User jvo - MathOverflow

Answer by jvo for A route towards understanding Shimura varieties?

2011-10-08T15:54:10Z

I think the general wisdom is that Deligne's Travaux de Shimura and Milne's Introduction to Shimura Varieties are the most comprehensive references, with the latter being somewhat lighter on prerequisites (but heavier on examples).

I've heard it suggested by people who work in the area that the best way to learn the theory is via special cases and examples, motivated by focused research problems. I suppose this is true of many things, though.

It might also help to learn about quaternionic Shimura curves first, assuming that you know a bit about modular curves.

Vanishing cycles in a nutshell?

2011-02-20T17:24:06Z

To quote one source among many, "the general reference for vanishing cycles is [SGA 7] XIII and XV". Is there a more direct way to learn the main principles of this theory (i.e. without the language of derived categories), in particular as it applies to the study of certain integral models of curves?

Characterization of algebraic points on Shimura varieties?

2011-03-24T21:22:45Z

Is there any (conjectural) characterization of $\overline{\bf{Q}}$-points on Shimura varieties?

The question of course does not always make sense for ${\bf{Q}}$-points: a theorem of Shimura shows that a quaternionic Shimura curve has no ${\bf{R}}$-points, and a theorem of Mazur shows that the modular curve $Y_0(N)$ has no ${\bf{Q}}$-points for $N$ sufficiently large. The question does however seem to make sense for certain abelian extensions of CM fields. (For instance, in the setting of a quaternionic Shimura curve $M$ defined over a totally real field $F$, if a totally imaginary quadratic extension $K/F$ embeds into the underlying quaternion algebra, then there is a supply of CM points on $M$ defined over certain abelian extensions of $K$). In particular, I should like to know more about the following questions:

(i) Over which number fields $k$ does a given Shimura variety $S(G, X)$ have a $k$-rational point?

(ii) For which such number fields $k$ will $S(G, X)(k)$ be Zariski dense?

(iii) To what extent are such $k$-rational points accounted for by CM points (or similar constructions)?

Sorry if these questions are imprecise or wrongly formulated, I would be happy to at least have an indication of where to look in the literature if someone has already thought about this.

Construction of abelian varieties from Hilbert modular forms?

2011-02-06T21:37:48Z

Some experts tell me that the construction of abelian varieties from Hilbert modular forms is an (apparently difficult) open problem. However, in view of the construction of $l$-adic Galois representations due to Carayol for instance, it is not clear what exactly the obstruction to the usual method of taking the quotient of the jacobian (of the associated quaternionic Shimura curve) by the `annihilator' of the associated quaternionic eigenform would be.

To be slightly more precise, consider the setting of Carayol "Sur les représentations galoisiennes modulo $l$ attachées aux formes modulaires" (Duke Math. Journal, 1986). That is, let $F$ be a totally real field of degree $d$, with set of real places $\lbrace \tau_1, \ldots, \tau_d \rbrace$. Fix integers $k \geq 2$ and $w$ having the same parity. Let $D_{k,w}$ denote the representation of $\operatorname{GL_2}({\bf{R}})$ that occurs via unitary induction as $\operatorname{Ind}(\mu, \nu)$, where $\mu$ and $\nu$ are the characters on ${\bf{R}}^{\times}$ given by

\begin{align*} \mu(t) &= \vert t \vert ^{\frac{1}{2}(k-1-w)}\operatorname{sgn}(t)^k; ~~ \nu(t) = \vert t \vert ^{\frac{1}{2}(-k+1-w)}. \end{align*} Fix integers $k_1, \ldots k_d$ all having the same parity. Let $\pi \cong \bigotimes_v \pi_v$ be a cuspidal automorphic representation of $\operatorname{GL_2}({\bf{A}}_F)$ such that for each real place $\tau_i$ of $F$, there is an isomorphism $$\pi_{\tau_i} \cong D_{k_i, w}.$$ It is well know that such representations correspond to holomorphic Hilbert modular forms of weight ${\bf{k}}=(k_1, \ldots, k_d)$. If $d$ is even, then assume additionally that there exists a finite prime $v \subset \mathcal{O}_F$ where the local component $\pi_v$ is an "essentially square integrable" (i.e. special or cuspidal) representation of $\operatorname{GL_2}(F_v)$ . Let $B/F$ be a quaternion algebra that is ramified at $\lbrace \tau_2, \ldots, \tau_d \rbrace$ if $d$ is odd, and ramified at $\lbrace \tau_2, \ldots, \tau_d, v \rbrace$ if $d$ is even. Let $$G = \operatorname{Res}_{F/{\bf{Q}}}(B^{\times})$$ be the associated algebraic group over ${\bf{Q}}$ . Hence, we have an isomorphism $$G({\bf{R}}) \cong \operatorname{GL}({\bf{R}}) \times \left( \mathbb{H}^{\times} \right)^{d-1},$$ where $\mathbb{H}$ denotes the Hamiltonian quaternions. Let $\overline{D}_{k,w}$ denote the representation of $\mathbb{H}^{\times}$ corresponding to $D_{k,w}$ via Jacquet-Langlands correspondence. We then consider cuspidal automorphic representations $\pi' = \bigotimes_v \pi_v'$ of $G({\bf{A}}_F)$ such that $\pi_{\tau_1}' \cong D_{k_1, w}$ and $\pi_{\tau_i} \cong \overline{D}_{k_i, w}$ for $i = 2, \ldots, d$ . Such representations should (I believe) correspond to modular forms of weight ${\bf{k}} = (k_1, \ldots, k_d)$ on the indefinite quaternion algebra $B$ . To be slightly more precise, let $S_{\bf{k}}(\mathfrak{m})$ denote the finite dimensional ${\bf{C}}$ -vector space of quaternionic modular forms of weight ${\bf{k}}$ and level $\mathfrak{m} \subset \mathcal{O}_F$ on $B$ . Write $\mathfrak{d} =\operatorname{disc}(B)$ . The space $S_{\bf{k}}(\mathfrak{m})$ comes equipped with actions of the standard Hecke operators $T_v$ for all primes $v \nmid \mathfrak{m}\mathfrak{d}$ , and with Atkin-Lehner involutions for all prime powers $v^e \mid \mathfrak{m}\mathfrak{d}$ . The Jacquet-Langlands correspondence induces a "Hecke equivariant" isomorphism of spaces \begin{align*} S^B_{\bf{k}}(\mathfrak{m}) &\cong S_{\bf{k}}(\mathfrak{m}\mathfrak{d})^{\operatorname{\mathfrak{d}-new}}, \end{align*} where $S_{\bf{k}}(\mathfrak{m}\mathfrak{d})^{\operatorname{\mathfrak{d}-new}}$ denotes the space of cuspidal Hilbert modular forms of weight ${\bf{k}}$ that are new at primes dividing $\mathfrak{d}$ .

