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Aug
28 |
awarded | Popular Question |
Oct
26 |
comment |
Axiomatizing Gross-Zagier formulae
So nice. This gives support to an affirmative answer to Question 1 in a non-self dual situation: remember I'm not asking how to construct the points. Assuming the existence of the point (which in examples 2,3,4 is granted by BSD), I am asking what would one need to prove ( about the L-function and about the point) in order to prove a Gross-Zagier formula. Since the original proof of Gross-Zagier is a computation which, to a large extent, works independently on both sides, these are really two questions: what does one need to prove about $L(\pi,s)$ and what about the point or cycle? |
Oct
24 |
comment |
Axiomatizing Gross-Zagier formulae
Ok, I just managed. |
Oct
24 |
revised |
Axiomatizing Gross-Zagier formulae
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Oct
24 |
comment |
Axiomatizing Gross-Zagier formulae
But it's not displaying well, I don't understand what I'm typing wrong. |
Oct
24 |
revised |
Axiomatizing Gross-Zagier formulae
deleted 6 characters in body |
Oct
24 |
revised |
Axiomatizing Gross-Zagier formulae
added 68 characters in body; added 13 characters in body |
Oct
23 |
comment |
Axiomatizing Gross-Zagier formulae
I just edited the question in order to focus it on the aspects I would like to learn more about. |
Oct
23 |
awarded | Editor |
Oct
23 |
revised |
Axiomatizing Gross-Zagier formulae
I just edited my question, to focus it on the aspects I'd like to learn about.; edited title |
Oct
19 |
awarded | Yearling |
Oct
9 |
awarded | Nice Question |
Oct
8 |
comment |
Axiomatizing Gross-Zagier formulae
Thanks, Olivier, this is the kind of ideas I was looking forward to discuss. The reasons you pose make me agree with you that "it is not straight-forward" to extend the current circle of ideas to non self-dual settings. And I agree even more with you in that it would be fun to learn whether some other ideas can be exploited to push these GZ formulae to some non self-dual scenario. On the geometric side of the formula, notice that I don't require the point to be a Heegner point. And I am an optimist as for whether some sort of GZ formula should hold (not that it'd be easy to prove!). |
Oct
8 |
comment |
Axiomatizing Gross-Zagier formulae
But nevertheless it just happens some times that $L(f/\mathbb{Q},\chi,1)$ or $L(f/K,\psi,1)$ vanish and the first derivative does not, and BSD is still in force! In the latter case, say, it predicts that the $\psi$-eigenpart of $E(H)\otimes \mathbb{C}$ has rank $1$ over $\mathbb{C}$, and is therefore generated by some point. The point $P_\psi$ of my question is one natural candidate, and there could be some other alternative constructions. In any case, one could aim to prove a Gross-Zagier formula showing that its height is related to $L'(f/K,\psi,1)$. |
Oct
8 |
comment |
Axiomatizing Gross-Zagier formulae
Hi again! In response to David and François, sure, in the non self-dual setting there is not such a neat criterion for the L-function to vanish at $s=1$. The reason is of course that $s=1$ is not any more the central critical value of the L-function, and the root number is just a complex number of absolute value $1$. If the answer to my question is "we do not expect a Gross-Zagier formula to hold", these will surely be the conceptual reasons why. |
Oct
8 |
asked | Axiomatizing Gross-Zagier formulae |
Oct
8 |
accepted | Is the square of a curve minus its diagonal affine? |
Sep
17 |
awarded | Nice Question |
Sep
6 |
comment |
Examples of q-expansions in a Hida family
Thanks! I'll wait for your paper for more details then! |
Sep
6 |
accepted | Examples of q-expansions in a Hida family |