Statement
Let E be an elliptic curve with integer coefficients in a Neron minimal model. Suppose that the discriminant Δ of E is written as the product Πpδp of prime powers pδp and similarly the conductor N of E is the product Πpnp of prime powers. Suppose that E is a modular elliptic curve, then we can perform a level descent modulo primes ℓ dividing one of the exponents δp of a prime dividing the discriminant. If pδp is an odd prime power factor of Δ and if p divides N only once (i.e. np=1), then there exists another elliptic curve E', with conductor N' = N/p, such that the coefficients of the L-series of E are congruent modulo ℓ to the coefficients of the L-series of E' .
The epsilon conjecture is a relative statement: assuming that a given elliptic curve E over Q is modular, it predicts the precise level of E.
Read more about this topic: Ribet's Theorem
Famous quotes containing the word statement:
“After the first powerful plain manifesto
The black statement of pistons, without more fuss
But gliding like a queen, she leaves the station.”
—Stephen Spender (1909–1995)
“The parent is the strongest statement that the child hears regarding what it means to be alive and real. More than what we say or do, the way we are expresses what we think it means to be alive. So the articulate parent is less a telling than a listening individual.”
—Polly Berrien Berends (20th century)
“A sentence is made up of words, a statement is made in words.... Statements are made, words or sentences are used.”
—J.L. (John Langshaw)