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2y

*Research talks*

Let $X$ be a projective variety over a field $k$. Chow groups are defined as the quotient of a free group generated by irreducible subvarieties (of fixed dimension) by some equivalence relation (called rational equivalence). These groups carry many information on $X$ but are in general very difficult to study. On the other hand, one can associate to $X$ several cohomology groups which are "linear" objects and hence are rather simple to understand. One then construct maps called "cycle class maps" from Chow groups to several cohomological theories.

In this talk, we focus on the case of a variety $X$ over a finite field. In this case, Tate conjecture claims the surjectivity of the cycle class map with rational coefficients; this conjecture is still widely open. In case of integral coefficients, we speak about the integral version of the conjecture and we know several counterexamples for the surjectivity. In this talk, we present a survey of some well-known results on this subject and discuss other properties of algebraic cycles which are either proved or expected to be true. We also discuss several involved methods.
Let $X$ be a projective variety over a field $k$. Chow groups are defined as the quotient of a free group generated by irreducible subvarieties (of fixed dimension) by some equivalence relation (called rational equivalence). These groups carry many information on $X$ but are in general very difficult to study. On the other hand, one can associate to $X$ several cohomology groups which are "linear" objects and hence are rather simple to ...

14C25 ; 14G15 ; 14J70 ; 14C15 ; 14H05

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y

*Research talks*

We construct curves over finite fields with properties similar to those of classical elliptic or Drinfeld modular curves (as far as elliptic points, cusps, ramification, ... are concerned), but whose coverings have Galois groups of type $\mathbf{GL}(r)$ over finite rings $(r\ge 3)$ instead of $\mathbf{GL}(2)$. In the case where the finite field is non-prime, there results an abundance of series or towers with a large ratio "number of rational points/genus". The construction relies on higher-rank Drinfeld modular varieties and the supersingular trick and uses mainly rigid- analytic techniques.
We construct curves over finite fields with properties similar to those of classical elliptic or Drinfeld modular curves (as far as elliptic points, cusps, ramification, ... are concerned), but whose coverings have Galois groups of type $\mathbf{GL}(r)$ over finite rings $(r\ge 3)$ instead of $\mathbf{GL}(2)$. In the case where the finite field is non-prime, there results an abundance of series or towers with a large ratio "number of rational ...

11G09 ; 11G20 ; 14G15

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y

*Research talks*

This talk will focus on the last step of the number field sive algorithm used to compute discrete logarithms in finite fields. We consider here non-prime finite fields of very small extension degree: $1 \le n \le 6$. These cases are interesting in pairing-based cryptography: the pairing output is an element in such a finite field. The discrete logarithm in that finite field must be hard enough to prevent from attacks in a given time (e.g. $10$ years). Within the CATREL project we aim to compute DL records in finite fields of moderate size (e.g. in $GF(p^n$) of global size from $600$ to $800$ bits) to estimate more tightly the hardness of DL in fields of cryptographic size ($2048$ bits at the moment). The best algorithm known to compute discrete logarithms in large finite fields (with small $n$) is the number field sieve (NFS):

(1) polynomial selection: select two distinct polynomials $f,g$ defining two number fields, such that they share modulo $p$ an irreducible degree $n$ factor, and have additional properties to improve the next two steps.

(2) sieving: sieve over elements that satisfy relations, to build the factor basis made of prime ideals of small norm.

(3) linear algebra: compute the kernel of a large matrix computed the step before. Then the logarithm of each element in the factor basis is known.

(4) individual logarithm: for a given element $s \in GF(p^n)$, decompose it over the factor basis to finally compute its discrete logarithm.

The most time consuming steps are the second and third: sieving and linear algerbra. After the sieve and the linear algebra, the logarithms of the prime ideals of small norm are known. To finally compute the discrete logarithm of the given element $s$, we lift $s$ in one of the number fields and factor it in prime ideals as with “small” elements in the sieve step. However here, $s$ does not have a small norm (bounded by $B \ll Q$). Its norm is very large, in particular, larger than $Q$. The usual way is to test for many $s' = s \cdot g^e$ with $g$ the given generator of $GF(p^n)$ until the norm of $s'$ is smooth enough. The time spent to find a good $e$ is asymptotically less than the sieving time. In practice, another modification of $s'$ is computed to reduce its norm. In [?], the authors write $s' = a(x) / b(x)$ with $a, b$ of coefficients of size $\sim p^{1/2}$ instead of $p$. With $n = 4$ the norm of $s$ is $O(p^{11/2})$. Their method compute $a,b$ of norm $O(p^{7/2})$. One need to factor into small prime ideals two elements $a,b$ instead of one $s'$.