Anyhow, at least when we assume ${\bf{k}} = (2, \ldots, 2)$ , a standard argument shows that there is a $G({\bf{A}}_f)$ -equivariant isomorphism $\Gamma(\omega) \cong S^B_{\bf{k}}(\mathfrak{m})$ , where $\omega$ is the sheaf of homomorphic $1$ -forms on the complex Shimura curve \begin{align*} M(\bf{C}) &= G({\bf{Q}}) \backslash G({\bf{A}}_f) \times X/H.\end{align*} Here, $X = {\bf{C}} - {\bf{R}}$ , and $H \subset G({\bf{A}}_f)$ is a compact open subgroup of level $\mathfrak{m}$ . Let $M$ denote Shimura's canonical model of this curve (defined over $F$ ). Let $J$ denote the Jacobian of $M$ . Let ${\bf{T}}$ denote the subalgebra of $\operatorname{End}(J)$ generated by Hecke correspondences on $M$ . My question is whether or not the following construction can or has been made rigorous. Namely, in the setup above, start with a Hilbert modular eigenform ${\bf{f}} \in \pi$ , and consider an associated quaternionic eigenform $\Phi \in \pi'$ . Viewing $\Phi$ as an eigenform for the Hecke algebra ${\bf{T}}$ , consider the homomorphism $\theta_{\Phi}:{\bf{T}} \longrightarrow E$ that sends a Hecke operator acting on $\Phi$ to its corresponding eigenvalue. Here, $E = E_{\Phi}$ denotes the extension of ${\bf{Q}}$ generated by all of the eigenvalues of $\Phi$ . Let $I_{\Phi} = \ker{\theta_{\Phi}}$ . Consider the quotient \begin{align*} A &= J/I_{\Phi}J. \end{align*} Is $A$ not an abelian variety associated to the Hilbert modular eigenform ${\bf{f}}$ ? Or is this completely trivial, with the subtle part being the task of showing that $\dim(A) = [E: {\bf{Q}}]$ ?

A more naive question to ask is why Shimura's construction cannot be generalized directly for a cuspidal Hilbert modular form ${\bf{f}} \in S_{\bf{k}}(\mathfrak{m})$ . Also, how does taking weight ${\bf{k}} = {\bf{2}}$ make the problem simpler? Apologies if parts of this question were somewhat vague, I have sketched matters for simplicity/space.

"Bad" reduction of Shimura curves via dual graphs

2011-01-13T09:01:58Z

I have the following naive (and inexpert) question about the reduction of Shimura curves at primes dividing the discriminant of the underlying quaternion algebra. It requires some background to state. That is, let $F$ be a totally real field of degree $d$. Fix a real place $\tau_1$ in the set of real places $\lbrace \tau_1, \ldots, \tau_d \rbrace$ of $F$. Let $B$ be a quaternion algebra over $F$ that is split at $\tau_1$ and ramified at $\tau_2, \ldots, \tau_d$. Let $H \subset \widehat{B}~(= B \otimes \widehat{Z})$ be a compact open subgroup. Let $M_H$ denote the Shimura curve over $F$ of level $H$, with complex points given by \begin{align*} M_H({\bf{C}}) &= B^{\times}\backslash \widehat{B}^{\times} \times \left({\bf{C}}-{\bf{R}} \right)/H.\end{align*} Fix a prime $v \subset \mathcal{O}_{F}$. Assume that $H$ can be factored as $H^v \times H_v$, with $H_v \subset B_v^{\times}~(= B^{\times} \otimes F_v)$ maximal. If $v$ does not divide the discriminant of $B$, then it is known by work of Morita and Carayol that $M_H$ has good reduction over $v$, hence that there exists a smooth model ${\bf{M}}_H$ of $M_H$ over $\mathcal{O}_{(v)}$ . If $v$ divides the discriminant of the quaternion algebra $B$, then it is known by work of Varshavsky for instance that there exists an integral model ${\bf{M}}_{H}^V$ of $M_H$ over $\mathcal{O}_{(v)}$ . (N.B. there is apparently also a model due to Drinfeld, described extensively in the literature for the case of $F={\bf{Q}}$, though it is not clear to me why Drinfeld's work, which seems to require a moduli theoretic description of $M_H$ , extends to the general totally real fields setting). Anyhow, let $F_v$ denote the completion of $F$ at $v$, with $\kappa_v$ the residue field and $\pi_v$ a uniformizer. By Cerednik's theorem, the completion of ${\bf{M}}_H^V$ along its closed fibre is canonically isomorphic to the product \begin{align*} GL(F_v)\backslash \widehat{\Omega}^{unr} \times D^{\times}\backslash \widehat{D}^{\times}/\overline{H}^v. \end{align*} Here, $\widehat{\Omega}^{unr}$ denotes the product $\widehat{\Omega} \times_{\operatorname{Spf}\mathcal{O}_{F_v}} \operatorname{Spf} \mathcal{O}_{F_v}^{unr}$ , where $\widehat{\Omega}$ denotes the $v$-adic upper half plane (viewed as a formal scheme), and $\mathcal{O}_{F_v}^{unr}$ the ring of Witt vectors with coefficients in $\overline{\kappa}_v$ . The action of $\gamma \in GL(F_v)$ on $\widehat{\Omega}^{unr}$ is via the image of $\gamma$ in $PGL(F_v)$ on the component $\widehat{\Omega}$, and via multiplication by $\operatorname{Frob}_v^{n(\gamma)}$ on $\widehat{\mathcal{O}}_{F_v}^{unr}$ , where $n(\gamma) = - ord_v \left( \det(\gamma) \right)$. As well, $D$ denotes the totally definite quaternion algebra over $F$ obtained from $B$ by switching invariants at $v$ and $\tau_1$, with $\overline{H}^v$ the compact open subgroup of $\widehat{D}^{\times v}$ corresponding to $H^v$ under a fixed isomorphism $B^{\times v} \cong D^{\times v}$. The theory of Mumford-Kurihara unifomization then gives the following information about this curve ${\bf{M}}_{H}^V$ :

