User flor.ian sprung - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:55:06Z http://mathoverflow.net/feeds/user/5730 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105190/power-series-expansions-of-l-series Power series expansions of $L$-series flor.ian sprung 2012-08-21T20:06:47Z 2012-10-09T19:34:55Z <p>Let $\zeta_K(s)$ be the Dedekind zeta function for a number field $K$. We can understand the first non-vanishing coefficient of its Laurent series via the class number formula. Is anything known/conjectured about the next term?</p> <p>On a related note, the BSD conjecture predicts the value of the first non-vanishing Taylor coefficient of the Hasse-Weil $L$-function of (say) an elliptic curve. Are there any conjectures about the coefficients after that?</p> http://mathoverflow.net/questions/62070/niemeier-lattices-and-theta-functions Niemeier lattices and theta functions flor.ian sprung 2011-04-18T01:08:05Z 2011-04-18T21:31:34Z <p>I have an extremely elementary question. Let's say someone randomly hands you a theta function associated to a Niemeier lattice (unimodular even, n=24). What can you say about which Niemeier lattice gave you this theta function in the first place?</p> http://mathoverflow.net/questions/16600/bad-reduction-for-elliptic-curves bad reduction for elliptic curves flor.ian sprung 2010-02-27T13:14:36Z 2011-04-07T12:41:24Z <p>Why do elliptic curves have bad reduction at some point if they are defined over Q, but not necessarily over arbitrary number fields?</p> http://mathoverflow.net/questions/42726/bsd-for-modular-forms BSD for modular forms flor.ian sprung 2010-10-19T00:47:39Z 2011-02-08T19:30:43Z <p>Given a modular form, what is the precise formulation of BSD (in particular, the residue formula for the $L$-function at special values)? And what about the special values if the $L$-function is twisted by some character? Does there exist a good reference?</p> http://mathoverflow.net/questions/42728/fourier-coefficient-of-a-modular-form Fourier coefficient of a modular form flor.ian sprung 2010-10-19T00:54:49Z 2010-10-21T06:11:02Z <p>If someone hands you a prime number $p$, and an algebraic number $x$ inside the Hasse-Weil bound, is there a normalized newform (say of weight two) so that $a_p=x$, where $a_p$ is the $p$th Fourier coefficient?</p> http://mathoverflow.net/questions/35139/fourier-coefficients-for-elliptic-curves-on-average Fourier coefficients for elliptic curves on average flor.ian sprung 2010-08-10T16:38:02Z 2010-08-11T20:17:35Z <p>Fix a prime p, and look at elliptic curves in some family (e.g. all elliptic curves ordered by height). How often do the Fourier coefficients a_p occur? Are there any conjectures?</p> http://mathoverflow.net/questions/25948/non-commutative-iwasawa-theory non-commutative iwasawa theory flor.ian sprung 2010-05-25T23:07:19Z 2010-06-08T14:37:48Z <p>In commutative Iwasawa theory, the main conjecture states that the p-adic L-function generates the characteristic ideal of an algebraic object. Non-commutative Iwasawa theory seems to mimik this - except that the existence of the object on the analytic side (to my knowledge) is still conjectural. My question is: why is it so hard to define a "non-commutative" p-adic L-function? And on a more technical note: In Coates-Fukaya-Kato-Sujatha-Venjakob, this conjectural function only exists for primes of good ordinary reduction. This seems to imply that in the excluded cases, things go horribly wrong. Does anybody know what or why?</p> http://mathoverflow.net/questions/22301/p-adic-l-functions p-adic L-functions flor.ian sprung 2010-04-23T05:21:41Z 2010-04-30T08:41:12Z <p>For modular forms, it is known that you can construct p-adic L-functions by interpolating (p-power conductor) twists of their associated L-functions at special values. Similarly, Kubota-Leopoldt's p-adic L-function can be constructed by interpolating Dirichlet L-functions at negative integers. My question is: how general is this method? For example, for which (L-functions of) automorphic representations have p-adic L-functions been constructed?</p> http://mathoverflow.net/questions/22301/p-adic-l-functions/23071#23071 Answer by flor.ian sprung for p-adic L-functions flor.ian sprung 2010-04-30T04:24:21Z 2010-04-30T04:24:21Z <p>@Emmerton: Thanks so much for your answer! So has GL_n been done over Q, or is this also part of your paper? </p> http://mathoverflow.net/questions/105190/power-series-expansions-of-l-series Comment by flor.ian sprung flor.ian sprung 2012-08-24T23:34:31Z 2012-08-24T23:34:31Z thanks to both of you for the references! http://mathoverflow.net/questions/42728/fourier-coefficient-of-a-modular-form Comment by flor.ian sprung flor.ian sprung 2010-10-19T02:26:49Z 2010-10-19T02:26:49Z Dear Ben and Alex, that's right, I was being imprecise. I meant normalized newform. http://mathoverflow.net/questions/35139/fourier-coefficients-for-elliptic-curves-on-average/35155#35155 Comment by flor.ian sprung flor.ian sprung 2010-08-11T19:29:40Z 2010-08-11T19:29:40Z Wow, thanks! David's and Papalardi's paper seems to answer the question for a fixed prime, too (section 4). It's in term of the Kronecker class number, as David wrote above. http://mathoverflow.net/questions/35139/fourier-coefficients-for-elliptic-curves-on-average/35144#35144 Comment by flor.ian sprung flor.ian sprung 2010-08-11T19:22:18Z 2010-08-11T19:22:18Z Thanks so much! Printing it out now...