Elliptic curves and automorphic representations
نویسندگان
چکیده
منابع مشابه
Galois Representations and Elliptic Curves
An elliptic curve over a field K is a projective nonsingular genus 1 curve E over K along with a chosen K-rational point O of E, which automatically becomes an algebraic group with identity O. If K has characteristic 0, the n-torsion of E, denoted E[n], is isomorphic to (Z/nZ) over K. The absolute Galois group GK acts on these points as a group automorphism, hence it acts on the inverse limit l...
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When Linnik introduced the classical large-sieve in 1941 [Lin], he was motivated by the following problem: given a non-trivial primitive character χ modulo q, how large (compared to q) can be the first n such that χ(n) = 1? From the Riemann Hypothesis one can deduce (see [Mon] chapter 13 for instance) n (log q) 2 and the (weaker) conjecture n q ε for all ε > 0 is known as Vinogradov's conjectur...
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One of the most classical problems in number theory is that of determining whether a given rational integer is the sum of two cubes of rational numbers. This “sum of two cubes” problem has been attacked from a variety of both classical and modern viewpoints; it would be nearly impossible to list here all of the various approaches taken and results obtained. An extensive compilation of the older...
متن کاملMod 4 Galois Representations and Elliptic Curves
Galois representations ρ : GQ → GL2(Z/n) with cyclotomic determinant all arise from the n-torsion of elliptic curves for n = 2, 3, 5. For n = 4, we show the existence of more than a million such representations which are surjective and do not arise from any elliptic curve.
متن کاملElliptic curves with nonsplit mod 11 representations
We calculate explicitly the j-invariants of the elliptic curves corresponding to rational points on the modular curve X+ ns(11) by giving an expression defined over Q of the j-function in terms of the function field generators X and Y of the elliptic curve X+ ns(11). As a result we exhibit infinitely many elliptic curves over Q with nonsplit mod 11 representations.
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ژورنال
عنوان ژورنال: Advances in Mathematics
سال: 1976
ISSN: 0001-8708
DOI: 10.1016/s0001-8708(76)80001-1