The curve ${\bf{M}}_{H}^V$ is an admissible curve over $\mathcal{O}_{F_v}$ in the sense of Jordan-Livne, i.e.

(i) ${\bf{M}}_H^V$ is a flat, proper curve over $\mathcal{O}_{F_v}$ with a smooth generic fibre.

(ii) The special fibre of ${\bf{M}}_H^V$ is reduced; the normalization of each of its irreducible components is isomorphic to ${\bf{P}}^1_{\kappa_v}$ , and its only singular points are $\kappa_v$ -rational, ordinary double points.

(iii) The local ring ${\bf{M}}_{H, x}$ at any singular point $x$ of the special fibre is isomorphic as an $\mathcal{O}_{F_v}$ -algebra to $\mathcal{O}_{F_v}[[X,Y]]/(XY - \pi_v^{m(x)})$ , for $m(x) \geq 1$ a uniquely determined integer.

The dual graph $\mathcal{G}({\bf{M}}_H^V) = (\mathcal{V}({\bf{M}}_H^V),\mathcal{E}({\bf{M}}_H^V))$ of the special fibre of ${\bf{M}}_H^V$ is
isomorphic to $GL(F_v)^{+} \backslash \left( \Delta \times D^{\times}\backslash \widehat{D}^{\times}/\overline{H}^v \right)$ , minus any loops. Here, $GL(F_v)^+ \subset GL(F_v)$ denotes the collection of matrices with determinant having even $v$-adic valuation, and $\Delta = (\mathcal{V}(\Delta), \mathcal{E}(\Delta))$ the Bruhat-Tits tree of $SL(F_v)$.

My question is the following: why is the edgeset $\mathcal{E}({\bf{M}}_H^V)$ nonempty? The dual graph $\mathcal{G}({\bf{M}}_H^V)$ is clearly disconnected, and seen easily to be given by the disjoint union of connected graphs \begin{align*} \coprod_i \mathcal{G}_i &= \coprod_i \overline{\Gamma}_i \backslash \Delta. \end{align*} Here, each $\overline{\Gamma}_i$ denotes the image in $PGL(F_v)$ of a suitable arithmetic subgroup $\Gamma_i \subset D^{\times} \cong D_v^{\times} \cong GL(F_v)$. Each component graph $\mathcal{G}_i = (\mathcal{V}_i, \mathcal{E}_i)$ is connected. Now, since $\Delta$ is a tree, each component graph $\mathcal{G}_i = \overline{\Gamma}_i \backslash \Delta$ is a tree. It is then well known that each (first) Betti number $\beta(\mathcal{G}_i) := \vert \mathcal{E}_i \vert - \vert \mathcal{V}_i \vert + 1$ must vanish, i.e. $\vert \mathcal{E}_i \vert = \vert \mathcal{V}_i \vert -1 = 0$. If so, then the cardinality of the edgeset $\mathcal{E}({\bf{M}}_H^V)$ must also equal zero. i.e. the special fibre of ${\bf{M}}_H^V$ would have no singular points ... what have I missed here?

Soft proof of multiplicity one for character groups of Shimura curves?

2011-02-13T21:50:32Z

Is it not possible to prove mutiplicity one type statements for character groups of quaternionic Shimura curves by simply using Raynaud's description for character groups at primes dividing the underlying discriminant?

