Impact
~9k
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- 0 posts edited
- 2 helpful flags
- 11 votes cast
Nov
22 |
awarded | Curious |
Nov
21 |
asked | How much extra ramification in a residual representation |
Oct
27 |
comment |
Residual Representation of a Motive
If you give me explicit algebraic equations for the addition law on the abelian variety, then the n-torsion is easy to write down as a variety: it's the preimage of a point under certain polynomials. That's the first step in finding the trace of Frobenius with Schoof-Pila, by working over the n-torsion symbolically. |
Oct
27 |
comment |
Residual Representation of a Motive
What is the number field corresponding to the 2-adic residual rep of the hypergeometric motive H([2,2,2,2],[1,1,1,1],-1/256) then? |
Oct
26 |
awarded | Yearling |
Oct
26 |
comment |
Residual Representation of a Motive
That's the first step in Schoof-Pila. See people.maths.ox.ac.uk/pila/Frobenius.pdf for the details of how we do much more over an abelian variety. |
Oct
26 |
comment |
Residual Representation of a Motive
I want a number field K, specified as a polynomial, and an explicit isomorphism of the Galois group onto a matrix group, such that the image of Frob under this element is the image of Frob in the representation, up to conjugacy of course. |
Oct
25 |
answered | Computing millions of coefficients of non self-dual modular forms |
Oct
25 |
comment |
Residual Representation of a Motive
Yes, they are computable basically instantly with Magma for most smallish primes. |
Oct
24 |
comment |
Computing millions of coefficients of non self-dual modular forms
What algorithms are you using in Sage and Magma? There are several different approaches, and if we knew which ones it might help in starting to look for solutions. |
Oct
24 |
asked | Residual Representation of a Motive |
Oct
13 |
comment |
Equivalence between Diffie Hellman and Discrete Log
You did, but the results are considerably weaker than what is commonly assumed true. So when someone gives an explicit reduction to CDH, the protocol won't be as secure if they were to use the above paper instead of the standard conjecture, and the difference is significant. |
Oct
13 |
comment |
Equivalence between Diffie Hellman and Discrete Log
Beware of asymptotic equivalences! For groups of realistic size, say $|G| \approx 2^{256}$, standard conjectures on the difficulty of the computational Diffie-Hellman problem assume $2^{128}$ difficulty, same as discrete log. Applying the reduction above requires $2^16$ calls to the DH oracle. As a result we only get out that DH is as hard as $2^102$, which is weaker than the commonly used conjecture, and a substantial gap from our best algorithms. So while the above paper suggests an asymptotic equivalence, the constant factors make it hard to apply in practice. |
Oct
12 |
answered | Why should curves be two-dimensional? |
Aug
11 |
awarded | Commentator |
Aug
11 |
comment |
Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes
A congruence class is determined by its traces in all representations. So we may need several modular forms, but once we have them we can say "a prime $p$ has Frob in this congruence class if and only if the coefficients have certain values". You might not be satisfied with this answer, and in some ways I'm not either, but this is analogous to what class field theory gives us. |
Aug
10 |
answered | Explicit Chebotarev and Langlands - irreducibility of X^5-X-1 mod primes |
Jul
21 |
answered | Optimal covering and CSPNG |
Jun
4 |
asked | Uniqueness of cohomological holomorphic discrete series representation |
May
10 |
comment |
genus 2 Siegel theta series of 3-dimensional lattices
I think Kudla's conjecture on towers tells you this, but I don't know if this proven. Also, even if there is a kernel on the space of forms, it could still be injective for lattices. Maybe this automorphic approach isn't the right one. |