16
votes

Accepted

### What are $L$-functions?

You seem to be looking for a single unifying idea that explains why $L$-functions are the way they are. I don't think this is possible. $L$-functions are more like an elephant, with different ...

13
votes

Accepted

### Understanding the Hodge filtration

The naive Hodge filtration of a smooth affine variety is, indeed, the whole thing. We always have the short exact sequence of complexes:
$$0 \to \Omega^{\bullet, \geq p} \to \Omega^{\bullet} \to \...

10
votes

Accepted

### Rational isogenies of prime degree $p\in\{11,17,19,37,43,67,163\}$

Five of these $j(E)$ values, one each for $p=11,19,43,67,163$,
are for curves $E$ with complex multiplication by the ring of integers
in the quadratic field of discriminant $-p$. All of them are ...

8
votes

Accepted

### Bad prime of torsor and original elliptic curve ; Definition of Tate–Shafarevich group $Ш(E/K)$

I fear you wish for too much here.
If $Ш$ is finite, then we can represent each element by a torsor; each torsor has good reduction away from a finite set and the union of all bad places would then be ...

7
votes

Accepted

### Action of complex conjugation on etale cohomology

The Weil pairing (or Poincaré duality in étale cohomology) gives a Galois-equivariant symplectic form $$H^1(\overline{X}, \mathbb Q_p) \times H^1(\overline{X}, \mathbb Q_p) \to \mathbb Q_\ell(-1).$$
...

6
votes

Accepted

### What's the relation between analytic stacks and higher complex/non-archimedean analytic stacks?

One way to think about categories of stacks (or more generally, (higher) topoi) is that one has some class of generating objects, and some class of allowed gluings; and then one builds the full ...

4
votes

### The definition of ${}^{\circ} \mathcal{HT}^{\mathcal{D}\text{-}\Theta^{\pm \mathscr{ell}}}$ in Inter-universal Teichmüller theory

Let me triage my questions. I think Dr. David Roberts and me agree Mochizuki's definition is not rigorous. The point being disagreed with is how math papers having non-strict arguments should be ...

4
votes

Accepted

### Bounding $H^4_{\text{ėt}}$ of a surface

Are you absolutely sure you want to compute $p$-adic etale cohomology for a smooth proper $\mathbb{Z}[1/S]$-scheme with $p \notin S$, so $p$ is not invertible on $X$? This will be painful, and I ...

3
votes

Accepted

### Difficulties in the proof of finiteness of n-Selmer group using cohomology

(Not sure any of these questions are at the right level for this forum, but here the comments that may help.)
question : Inflation-restriction sequence.
question : The target can be identified with ...

3
votes

Accepted

### Zeta function of variety over positive characteristic function field vs. local zeta factor of variety over $\mathbb{F}_p$

The two zeta functions are the same. This is an immediate corollary of Milne, Etale Cohomology, proposition 13.8(c).
Reference:
Milne, J. S. Etale Cohomology (PMS-33). Princeton University Press, 1980....

2
votes

Accepted

2
votes

### Bloch–Beilinson conjecture for varieties over function fields of positive characteristic

This may not be precisely what you want, but a function field analogue of Beilinson's conjectures is formulated in R. Sreekantan, Non-Archimedean regulator maps and special values of $L$-functions, ...

2
votes

Accepted

### Grössencharakter or Galois representation associated to a CM elliptic curve in characteristic $p$

The construction of the Galois representation works equally well over every field. It is the Galois action on the $\ell$-adic Tate module, viewed as a rank one free module over $\mathbb Q_\ell \...

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