Local-global compatibility and modular curves

I have been told by some people that local-global Langlands compatibility for $ GL_2$ (the vanilla version, not the one being developed in this decade by Emerton and others) implies Shimura conjecture on reduction of the quotient of Jacobians of modular curves. How does this implication work? There are also some stronger results of Katz–Mazur, I think. Can Langlands be applied to prove them?

Degree-3 curves on the Calabi–Yau quintic

Robbert Dijkgraaf said,1 concerning the simplest Calabi–Yau space, the quintic:

“A classical result from the 19th century states that the number of lines — degree-one curves — is equal to 2,875. The number of degree-two curves was only computed around 1980 and turns out to be much larger: 609,250. But the number of curves of degree three required the help of string theorists.”

I gather from OEIS sequence A076912 that the number is 317,206,375.

Q. Can anyone point me to (or describe) how the degree-3 count was settled with “the help of string theorists”?

1Robbert Dijkgraaf. “Quantum Questions Inspire New Math.” Quanta. The Best Writing in Mathematics 2018, Ed. M. Pitici. p.80. Princeton. Publisher’s link.

ParametricPlot of multiple curves Plot y[z][t] versus x[z][t] with both change with t (continuous variable) and z (represent different groups)

Two functions both change with the same independent variable t, there is another variable represent different groups (z). how to plot y[t] versus x[t] with each group of z represented by one independent curve. See example equation below, with z as a separate integer variable that varies while t is a continuous variable. Please note I do not plan to use Manipulate function.

x[z_][t_] := z/5*Exp[-0.1*t] y[z_][t_] := 500/(2 + x[t]) z={1,2,5,10,20} ParametricPlot[{Evaluate[x[z][t]],Evaluate[y[z][t]},{x,0,100},{z,zlist},AspectRatio -> 1] 

How to prove two curves in the frame bundle to project to the same curve on base manifold?

There is a problem about Cartan’s development, arising from the paper ‘Kinetic Brownian motion on Riemannian manifolds’, Subsection 2.4.1. To be precise, let $ (M,g)$ be a $ d$ -dimensional complete Riemannian manifold, $ \pi:OM\to M$ be its orthonormal frame bundle with structure group $ O(d)$ . Denote by $ H_v$ the standard horizontal vector field on $ P$ corresponding to $ v\in\mathbf R^d$ , uniquely characterized by the property that $ \pi_*(H_v(z)) = e(v)$ for all $ z = (x,e) \in OM$ . Let $ \{\epsilon_1,…,\epsilon_d\}$ be the canonical basis of $ \mathbf R^d$ , with dual basis $ \{\epsilon_1^*,…,\epsilon_d^*\}$ . Denote by $ V_i, 1\le i\le d$ the vertical vector field induced by $ a_i=\epsilon_i \otimes \epsilon_1^* – \epsilon_1 \otimes \epsilon_i^* \in SO(d)$ .

Given a smooth curve $ \{m_t\}_{0\le t\le1}$ , define the Cartan’s development of $ \gamma$ on $ OM$ as the solution to the ODE on $ OM$ , \begin{equation}\tag{1} \dot z_t = H_{\dot m_t}(z_t), \quad z_0 = (x_0,e_0)\in OM. \end{equation}

Now Assume $ \{m_t\}_{0\le t\le1}$ is run at unit speed, i.e., $ |\dot m_t|\equiv 1$ . Then, given an orthonormal basis $ f_0$ of $ \mathbf R^d$ with $ f_0(\epsilon_1) = \dot m_0$ , solve the following ODE on $ SO(d)$ , \begin{equation} \dot f_t = \sum_{i=2}^d (f_t(\epsilon_i),\ddot m_t)a_i(f_t), \end{equation} started from $ f_0$ , and define the $ \mathbf R^{d-1}$ -valued path $ \{h_t\}_{0\le t\le1}$ , starting from zero, by the ODEs \begin{equation} \dot h^i_t = (f_t(\epsilon_i),\ddot m_t), \quad 2\le i\le d. \end{equation} Consider the following ODE on $ OM$ , \begin{equation}\tag{2} \dot{\tilde z}_t = H_{\epsilon_1}(\tilde z_t)+ \sum_{i=2}^d V_i(\tilde z_t) \dot h_t^i, \quad \tilde z_0 = (x_0,e_0)\in OM. \end{equation} Then the paper, mentioned in the very beginning, has the following claim:

Claim: $ \pi(\tilde z_t) = \pi(z_t)$ .