To be slightly more precise, let $F$ be a totally real field of degree $d$. Fix ideals $\mathfrak{M}^+, \mathfrak{M}^{-} \subset \mathcal{O}_F$ . Suppose that $\mathfrak{M}^{-}$ is the squarefree product of a number of primes congruent to $d-1 \mod 2$ . Assume that $(\mathfrak{M}^+, \mathfrak{M}^{-}) =1$ . Let $M(\mathfrak{M}^+, \mathfrak{M}^{-})$ denote the Shimura curve of level $\mathfrak{M}^+$ associated to the indefinite quaternion algebra of discriminant $\mathfrak{M}^{-}$ . (Here, "indefinite" means ramified at all but one of the real places of $F$ , hence the condition on $\mathfrak{M}^{-}$ ). Fix a prime $v \subset \mathcal{O}_F$ that does not divide $\mathfrak{M}^+ \mathfrak{M}^{-}$ . Assume from now on that the level of $M(\mathfrak{M}^+, \mathfrak{M}^{-})$ is maximal at $v$ . Then, there exists a nice integral model ${\bf{M}}(\mathfrak{M}^{+}, \mathfrak{M}^{-})$ of $M(\mathfrak{M}^+, \mathfrak{M}^{-})$ over $\mathcal{O}_{F_v}$ (due to Carayol in the case that $v \nmid \mathfrak{M}^{-}$ , or due to combined works of Cerednik and Varshavsky in the case that $v \mid \mathfrak{M}^{-}$ ). Let $J(\mathfrak{M}^{+}, \mathfrak{M}^{-})$ denote the jacobian of $M(\mathfrak{M}^{+}, \mathfrak{M}^{-})$ , with ${\bf{J}}(\mathfrak{M}^{+}, \mathfrak{M}^{-})$ its Neron model over $\mathcal{O}_{F_v}$, and ${\bf{J}}_v^0(\mathfrak{M}^{+}, \mathfrak{M}^{-})$ the component of the identity of its special fibre. Let $\mathcal{X}_v(\mathfrak{M}^{+}, \mathfrak{M}^{-})$ denote the character group of the maximal torus of ${\bf{J}}_v^0(\mathfrak{M}^{+}, \mathfrak{M}^{-})$. Additionally, given an ideal $\mathfrak{m} \subset \mathcal{O}_F$ that does not divide $\mathfrak{M}^+ \mathfrak{M}^{-}$, let $M(\mathfrak{m}; \mathfrak{M}+, \mathfrak{M}^{-})$ denote the Shimura curve $M(\mathfrak{M}^{+}, \mathfrak{M}^{-})$ with maximal level structure at primes dividing $\mathfrak{m}$ inserted (so, $M(\mathfrak{m}\mathfrak{M}^+, \mathfrak{M}^{-})$ is the Shimura curve of $\mathfrak{m}\mathfrak{M}^{+}$-level structure associated to the indefinite Shimura curve of discriminant $\mathfrak{M}^{-}$, with the extra condition that the level be maximal at primes dividing $\mathfrak{m}$).

Suppose now that we have ideals $\mathfrak{N}^+, \mathfrak{N}^{-} \subset \mathcal{O}_F$ such that $(\mathfrak{N}^+, \mathfrak{N}^{-}) =1$ and $\mathfrak{N}^{-}$ is the squarefree product of a number of primes congruent to $d \mod 2$. Given a ring $\mathcal{O}$, let $\mathbb{S}_2(\mathfrak{N}^{+}, \mathfrak{N}^{-}; \mathcal{O})$ denote the space of $\mathcal{O}$-valued automorphic forms of weight $2$ and level $\mathfrak{N}^+$ on the totally definite quaternion algebra of discriminant $\mathfrak{N}^{-}$ over $F$. Let $\mathbb{T}(\mathfrak{N}^+, \mathfrak{N}^{-})$ denote the associated algebra of Hecke operators. Given an ideal $\mathfrak{n} \subset \mathcal{O}_F$ that does not divide $\mathfrak{N}^+ \mathfrak{N}^{-}$, let $\mathbb{S}_2(\mathfrak{n}; \mathfrak{N}, \mathfrak{N}^{-}; {\bf{Z}})$ denote the space of forms of level $\mathfrak{n}\mathfrak{N}^+$, with the level being maximal at primes dividing $\mathfrak{n}$. Fix a prime $v \subset \mathcal{O}_F$ that does not divide the product $\mathfrak{N}^+ \mathfrak{N}^{-}$. Let us now take $\mathfrak{M}^+ = \mathfrak{N}^+$ and $\mathfrak{N}^{-} = v \mathfrak{N}^{-}$ in the Shimura curves setup above. In particular, consider the Shimura curve $M(\mathfrak{N}^+, v\mathfrak{N}^{-} )$, with $\mathcal{X}_v(\mathfrak{N}^+, v\mathfrak{N}^{-})$ the associated character group. Raynaud's theory gives a combinatorial description of $\mathcal{X}_v(\mathfrak{N}^+, v\mathfrak{N}^{-})$ in terms of the dual graph $\mathcal{G}_v = (\mathcal{V}(\mathcal{G}_v), \mathcal{E}(\mathcal{G}_v))$ of the special fibre of ${\bf{M}}(\mathfrak{N}^+, v\mathfrak{N}^{v})$ . That is, fix an orientation $s, t: \mathcal{E}(\mathcal{G}_v) \longrightarrow \mathcal{V}(\mathcal{G}_v)$ of the dual graph. Then, by Raynaud's theory, there is a short exact sequence of ${\bf{Z}}$ -modules

\begin{align*}\mathcal{X}_v(\mathfrak{N}^+, v\mathfrak{N}^{-}) \longrightarrow {\bf{Z}}[\mathcal{E}(\mathcal{G}_v)] \longrightarrow {\bf{Z}}[\mathcal{V}(\mathcal{G}_v)]^0. \end{align*} that identifies the character group $\mathcal{X}_v(\mathfrak{N}^+, v\mathfrak{N}^{-})$ with the kernel of the coboundary/degeneracy map $d_* = s_* - t_*$ . Let us now assume that $\mathfrak{N}^{-}\neq \mathcal{O}_F$ . Fix a prime divisor $\mathfrak{q} \mid \mathfrak{N}^{-}$ . Fixing a suitable orientation of the dual graph, and using various identifications (from the theory of Mumford-Kurihara uniformization on the bottom, then from Carayol's description supersingular points on the top), one can obtain the following diagram, where the vertical arrows are all isomorphisms:

