85 research outputs found
L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields
The goal of this paper is to explain how a simple but apparently new fact of
linear algebra together with the cohomological interpretation of L-functions
allows one to produce many examples of L-functions over function fields
vanishing to high order at the center point of their functional equation. The
main application is that for every prime p and every integer g>0 there are
absolutely simple abelian varieties of dimension g over Fp(t) for which the BSD
conjecture holds and which have arbitrarily large rank.Comment: To appear in Inventiones Mathematica
Explicit Methods in Number Theory
These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes asymptotics for field extensions and class numbers, random matrices and L-functions, rational points on curves and higher-dimensional varieties, and aspects of lattice basis reduction
SATO-TATE EQUIDISTRIBUTION OF CERTAIN FAMILIES OF ARTIN L-FUNCTIONS
We study various families of Artin L-functions attached to geometric parametrizations of number fields. In each case we find the Sato-Tate measure of the family and determine the symmetry type of the distribution of the low-lying zeros
Modular Koszul duality
We prove an analogue of Koszul duality for category of a
reductive group in positive characteristic larger than 1 plus the
number of roots of . However there are no Koszul rings, and we do not prove
an analogue of the Kazhdan--Lusztig conjectures in this context. The main
technical result is the formality of the dg-algebra of extensions of parity
sheaves on the flag variety if the characteristic of the coefficients is at
least the number of roots of plus 2.Comment: 62 pages; image displays best in pd
Explicit Methods in Number Theory
The aim of the series of Oberwolfach meetings on ‘Explicit methods in number theory’ is to bring together people attacking key problems in number theory via techniques involving concrete or computable descriptions. Here, number theory is interpreted broadly, including algebraic and analytic number theory, Galois theory and inverse Galois problems, arithmetic of curves and higher-dimensional varieties, zeta and -functions and their special values, and modular forms and functions
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