on the Zeroes of Hasse -weil L-function - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T23:17:06Z http://mathoverflow.net/feeds/question/71061 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71061/on-the-zeroes-of-hasse-weil-l-function on the Zeroes of Hasse -weil L-function Trust God 2011-07-23T11:51:28Z 2011-07-23T20:54:42Z <p>my question is that </p> <p>already we know that the Birch and Swinnerton Dyer conjecture ,formally conjectures that the Hasse-weil L-function should have a zero at $s=1$ when curves have infinitely many rational points on it,</p> <p>so my question is that imagine an elliptic curve $E/\mathbb{Q}$ which has rank $r>0$ and with $\left|{E(\mathbb{Q})}\right|=\infty$ so we find that $L(E/\mathbb{Q},s)_{s=1}=0$</p> <p>but i am interested in the zeroes on the line $s=1+it$, we know that above curve has got a zero at $s=1$ so are there any zeroes on the line $s=1+it$ ,if so tell me the cardinality of set of zeroes ,i mean whether there are finite zeroes or infinitely many zeroes </p> <p>and if i get the answer there is a deep intuition behind the answer and properties of elliptic curves,</p> <p>and may be someone can conjecture still more things knowing the zeroes there on the line </p> <p>thank you, touch everyone's feet who helped me,by suggesting books,and resources and making me what i am today by studying privately</p> http://mathoverflow.net/questions/71061/on-the-zeroes-of-hasse-weil-l-function/71062#71062 Answer by Micah Milinovich for on the Zeroes of Hasse -weil L-function Micah Milinovich 2011-07-23T12:13:36Z 2011-07-23T13:50:46Z <p>The $L$-function has about $\displaystyle{\frac{T}{\pi} \log T \ }$ zeros in the strip with $0 &lt; t &lt; T$. See section 5.3 of Iwaniec and Kowalski's "Analytic Number Theory," in particular Theorem 5.8.</p> <p>It should be possible, if it hasn't been done already, to show that a positive proportion of these zeros are on the critical line using Selberg's method. Hafner extended Selberg's method to various families of degree 2 $L$-functions in a series of papers in the 1980s.</p> http://mathoverflow.net/questions/71061/on-the-zeroes-of-hasse-weil-l-function/71084#71084 Answer by Frank Thorne for on the Zeroes of Hasse -weil L-function Frank Thorne 2011-07-23T20:54:42Z 2011-07-23T20:54:42Z <p>+1 to Micah, read his answer first!</p> <p>As an addendum, the $L$-function of an elliptic curve looks like (and is) an $L$-function of degree 2. It has a certain <em>conductor</em>, which is defined in terms of the primes of bad reduction and which affects the analytic behavior of the $L$-function.</p> <p>But beyond that, there is unfortunately not a lot that is known from an analytic perspective for elliptic curve $L$-functions in particular. Once it is proved that they have analytic continuation and a functional equation, knowledge of where the $L$-function came from seems to be disappointingly useless for proving theorems. (With some exceptions, e.g., the value of the critical point; there is also the modularity theorem, etc. but I am not regarding that as "analytic".) There is a lot of general machinery, which is very well explained in Chapter 5 of Iwaniec and Kowalski for general $L$-functions in general. Beyond that, my impression (which could be mistaken) is that not a whole lot is known.</p>