Some details aside (for brevity -- and compilation issues), the important extraction to make here is that of the induced short exact sequence of $\mathbb{T}(v ; \mathfrak{N}^+, \mathfrak{N}^{-})$ -modules

\begin{align*} \mathcal{X}_v(\mathfrak{N}^{+}, v\mathfrak{N}^{-}) \longrightarrow \mathcal{X}_{\mathfrak{q}}(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q}) \longrightarrow \mathcal{X}_{\mathfrak{q}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})^2.\end{align*} In particular, via the induced isomorphism $\mathcal{X}_{\mathfrak{q}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q}) \cong \mathbb{S}_2(\mathfrak{N}^+, \mathfrak{N}^{-}; {\bf{Z}})$ , the character group $\mathcal{X}_{\mathfrak{q}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})$ inherits the structure of a $\mathbb{T}(\mathfrak{N}^+, \mathfrak{N}^{-})$ module. So, is it not clear (e.g. by Jacquet-Langlands plus multiplicity one for $\operatorname{GL}_2$ ) that $\mathcal{X}_{\mathfrak{q}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})$ is free of rank $1$ over $\mathbb{T}(\mathfrak{N}^+, \mathfrak{N}^{-})$ ? Moreover, suppose we take any maximal ideal $\mathfrak{m} \subset \mathbb{T}(\mathfrak{N}^+, \mathfrak{N}^{-})$ . Let $\mathcal{X}_{\mathfrak{q}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})_{\mathfrak{m}}$ denote the localization of $\mathcal{X}_{\mathfrak{q}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})$ at $\mathfrak{m}$ . Let $\mathbb{T}(\mathfrak{N}^+, \mathfrak{N}^{-})_{\mathfrak{m}}$ denote the localization of $ \mathbb{T}(\mathfrak{N}^+, \mathfrak{N}^{-})$ at $\mathfrak{m}$ . Is it not also clear that $\mathcal{X}_{\mathfrak{q}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})_{\mathfrak{m}}$ is free of rank $1$ over $\mathbb{T}(\mathfrak{N}^+, \mathfrak{N}^{-})_{\mathfrak{m}}$ ?

Selmer of an abelian variety versus that of its dual.

2011-03-09T14:18:20Z

What is the precise relationship between the Selmer group of an abelian variety and that of its dual? For instance, does the vanishing of one not imply the same for the other?

To fix ideas, let $A$ be an abelian variety defined over a number field $K$, with $A^t$ denote the corresponding dual abelian variety. Fix a rational prime $p$. Suppose for instance that we consider the compactified Selmer group \begin{align*} \mathfrak{S}(A/K) &= \ker \left( H^1(G_S(K), A_{p^n}) \longrightarrow \bigoplus_{v \in S} H^1(K_v, A(\overline{K}_v) )(p) \right).\end{align*} Here, $S$ is any finite set of primes of $K$ containing the primes above $p$ and the primes where $A$ has bad reduction; $G_S(K) = \operatorname{Gal}(K^S/K)$, where $K^S$ is the maximal extension of $K$ unramified outside of $S$ and the archimedean primes of $K$, and $A_{p^n} = \ker \left([p^n]:A \longrightarrow A\right)$ denotes the $p^n$ torsion of $A$. Does $\mathfrak{S}(A/K) =0$ if and only if $\mathfrak{S}(A^t/K)=0$? Clearly this will be the case if $A$ is principally polarized (in which case there is an isomorphism $A \cong A^t$). However, the general case seems tricky to prove by inspection. I am aware that if the $p$-primary part of the Tate-Shafarevich group $\operatorname{Sha}(A/K)$ is finite, then its follows from basic properties of the Cassels-Tate pairing that $\mathfrak{S}(A/K) \cong \mathfrak{S}(A^t/K)$ if and only if $A(K)_{p^{\infty}} \cong A^t(K)_{p^{\infty}}$ (where $A(K)_{p^{\infty}} = \bigcup_{n \geq 0} A(K)_{p^n}$ ). But, when is it the case that this latter condition is known (not) to be true? Is there not a better, perhaps unconditional, deduction? Surely this ought to be classical ...

Status of Ihara's lemma for Shimura curves over totally real fields?

2011-02-20T22:20:05Z

What is the status of Ihara's lemma for Shimura curves over totally real fields? In particular, why is it not implicit in the exact sequence of Rajaei, "On the levels of mod $l$ Hilbert modular forms" (Crelle 2001), Theorem 3 (3.18)?