But why?

I try to prove this claim. But I am not able to finish that.

Use the coordinate system $ (x^i,e_l^k)$ on $ OM$ . Then \begin{align} H_v &= v^j e_j^i \partial_{x^i} – v^r \Gamma^k_{ij} e_l^j e_r^i \partial_{e_l^k}, \ V_i &= e_i^k \partial_{e^k_1} – e_1^k \partial_{e_i^k}. \end{align} On the one hand, Eqn. (1) is represented as \begin{equation}\left\{ \begin{aligned} \dot{x}^i &= e_j^i \dot m^j, \ \dot e_l^k &= -\Gamma^k_{ij} e_l^j e_r^i \dot m^r, \end{aligned} \right. \end{equation} where $ \Gamma^k_{ij}$ are the Christoffel’s symbols of the metric $ g$ . We can obtain \begin{equation} \ddot x^i = \dot e_j^i \dot m^j + e_j^i \ddot m^j = -\Gamma^i_{kl} e_j^l e_r^k \dot m^r \dot m^j + e_j^i \ddot m^j = -\Gamma^i_{kl} \dot x^l \dot x^k + e_j^i \ddot m^j, \end{equation} that is, \begin{equation}\tag{1*} \frac{\nabla \dot x^i}{dt} = e_j^i \ddot m^j. \end{equation} On the other hand, Eqn. (2) is represented as \begin{equation}\left\{ \begin{aligned} \dot{\tilde x}^i &= \tilde e_1^i, \ \dot{\tilde e}_1^k &= -\Gamma^k_{ij} \tilde e_1^j \tilde e_1^i + \sum_{i=2}^d \tilde e_i^k \dot h^i, \ \dot{\tilde e}_l^k &= -\Gamma^k_{ij} \tilde e_l^j \tilde e_1^i – \tilde e_1^k \dot h^l, \quad 2\le l \le d. \end{aligned} \right. \end{equation} We have \begin{equation} \ddot{\tilde x}^i = \dot{\tilde e}_1^i = -\Gamma^i_{kj} \tilde e_1^j \tilde e_1^k + \sum_{k=2}^d \tilde e_k^i \dot h^k = -\Gamma^i_{kj} \dot{\tilde x}^j \dot{\tilde x}^k + \sum_{k=2}^d \tilde e_k^i \dot h^k, \end{equation} that is, \begin{equation}\tag{2*} \frac{\nabla \dot{\tilde x}^i}{dt} = \sum_{k=2}^d \tilde e_k^i \dot h^k. \end{equation} If (1*) and (2*) are the same ODE, then under the same initial condition $ x(0) = \tilde x(0) = x_0$ , we have $ x=\tilde x$ , which proves the claim. But I do not know how to compare (1*) and (2*).

Can anyone give some hints or reference? TIA…

PS: This is a crosspost from math.stackexchange.

Higher dimensional generalization of an identity between traces of Hecke operators and number of elliptic curves over finite fields?

In http://www.math.ubc.ca/~behrend/ladic.pdf, the author uses his generalization of Lefschetz trace formula to smooth algebraic stacks to prove an identity (Proposition 6.4.11.):

$ \sum_{k} \frac{1}{p^{k+1}} tr T_p |_{S_{k+2}}=1- \frac{1}{p^3-p} – \sum_{E / \mathbb F_p}\frac{1}{\#E(\mathbb F_p) \# Aut E(\mathbb F_p)}$ .

Here $ T_p$ is the $ p$ -th Hecke operator on the space of cusp forms $ S_{k+2}$ of weight $ k + 2$ . The sum on the right hand side extends over all isomorphism classes of elliptic curves over $ \mathbb F_p$ .

In the proof, he apply our trace formula to the algebraic stack of curves of genus one. Can we generalized this identity to other Shimura varieties ? For example, can we apply Behrend trace formula to some moduli stacks describing torsors of abelian varieties?