Interplay between Riemann and Swinnerton-Dyer - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T04:04:07Z http://mathoverflow.net/feeds/question/102119 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/102119/interplay-between-riemann-and-swinnerton-dyer Interplay between Riemann and Swinnerton-Dyer Shanmukha_Srinivasan 2012-07-13T09:04:37Z 2012-07-13T19:36:44Z <p>Hello everyone, </p> <p>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 <a href="http://en.wikipedia.org/wiki/Generalized_Riemann_hypothesis" rel="nofollow">GRH</a> imposes the location of finding zeroes of a $L$-function with $\chi$ character.</p> <p>So in particular Swinnerton-Dyer's conjecture states that </p> <blockquote> <p>The underlying Mordell-Weil group of an elliptic curve has an infinite cardinality if there is a zero at $s=1$. i.e. $$E(\mathbb{Q})=\infty \iff L(E,1)=0.$$ $$E(\mathbb{Q})&lt;\infty \iff L(E,1)\neq 1.$$ So can we predict that Hasse-Weil L-function of an elliptic-cruve satisfies the GRH. After normalizing can we put it this way : </p> <blockquote> <p>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 $L(E,s)\neq0$. </p> </blockquote> </blockquote> <p>I have some more set of questions concerning the significance of zeroes . They can be stated as </p> <ol> <li>Are there any zeroes existing in the critical strip of $L(E,1+it)$ ? . </li> <li>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 ? . </li> </ol> <p>Are there any interesting results that are published in this direction so far ? </p> <p>Thank you.</p> http://mathoverflow.net/questions/102119/interplay-between-riemann-and-swinnerton-dyer/102172#102172 Answer by Jamie Weigandt for Interplay between Riemann and Swinnerton-Dyer Jamie Weigandt 2012-07-13T19:36:44Z 2012-07-13T19:36:44Z <p>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 <a href="http://www.lmfdb.org/" rel="nofollow">L-Functions and Modular Forms Database</a>, 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.</p> <p>For example you can go to <a href="http://www.lmfdb.org/L/EllipticCurve/Q/389.a/" rel="nofollow">L-function of the elliptic curve 389a</a> 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.</p> <p>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.</p>