To be a little bit more precise (or at least to give the main idea), let $F$ be a totally real field of degree $d$. Let $M(\mathfrak{M}^+, \mathfrak{M}^{-})$ be the Shimura curve of level $\mathfrak{M}^+ \subset \mathcal{O}_F$ associated to the indefinite quaternion algebra of discriminant $\mathfrak{M}^{-} \subset \mathcal{O}_F$, and $M(\mathfrak{m};\mathfrak{M}^+, \mathfrak{M}^{-})$ the Shimura curve with maximal level at primes dividing $\mathfrak{m} \subset \mathcal{O}_F$. Fix two coprime ideals $\mathfrak{N}^+, \mathfrak{N}^{-} \subset \mathcal{O}_F$ such that $\mathfrak{N}^{-}$ is the squarefree product of a number of primes ideals congruent to $d \mod 2$. Suppose that $\mathfrak{N}^{-}$ has at least one prime divisor $\mathfrak{q}$ say. Fix a prime $v \subset \mathcal{O}_F$ that does not divide $\mathfrak{N}^+\mathfrak{N}^{-}$ . Let ${\bf{M}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}/\mathfrak{q})$ and ${\bf{M}}(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})$ be the good reduction integral models over $\mathcal{O}_{(\mathfrak{q})}$ of $\bf{M}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}/\mathfrak{q})$ and $M(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})$ respectively. Ihara's lemma for Shimura curves over totally real fields, at least as I understand it, is the assertion that for any non-Eisenstein maximal ideal $\mathfrak{m}$ in the algebra $\mathbb{T}(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-})$ generated by Hecke operators acting on $M(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})$ , the map \begin{align*} H^1({\bf{M}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}/\mathfrak{q}), \mathcal{F})_{\mathfrak{m}}^{2} &\longrightarrow H^1({\bf{M}}(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q}), \mathcal{F})_{\mathfrak{m}} \\ (f_1, f_2) &\longmapsto 1_* f_1 +\eta_{v, *} f_2 \end{align*} is injective, where $\eta_v = \left( \begin{array}{cccc} 1 & 0 \\ 0 & \pi_v \end{array}\right)$ . Here, $\mathcal{F}$ is the usual sheaf defined by Carayol and Jarvis, and $\pi_v$ is a fixed uniformizer at $v$ . Rajaei shows that there is an injection of the associated dual character groups, \begin{align*} \widehat{\mathcal{X}}_{\mathfrak{q}}(\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}/\mathfrak{q}))_{\mathfrak{m}}^{2} &\longrightarrow \widehat{\mathcal{X}}_{\mathfrak{q}}(v\mathfrak{q}; \mathfrak{N}^+, \mathfrak{N}^{-}/\mathfrak{q})_{\mathfrak{m}} \\(f_1, f_2) &\longmapsto 1_* f_1 +\eta_{v, *} f_2 .\end{align*} Sorry if the question is naive! It is related to a previous question I asked here (http://mathoverflow.net/questions/55347/soft-proof-of-multiplicity-one-for-character-groups-of-shimura-curves), to the effect of whether or not proofs of these types of results can be streamlined for the case of parallel weight $2$. In particular, consider the Ribet/Rajaei exact sequence \begin{align*} \mathcal{X}_{v_2}(\mathfrak{N}^+, \mathfrak{p}v_1 v_2 \mathfrak{N}^{-}) \longrightarrow \mathcal{X}_{\mathfrak{p}}(\mathfrak{p}v_2; \mathfrak{N}^+; v_1 \mathfrak{N}^{-}) \longrightarrow \mathcal{X}_{\mathfrak{p}}(\mathfrak{p}; \mathfrak{N}^+, v_1 \mathfrak{N}^{-})^2.\end{align*} Here, in the notations of the previous question, $v_1, v_2, \mathfrak{p} \subset \mathcal{O}_F$ are distinct primes that do not divide the product $\mathfrak{N}^+ \mathfrak{N}^{-}$. Rajaei shows that the localization of the natural map \begin{align*} \widehat{\mathcal{X}}_{\mathfrak{p}}(v_2; \mathfrak{N}^+, v_1 \mathfrak{N}^{-})^2 &\longrightarrow \widehat{\mathcal{X}}_{\mathfrak{p}}(\mathfrak{p}v_2; \mathfrak{N}^+, v_1 \mathfrak{N}^{-}) \end{align*} given by $1_* \oplus {\eta_{\mathfrak{p}}}_*$ at any non-Eisenstein maximal ideal of the Hecke algebra $\mathbb{T}(\mathfrak{p}v_2; \mathfrak{N}^+, v_1\mathfrak{N}^{-})$ is injective. On the other hand, given the diagram in the previous question (so beautifully compiled by Dror Speiser), we have identifications \begin{align*} \mathcal{X}_{\mathfrak{p}}(\mathfrak{p}v_2; \mathfrak{N}^+, v_1 \mathfrak{N}^{-}) &\cong \operatorname{Div}^0 \left({\bf{M}}(\mathfrak{p}v_2; \mathfrak{N}^+, v_1 \mathfrak{N}^{-})^{ss} \otimes \kappa_{\mathfrak{p}} \right) \\ \mathcal{X}_{\mathfrak{p}}(\mathfrak{p}; \mathfrak{N}^+, v_1 \mathfrak{N}^{-}) &\cong \operatorname{Div}^0 \left({\bf{M}}(\mathfrak{p}; \mathfrak{N}^+, v_1 \mathfrak{N}^{-})^{ss} \otimes \kappa_{\mathfrak{p}} \right) \end{align*} Can we not then just take ${\bf{Z}}$-duals to deduce the result? This is the naive question.

Tamagawa numbers of abelian varieties and torsion.

2011-03-13T21:21:23Z

Let $A$ be an abelian variety defined over a number field $K$. Fix a prime $v \subset \mathcal{O}_K$, with underlying rational prime $p$. What relationship, known or conjectural (if any), should there be between the local Tamagawa number $c_v(A) = [A(K_v): A_0(K_v)]$ and the cardinality of the $p$-primary subgroup $A(K_v)(p)$ of $A(K_v)$?

I am aware of the recent work of Lorenzini, which considers possible cancellations of the ratio \begin{align*} \frac{\prod_{v \subset \mathcal{O}_K} c_v(A)}{\vert A(K)_{\operatorname{tors}}\vert}, \end{align*} and shows for instance this ratio for $A$ an elliptic curve defined over $K ={\bf{Q}}$ is always greater than or equal to $\frac{1}{5}$.

