28
votes
0answers
611 views

“Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...
22
votes
2answers
1k views

Why should I care about topological modular forms?

There seems to be a lot of recent activity concerning topological modular forms (TMF), which I gather is an extraordinary cohomology theory constructed from the classical theory of modular forms on ...
2
votes
0answers
164 views

“Extended” Weil Cohomology Theories

According to Wikipedia, a Weil cohomology theory is a functor from the category of smooth projective varieties over a field $k$, to graded algebras over a field $K$ of characteristic zero, together ...
5
votes
2answers
257 views

How does Tate cohomology fit into a derived categories framework?

I've read through one class field theory text after another, but there's something very non-intuitive for me about cohomology that makes it hard for me to understand why Tate cohomology was invented. ...
6
votes
1answer
491 views

Equivalence between statements of Hodge conjecture

Dear everyone, I was unable to obtain the equivalence between the two statements of the Hodge conjecture. I searched for some previous questions that others asked here, to check whether someone has ...
11
votes
0answers
296 views

Cohomological interpretations of quadratic form invariants over rings?

The standard approach to classifying of quadratic forms over $\Bbb Q$ is to use the Hasse (local-global) principle together with a system of standard invariants of quadratic forms over the local ...
10
votes
2answers
560 views

Examples of Galois-invariant central simple algebras which aren't base change?

Suppose $L/K$ is a Galois extension of number fields, with Galois group $G_{L/K}$. Write $\mathrm{Br}(L)^{G_{L/K}}$ for the subgroup of central simple algebras $A/L$ which are Galois-invariant; ...
18
votes
1answer
1k views

De Rham cohomology of formal groups

Let $G$ be some (dimension $1$, to simplify) formal group over a characteristic $0$ field $K$. The law of $G$ is denoted by $\oplus$. If $w(X) \in K[[X]] dX$ is a differential form, let $F_w(X)$ be ...
3
votes
5answers
999 views
1
vote
0answers
210 views

cohomological dimension for tamely ramified extenstions

Let $K$ be a local field with $K^{ur}$ the maximal unramified extension and $K^{tr}$ the maximal tamely ramified extension. Assume that the characteristic of the residue field of $K$ is $q$. If $p$ ...
12
votes
2answers
1k views

Why does the Gamma function satisfy a functional equation?

In question #7656, Peter Arndt asked why the Gamma function completes the Riemann zeta function in the sense that it makes the functional equation easy to write down. Several of the answers were from ...
15
votes
0answers
392 views

Cohomological characterization of CM curves

In his 1976 classical Annals paper on $p$-adic interpolation, N. Katz uses the fact that if $E_{/K}$ is an elliptic curve with complex multiplications in the quadratic field $F$, up to a suitable ...
3
votes
2answers
306 views

Can an abelian variety be represented as the cohomology of some other object?

Question Given an abelian variety $V$ and an integer $n$, is there a natural abelian category with a natural object $X$ and natural coefficients $F$ so that $V\simeq H^n (X,F)$? Motivation Studying ...
1
vote
1answer
754 views

Neukirch's class field axiom and cohomology of units for unramified extension

This question may be too detailed but perhaps somebody knows the answer: Neukirch proofs in his algebraic number theory book in chapter IV, proposition 6.2, that his class field axiom implies that the ...
9
votes
4answers
748 views

Abel's equation for the dilog

Abel's identity for the dilogarithm (see the wikipedia page about polylogarithms) plays a role in web geometry as it is one of the abelian relations of the first example of exceptional web (Bol's ...