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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?

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In the case of modular forms, the circle of conjectures is called "Bloch-Kato conjecture" rather than BSD. More precisely, you want to explicate the Bloch-Kato conjecture for the motif attached to your modular (new)form. Not sure if this has been done explicitly for modular forms (even my more elementary question about Artin L-functions is still unanswered), but if you are familiar with the Bloch-Kato formalism, then that's the place to look. – Alex B. Oct 19 '10 at 1:45
Dear Ian, In the case of a normalized eigenform over $\mathbb Q$ (which I gather is your case of interest), then everything can be done explicitly, and in fact has been. Two good references are K.Kato Iwasawa theory and p-adic Hodge theory (Kodaira math. 1993) and $p$-adic Hodge theory and values of zeta functions of modular forms (Asterisque 2004). That said, there is another place where everything is checked quite explicitly, and this is the PhD. thesis of M.Gealy. The first and third references are online (I think) ut if you have trouble finding them, you could always e-mail me. – Olivier Oct 19 '10 at 8:09
Dear Alex, your question wasn't more elementary, because you asked for some interpretation of the Tamagawa number conjecture in terms of natural objects. To find this might be a difficult task (especially since natural can be in the eye of the beholder). However, specializing Tamagawa Number Conjectures to the case of modular forms is an interesting exercise, not at all easy by the way, but certainly doable (and done 15 years ago). One last thing to Ian: the references I mentioned should also do for twists by characters but you should know that for some twists, things get much harder. – Olivier Oct 19 '10 at 8:14
up vote 7 down vote accepted

My comments are getting too long, so here is a tentative answer.

First, a general statement: conjectures predicting special values of $L$-functions are formulated for all motives over number fields. Because normalized eigenforms (even twisted by finite order characters $\chi$) are attached to motives (or vice-versa), there exists conjectures predicting the special values of $L(f,s,\chi)$.

Now what do they look like? For simplicity and because you are especially interested in the value of the residue, I'll restrict to the case of a critical value (so $s=0,\cdots,k-1$ if $f$ is of weight $k$ EDIT: $s=1,\cdots,k-1$ Thanks to David Loeffler for pointing this out). The general conjectures then imply that there exists a special element which is a basis of the determinant of the motivic cohomology which is sent to $L(f,s,\chi)$ by the realization morphism from motivic cohomology to Betti cohomology and to a specific basis of the determinant of étale cohomology by the realization morphism to $p$-adic étale cohomology (for any $p$). But forget about this, because when $s≠k/2$ (which I assume henceforth), then $L(f,s,\chi)$ is non-zero (by Jacquet) and in this case, K.Kato has constructed a candidate $z$ for this conjectural element in his article $p$-adic Hodge theory and values of zeta functions of modular forms. I can't really say that this $z$ lives in the right space, simply because I am unsure whether motivic cohomology is properly defined in this case, but at least it lives in something that has all the property you would wish for motivic cohomology (namely the second $K$-theory group $K_{2}$ of the modular curves) and it is sent to the right value through the realization morphism to Betti cohomology.

So now the Tamagawa Number Conjecture predicts that it should be sent to a specific basis of the determinant of the $p$-adic étale cohmology for any $p$. Unraveling what it means in this case, you get that $H^2(\textrm{Spec}\ \mathbb Z[1/p],T)$ is a finite group and the following conjectural equality: \begin{equation} \sharp H^2_{et}(\textrm{Spec}\ \mathbb Z[1/p],T)=[H^1_{et}(\textrm{Spec}\ \mathbb Z[1/p],T):z] \end{equation} Here $T$ is any lattice in the Galois representation of your modular form and $[-:-]$ denotes generalized index (so $[\mathbb Z_{p}:1/p]=p^{-1}$).

Now you have a perfectly valid expression of a generalized BSD conjecture, and this is how I think of conjectures about special values in this case. However, you might want to express this conjecture in a way that recovers usual BSD when $f$ comes from an elliptic curve $E$. This again is a doable exercise (albeit one I find non-trivial) which is done for instance in Burns-Flach Math. Ann. 305 (section 1.7) or O.Venjakob London Math. Soc. Lecture Note Ser., 320 (section 3.1). I generally recommend the latter article because I learnt a lot from it myself but beware that, on this very specific question, there is a typo in the definition of one of the crucial objects, if memory serves well, so I rather recommend using both articles in parallel.

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Is $s = 0$ critical for modular forms? I always thought the critical values were $s = 1, \dots, k-1$, i.e. the ones you can "get at" with modular symbols. – David Loeffler Oct 19 '10 at 9:12
You are right. Let me edit my post. – Olivier Oct 19 '10 at 9:33

Try looking in the book by Bellaiche and Chenevier, "Families of Galois representations and higher rank Selmer groups" (published in Asterisque, or available from the Arxiv here). They give a nice description (Conjecture 5.1.3 in the arxiv version) of the Bloch-Kato conjecture for the order of vanishing of the L-function for a general "geometric" Galois representation.

They don't state a conjecture for the exact leading term; I don't know what this would look like in general.

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This paper by Dummigan, me, and Watkins gives a detailed answer to your question. Plus it has some explicit examples of weight 4 modular forms so that the corresponding motive conjecturally has nontrivial (visible) Shaferevich-Tate group.

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