The question comes up naturally in a certain Euler characteristic computations (e.g. for the $p^{\infty}$ -Selmer group of $A$ defined over some profinite Galois extension $K_{\infty}$ of $K$ , where certain primes of $K$ are known to split completely, and hence where the torsion subgroup $A(K_{\infty})_{\operatorname{tors}}$ is known to be finite). In particular, granted the refined conjecture of Birch and Swinnerton-Dyer, it is apparent from these computations that $c_v(A)$ for a prime of bad (multiplicative) reduction $v$ is given essentially by the quotient \begin{align*}\frac{\vert H^1(G_w, A(K_{\infty, w}))(p)\vert }{\vert H^2(G_w, A(K_{\infty, w}))(p)\vert }. \end{align*} Here, $K_{\infty, w}$ is the union of all completions at primes above $v$ in $K_{\infty}$ , and $G_w$ denotes the Galois group $\operatorname{Gal}(K_{\infty, w}/K_v)$ . If $A$ has good ordinary reduction at all primes above $p$ in $K$ , then it is possible to use the Coates-Greenberg theory of deeply ramified extensions to characterize the local factor at $v$ in the Euler characteristic formula coming from this quotient as $\vert \widetilde{A}(\kappa_v)(p)\vert^2$ , where $\widetilde{A}$ is the reduction of $A$ mod $v$ , and $\kappa_v$ the residue field at $v$ (which is consistent with B+S-D). But, this characterization seems to break down when $A$ does not have good ordinary reduction at $v$ , and I have not found any calculations in the literature for this case of bad reduction. Hence why I ask about any possible known relation to torsion. Sorry if the question is silly!

Answer by jvo for A non-technical account of the Birch—Swinnerton-Dyer Conjecture

2011-03-12T09:57:06Z

In my opinion, the best non-technical overview has to be in the Clay Millenium Problems description of Wiles:

http://www.claymath.org/millennium/Birch_and_Swinnerton-Dyer_Conjecture/birchswin.pdf

The conjecture, far from being resolved (or perhaps even formulated correctly), is very much an active area of research. Moreover, there is not yet a good conceptual framework in which to view the problem, which perhaps explains the lack of non-technical or popular literature on the subject.

If you are looking to learn about B+S-D in a more serious way, or would like a nice (historical) overview, you might enjoy the paper of John Tate, "The Arithmetic of Elliptic Curves", Inventiones 23 (1974).

Answer by jvo for Status of Ihara's lemma for Shimura curves over totally real fields?

2011-03-02T10:06:53Z

Okay, so the answer to the naive question is most likely no for several reasons. For instance, the identifications explained in the second paragraph show that the injection in Rajaei is really an analogue of Ihara's lemma for totally definite quaternion algebras (via Carayol's description of supersingular points). As well, the cocharacter groups only give information about the p-new automorphic forms. In general, it seems very unlikely that Ihara's lemma for Shimura curves over totally real fields can ever be reduced to the totally definite case, at least via naive methods. As for the status of Ihara's lemma itself, there does appear to be at least one work towards a generalization to totally real fields of the method of Diamond and Taylor, for instance in the preprint of Chuangxun Cheng (Ph.D. student of Emerton) ...

Answer by jvo for Nonvanishing of central L-values of quadratic twists?

2011-02-25T12:05:54Z

Though it is perhaps not an "answer" as such, let me try to explain some intuition. In certain settings, it is possible to formulate subtle analogues of Mazur's conjecture for nonvanishing of central values (think Cornut-Vatsal) from the refined conjecture of Birch and Swinnerton-Dyer via Iwasawa theory. A general method is explained in section 4 of Coates-Fukaya-Kato-Sujatha, " Root numbers, Selmer groups, and non-commutative Iwasawa theory" (available at http://www.math.tifr.res.in/~sujatha/root.pdf). CFKS consider the setting of the so-called False-Tate curve extension, but a similar (and simpler) set of arguments can be used to deduce an analogous conjecture for the setting of the ${\bf{Z}}_p^2$ of an imaginary quadratic field. To be slightly more precise, fix a rational prime $p$. Fix an eigenform $f \in S_2(\Gamma_0(N))$ with $(N,p)=1$. Fix an imaginary quadratic field $k$ with discriminant prime to $pN$. Let $k_{\infty}$ denote the ${\bf{Z}}_p^2$-extension of $k$, which is the compositum of the cyclotomic ${\bf{Z}}_p$-extension $k^c$ with the anticyclotomic ${\bf{Z}}_p$-extension $k^a$. Write $\lambda_f(k)$ to denote the cyclotomic $\lambda$-invariant associated to $f$, with $\mu_f(k)$ the cyclotomic $\mu$-invariant. Let $\mathcal{W}$ be any finite order character of $\operatorname{Gal}(k_{\infty}/k)$. Such a character can always be written as a product of characters $\rho \cdot \chi$, where $\rho$ is a finite order character of $\operatorname{Gal}(k^a/k)$, and $\psi$ is a finite order character of $\operatorname{Gal}(k^c/k)$. What the CFKS conjecture predicts, very roughly, is the following assertion. Assume that $\mu_f(k)=0$, and fix a finite order character $\rho$ of $\operatorname{Gal}(k^a/k)$. Let $\Psi$ denote the set of finite order character of $\operatorname{Gal}(k^c/k)$. Then, assuming the refined Birch and Swinnerton-Dyer conjecture, \begin{align*} \sum_{\psi \in \Psi} \operatorname{ord}_{s =1}L(f \times \rho \cdot \psi, s) &\leq \lambda_f(k). \end{align*} So, what does this tell us? Well for one, it tells us that Rohrlich nonvanishing (at least in this setting) should be a general phenomenon. One can make this intuition slightly more precise via the following heuristic argument. View any finite extension of $k^c$ over $k$ as a totally imaginary quadratic extension of its maximal totally real subfield. Suppose that the nonvanishing theorem of Cornut-Vatsal were uniformly effective (in the sense that their $n$ sufficiently large could be replaced by some absolute $n_k$ that does not grow as we ascend the cyclotomic tower). Then, invoking their result systematically and decomposing via Artin formalism, we should (I think) expect the following behaviour. Let $\epsilon(f/k, s)$ denote the root number of the Rankin-Selberg $L$-function $L(f/k, s)$. Given $\rho$ a finite order character of $\operatorname{Gal}(k^a/k)$ of conductor greater that $p^{n_k}$, we should have:

