Forschungsseminar "Arithmetische Geometrie"

Sommersemester 2009


28.04.2009 Hartwig Mayer (HU Berlin)
Arithmetic self-intersection number of the dualizing sheaf on modular curves
05.05.2009 Anna Posingies (HU Berlin)
A Kronecker Limit Formula for Fermat Curves
19.05.2009 Shabnam Akhtari (MPI Bonn)
Representation of Integers by Binary Forms
Abstract: Suppose F(x,y) is an irreducible binary form with integral coefficients, degree n >= 3 and discriminant D_F \neq 0. Let h be an integer. The equation F(x,y)=h has finitely many solutions in integers x and y. I shall discuss some different approaches to the problem of counting the number of integral solutions to such equations. I will give upper bounds upon the number of solutions to the Thue equation F(x,y)= h. These upper bounds are derived by combining methods from classical analysis and geometry of numbers. The theory of linear forms in logarithms plays an essential rule in this study.
26.05.2009 Anil Aryasomayajula (HU Berlin)
A fundamental identity of metrics on hyperbolic Riemann surfaces of finite volume
02.06.2009 Ronald van Luijk (Leiden)
Explicit two-coverings of Jacobians of genus two
23.06.2009 Remke Nanne Kloosterman (HU Berlin)
Average rank of elliptic n-folds
Abstract: For elliptic curves over number fields it is conjectured that the half the curves have rank 1 and half the curves have rank 0. Similarly, if C/F_q is a curve then it is conjectured the half the elliptic curves over F_q(C) have rank 0 and half the curves have rank 1.
In this talk we show that the situation is different if one considers elliptic curves over F_q(V), with dim(V)>1.
07.07.2009 Anna von Pippich (HU Berlin)
The arithmetic of elliptic Eisenstein series
14.07.2009 H. Shiga (Chiba University)
A Jacobi type formula in two variables with application to a new AGM
J. Estrada Sarlabous (Havana)
Non-hyperelliptic curves of genus 3 and the DLP
Abstract: The index calculus algorithm of Gaudry, Thom$\acute{e}$, Th$\acute{e}$riault, and Diem makes possible to solve the Discrete Logarithm Problem (DLP) in the Jacobian varieties of hyperelliptic curves of genus 3 over $F_q$ in $O(q^{4/3})$ group operations. On the other hand, applied to Jacobian varieties of non-hyperelliptic curves of genus 3 over $F_q$, the index calculus algorithm of Diem requires only $O(q)$ group operations to solve the DLP.
This vulnerability to faster index calculus attacks of the non-hyperelliptic curves of genus 3 has discouraged the use of Jacobian varieties of non-hyperelliptic curves of genus 3 as a basis of DLP-based cryptosystems.
A recent work of B. Smith introduces the idea of exploiting this vulnerability to faster index calculus attacks of the non-hyperelliptic curves of genus 3 to discard a non-negligable subset of hyperelliptic curves of genus 3 over $F_q$.
I would like to expose some highlights of this approach of B. Smith (the mathematical ingredients are nice: isogenies of Jacobian varieties, Recilla's trigonal construction, etc.) and to discuss the interest of studying non-hyperelliptic curves of genus 3 in the context of DLP-based cryptosystems.