for our record computations of discrete logarithms in $\mathbb{F}_pn$ with $2 \leqslant n \leqslant 6$, we improve the preparation of $s$, so that its norm in the number field is less than $Q$. This improves its smoothness property. Assume that we want to compute the discrete logarithm of $s$ in the larger subgroup of prime order $\ell$ of $GF(p^n)$, with $\ell$ $|$ $\Phi_np$. We decompose $s$ in $\epsilon \cdot s'$ with $\epsilon$ in a subfield or in a subgroup of order prime to $\ell$ and $s?$ with reduced coefficient size. We still have $log_g s = log_g s'$ mod $\ell$. We use a tower representation of $GF(p^n)$ with subfields for our purpose. We reduce the norm of $s \in \mathbb{F}_{p4}$ from $O(p^{11/2})$ to $O(p^{7/2}), s \in GF(p^3)$ from $O(p^6)$ to $O(p^2)$ and $s \in \mathbb{F}_{p2}$ from $O(p^4)$ to $O(p)$. This does not change the asymptotic complexity of this last step but this improves a lot its running time for small $n$.
This talk will focus on the last step of the number field sive algorithm used to compute discrete logarithms in finite fields. We consider here non-prime finite fields of very small extension degree: $1 \le n \le 6$. These cases are interesting in pairing-based cryptography: the pairing output is an element in such a finite field. The discrete logarithm in that finite field must be hard enough to prevent from attacks in a given time (e.g. $10$ ...

11Y16 ; 11T71 ; 94A60

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y

*Research talks*

Let $E$ be an elliptic curve over a number field $K$. For each integer $n > 1$ the action of the absolute Galois group $G_K := Gal(\overline{K}/K)$ on the $n$-torsion subgroup $E [n]$ induces a Galois representation $\rho_{E,n}:G_K \rightarrow$ Aut$(E[n]) \backsimeq GL_2(\mathbb{Z} /n\mathbb{Z})$. The representations $\rho_{E,n}$ form a compatible system, and after taking inverse limits one obtains an adelic representation $\rho_E:G_K \rightarrow GL_2(\hat{\mathbb{Z}})$. If $E/K$ does not have $CM$, then Serre’s open image theorem implies that the image of $\rho_E$ has finite index in $GL_2(\hat{\mathbb{Z}})$; in particular, $\rho_{E,\ell}$ is surjective for all but finitely many primes $\ell$.

I will present an algorithm that, given an elliptic curve $E/K$ without $CM$, determines the image of $\rho_{E,\ell}$ in $GL_2(\mathbb{Z} /\ell\mathbb{Z})$ up to local conjugacy for every prime $\ell$ for which $\rho_{E,\ell}$ is non-surjective. Assuming the generalized Riemann hypothesis, the algorithm runs in time that is polynomial in the bit-size of the coefficients of an integral Weierstrass model for $E$. I will then describe a probabilistic algorithm that uses this information to compute the index of $\rho_E$ in $GL_2(\hat{\mathbb{Z}})$.
Let $E$ be an elliptic curve over a number field $K$. For each integer $n > 1$ the action of the absolute Galois group $G_K := Gal(\overline{K}/K)$ on the $n$-torsion subgroup $E [n]$ induces a Galois representation $\rho_{E,n}:G_K \rightarrow$ Aut$(E[n]) \backsimeq GL_2(\mathbb{Z} /n\mathbb{Z})$. The representations $\rho_{E,n}$ form a compatible system, and after taking inverse limits one obtains an adelic representation $\rho_E:G_K \...

11G05 ; 11Y16

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y

*Research talks*

The modular curve $Y^1(N)$ parametrises pairs $(E,P)$, where $E$ is an elliptic curve and $P$ is a point of order $N$ on $E$, up to isomorphism. A unit on the affine curve $Y^1(N)$ is a holomorphic function that is nowhere zero and I will mention some applications of the group of units in the talk.

The main result is a way of generating generators (sic) of this group using a recurrence relation. The generators are essentially the defining equations of $Y^1(N)$ for $n < (N + 3)/2$. This result proves a conjecture of Maarten Derickx and Mark van Hoeij.
The modular curve $Y^1(N)$ parametrises pairs $(E,P)$, where $E$ is an elliptic curve and $P$ is a point of order $N$ on $E$, up to isomorphism. A unit on the affine curve $Y^1(N)$ is a holomorphic function that is nowhere zero and I will mention some applications of the group of units in the talk.

The main result is a way of generating generators (sic) of this group using a recurrence relation. The generators are essentially the defining ...

11F03 ; 11B37 ; 11B39 ; 11G16 ; 14H52

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