\begin{align*} \sum_{\psi \in \Psi} \operatorname{ord}_{s=1} L(f \times \rho \cdot \psi, s) &= 0 \text{ if $\epsilon(f/k, 1) = 1;$} \end{align*}

\begin{align*} \sum_{\psi \in \Psi} \operatorname{ord}_{s=1} L(f \times \rho \cdot \psi, s) &= 1 \text{ if $\epsilon(f/k, 1) = -1;$} \end{align*}

Sorry to have skipped steps, or if this is perhaps somewhat unclear in places. The main idea is that there are subtle generalizations of Mazur's conjecture to the types of settings that you will likely want to consider. These generalizations suggest that one should expect generic nonvanishing à la Rohrlich, even in the case where the root number at the bottom is $-1$. And though it is not a priori clear, it might be possible to use these sorts of ideas to obtain the general formulation of Goldfeld's conjecture that you ask for.

Comment by jvo

2011-06-15T07:49:29Z

This is just a comment, because I have not read the through Katz's construction in detail. My impression is that the problem could be a very hard one, because there would be an implicit need to compare the p-adic period(s) in Katz with the period used in Hida/Perrin-Riou (the Petersson inner product of the underlying eigenform). Also, note that the two-variable measure of Hida/Perrin-Riou can be seen to take integral values in the field of definition, as a consequence of the fact that Hida's bounded linear form sends integral-valued forms to integral-valued forms (which is easy to check).

Comment by jvo

2011-03-25T09:04:14Z

Many thanks for these excellent comments! I hadn't considered the issue with genus for modular curves, which is a good observation to make here.

Comment by jvo

2011-03-14T09:26:06Z

Hi again, and thanks for these too! I have already done and written the full calculation assuming $A$ has good ordinary reduction at all primes above $p$ in $K$ for this particular $K_{\infty}$ (which does not contain the cyclotomic ${\bf{Z}}_p$-extension of $K$). I will definitely have a look at the papers of Kobayashi (?), Iovita-Pollack and Perrin-Riou now. I usually avoid the supersingular setting for simplicity ... and yes, I am jvo :)

Comment by jvo

2011-03-14T09:00:28Z

Thanks, this is very helpful. I am of course not attempting the case of good non-ordinary reduction here! Thanks as well for the reference.

Comment by jvo

2011-03-10T07:12:59Z

Excellent answer, thanks!

Comment by jvo

2011-03-10T07:12:43Z

Ah yes, thanks for pointing this out!

Comment by jvo

2011-03-09T17:28:50Z

I think that any isomorphism should do. And I agree that if p divides the isogeny, then it is not at all clear whether or not such an isomorphism should hold!

Comment by jvo

2011-02-26T09:48:10Z

Yes, that is exactly what the notation means, and there is no mistake with the root number. Sorry to have compressed so much information into such a short paragraph. The deduction in fact requires far more justification than I have given. I wrote a paper about this while I was a grad student, but never submitted it. (The result seemed to be well known to experts, and has also been subsumed by subsequent results/projects). Perhaps I'll write a note about this now, as the deduction is perhaps not as well understood as I had assumed ... in which case, I will post a link here :)

Comment by jvo

2011-02-25T10:36:15Z

Kevin is absolutely right about the sign, as well as about Rohlich's cyclotomic nonvanishing theorem (not to be confused with Rohlich's many other nonvanishing theorems). However, as monodromy points out correctly, the nonvanishing of the associated p-adic L-function is in fact a consequence of Rohrlich. I do not know of any precise generalization of Goldfeld to the setting that you mention. However, if one assumes the refined conjecture of Birch and Swinnerton-Dyer, it is possible to use Iwasawa theory to make some very precise (and surprising) conjectures about nonvanishing of central values

Comment by jvo

2011-02-21T09:50:38Z

Great idea, thanks! I've only ever looked at the last few sections of that paper, I suppose I should have read it all ...

Comment by jvo

2011-02-21T09:31:56Z

Awesome, thanks!

Comment by jvo

2011-02-20T19:53:32Z

Excellent answer, thanks!

Comment by jvo

2011-02-20T18:44:35Z

Why, thanks for the tip! Sadly, of the two local copies of Freitag-Kiehl, one is on extended loan, and the other is "missing" ...

Comment by jvo

2011-02-14T10:07:57Z

@Emerton: Hi, thanks for these comments. Sorry the LaTeX is (or was) a bit of a mess, I was writing in a rush. I had meant to ask about either case, though I suppose the case of tensoring with ${\bf{Z}}$ is more relevant for the applications I have in mind. I would be happy to see the more detailed answer for the latter case. @Dror Speiser: Many thanks for the fixup!

Comment by jvo

2011-02-08T08:33:34Z

@David Hansen: Thanks for pointing this out!