Elliptic curve

About: Elliptic curve is a research topic. Over the lifetime, 13928 publications have been published within this topic receiving 255355 citations.

L-Functions and Tamagawa Numbers of Motives

Abstract: The notion of a motif was first defined and studied by A. Grothendieck, and this paper is an attempt to understand some of the implications of his ideas for arithmetic. We will formulate a conjecture on the values at integer points of L-functions associated to motives. Conjectures due to Deligne and Beilinson express these values “modulo Q* multiples” in terms of archimedean period or regulator integrals. Our aim is to remove the Q* ambiguity by defining what are in fact Tamagawa numbers for motives. The essential technical tool for this is the Fontaine-Messing theory of p-adic cohomology. As evidence for our Tamagawa number conjecture, we show that it is compatible with isogeny, and we include strong results due to one of us (Kato) for the Riemann zeta function and for elliptic curves with complex multiplication.

P-ADIC Properties of Modular Schemes and Modular Forms

Abstract: This expose represents an attempt to understand some of the recent work of Atkin, Swinnerton-Dyer, and Serre on the congruence properties of the q-expansion coefficients of modular forms from the point of view of the theory of moduli of elliptic curves, as developed abstractly by Igusa and recently reconsidered by Deligne. In this optic, a modular form of weight k and level n becomes a section of a certain line bundle \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } ^{ \otimes k} \) on the modular variety Mn which “classifies” elliptic curves with level n structure (the level n structure is introduced for purely technical reasons). The modular variety Mn is a smooth curve over ℤ[l/n], whose “physical appearance” is the same whether we view it over ℂ (where it becomes ϕ(n) copies of the quotient of the upper half plane by the principal congruence subgroup Г(n) of SL(2,ℤ)) or over the algebraic closure of ℤ/pℤ, (by “reduction modulo p”) for primes p not dividing n. This very fact rules out the possibility of obtaining p-adic properties of modular forms simply by studying the geometry of Mn ⊗ℤ/pℤ and its line bundles \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\omega } ^{ \otimes k} \); we can only obtain the reductions modulo p of identical relations which hold over ℂ.

The AdS 5 ×S 5 superstring worldsheet S matrix and crossing symmetry

Abstract: An S matrix satisfying the Yang-Baxter equation with symmetries relevant to the AdS{sub 5}xS{sup 5} superstring recently has been determined up to an unknown scalar factor. Such scalar factors are typically fixed using crossing relations; however, due to the lack of conventional relativistic invariance, in this case its determination remained an open problem. In this paper we propose an algebraic way to implement crossing relations for the AdS{sub 5}xS{sup 5} superstring worldsheet S matrix. We base our construction on a Hopf-algebraic formulation of crossing in terms of the antipode and introduce generalized rapidities living on the universal cover of the parameter space which is constructed through an auxillary, coupling-constant dependent, elliptic curve. We determine the crossing transformation and write functional equations for the scalar factor of the S matrix in the generalized rapidity plane.