elliptic curve 389a and see that the Z-function on the bottom appears to have a zero of multiplicity 2 at 0. (It actually does because this elliptic curve has rank 2!) You also see many zeros of the Z-function between 0 and 30, I count 34. So yes, there certainly a lot of zeros of this Hasse-Weil L-function on the critical line.

I don't know a lot about zeros of $L$-functions so I don't know if there order of vanishing at the zeros away from the central point means anything. I would guess that the probability that a randomly chosen zero is anything other than a simple zero is $0$. (Analogous to the conjecture that a randomly chosen elliptic curve has probably $0$ of having rank $> 1$. Experts in Random Matrix Theory would know better than me, and hopefully they will appear soon.