Maximal unramified quotient of $E[p]$ for the action of $G_{\mathbb{Q}_p}$

Let $ E$ be an elliptic curve with good and ordinary reduction at an odd prime $ p$ . Suppose $ E[p]$ denotes the $ p$ -torsion points of $ E$ and $ G_{\mathbb{Q}_p} := \text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ .

In the article `Selmer group and congruences (page 6)’, Greenberg says that one can characterize $ \widetilde{E}[p]$ as the maximal unramified quotient of $ E[p]$ for the action of $ G_{\mathbb{Q}_p}$ where $ \widetilde{E}$ denotes the reduction of $ E$ in $ \mathbb{F}_p$ .

This is so because $ p$ is assumed to be odd and therefore the action of the inertia subgroup of $ G_{\mathbb{Q}_p}$ on the kernel of the reduction map $ \pi: E[p] \longrightarrow \widetilde{E}[p]$ is nontrivial.

It will be every helpful if someone can explain how `$ p$ being odd’ is playing a role in proving the non trivial action of the inertia subgroup on the kernel of the reduction map $ \pi$ ?

How to record time for every action in Google Sheet

I am new to Google Sheet, I have just started to use recently. So I am not familiar with.

In Google Sheet, I want to record my timing for every actions while I start and end an action.

For example, if I have three columns, A1 = Start time; A2 = End time; A3 = Time taken (A2-A1, this we can use it in formula)

But I want to make the time recording by automatically, e.g., Before I start my action, I wish to click on A1, the cell will automatically shows the current time and it should not change afterwards. Then, after I completed my work, I wish to click on A2 to record end time that means to record end time of my action to calculate actual time taken for my action.

Is it possible to get the current time by clicking the cell and it should not change afterwards?

Sync Google Docs Action Items with Google Tasks

In Google Docs (as well as Sheets and Slides), one can assign an “action item” (e.g. some to-do) to another person by marking him/her in a comment (link).

Also, there is the “Google Tasks” feature available in Google’s Mail/Calendar service (link).

I am hoping to sync the two such that any “action” I have been assigned to in a Google Doc also shows up in my Tasks List as an item.

Is there a way to achieve this?

How to remove duplicated code when performing action to a Python list filtered by an if statement

I have written the following function for a CLI tool, it works as expected but it also contains duplicated code.

What would be an elegant solution to get rid of the duplicated part?

def populate_teams_tab(self):     teams = self.get_data(self.TEAMS_URL)     returned_list = []     for team in teams:         if self.IS_ROGUE:             if team['id'].upper() == "ROGUE":                 # Here I have a few lines of manipulation that are adding all the ROGUE teams to a list that will be returned, I will simplify with this example                 returned_list.append(team)         else:                 # Here I have the _same_ lines of manipulation that are adding all the teams to a list that will be returned, I will simplify with this example                 returned_list.append(team) 

Ways to attack with move action

So, I’m playing pathfinder with a warrior DEX based, I use power attack mythic in my attacks and try to attack as much as I can. So I have this friend that can give me mithic haste that give me a bonus move action, but I have the most hp in the group(that means the tanker) so I need to stay close of the enemies that reduce some actions to use if extra movement. So with that in mind there is a way to convert move action in attack?

What is the result of a Readied Action to move the target triggered by a melee attack attempt?


Fred the Fighter wants to survive another round in the ring with Bob the Barbarian. He’s a skilled fighter, and has taken Combat Maneuver feats. Bob moves in to attack, but Fred readied to perform a Combat Maneuver. He specifically says “when the Barbarian attempts to melee attack me, I Bull Rush him”. Bob is pushed outside of his reach from Fred.

What does Bob’s turn look like? He’s already moved, and was in the middle of attempting a melee attack (which no longer has a valid target).

Would any of these situations change the situation enough to justify another question?

  • Bob’s was already within reach, and it was his first attack in a Full Attack action
  • Fred’s Bull Rush leaves Bob near a different valid target (ally or not)
  • Fred ready-Grapples Bob instead

User Custom Action in Context Menu of Modern List

I am developing a user custom action to be used in modern list/library views. The issue is, as we cannot use JavaScript in context menu custom action, how should I form a valid URL. Now I have a URL like:{StandardTokens}&Action=test&ListId={ListId}&ItemId={ItemId}&Source={Source}

I want to encode the above URL in C# but in such a way that dynamic params like {ListId} works. It is easy to do this in JS by using escape but not working without javascript. I have already tried HttpUtility.JavaScriptStringEncode and few other encoding mechanisms but all are failing one way or other.

URL that works with JS (this works as ribbon but in context menu, this is not allowed):

string UrlAction = "javascript:window.location='" + BaseUrl + "&redirect_uri=' + escape('" + remoteWebUrl + "/Pages/Default.aspx?{StandardTokens}&Action=test&ListId={ListId}&ItemId={ItemId}&Source={Source}')"; 

When the action of the gauge group on the space of connections is free?

Let $ G$ be a compact Lie group. Let $ \mathcal{A}$ be the space of connections on the principal trivial $ G$ -bundle $ G\times \mathbb{R}^4$ possibly with some growth condition (to specify it is a part of the question). The gauge group $ \mathcal{G}:=Maps(\mathbb{R}^4\to G)$ acts on $ \mathcal{A}$ in the usual ways.

Can the action of $ \mathcal{G}$ on $ \mathcal{A}$ be free? E.g. for $ G=SU(2)$ ? If not, is it true that the set of connections with non-trivial stabilizers (or infinitesimal stabilizers) is ‘very small’ in some sense?

Remark. If $ G=U(1)$ then the action of $ \mathcal{G}$ on $ \mathcal{A}$ is free provided we impose a growth condition on connections such that they should vanish at infinity at least along a given direction.