[Paper Review] A visible factor of the special L-value

Amod Agashé|ArXiv.org|Oct 14, 2008

Algebraic Geometry and Number Theory14 references3 citations

TL;DR

This paper provides a formula for the ratio of the special L-value to the real period of a quotient A of J₀(N) attached to a newform f, expressing it as a rational number. It identifies an integer factor in the numerator linked to congruences between f and eigenforms of positive analytic rank, and using visibility theory, proves that if an odd prime q divides this factor, then q divides either the order of the Shafarevich-Tate group or a component group of A—offering theoretical support for the second part of the Birch and Swinnerton-Dyer conjecture.

ABSTRACT

Let~$A$ be a quotient of $J_0(N)$ associated to a newform $f$ such that the special $L$-value of $A$ (at $s=1$) is non-zero. We give a formula for the ratio of the special $L$-value to the real period of $A$ that expresses this ratio as a rational number. We extract an integer factor from the numerator of this formula; this factor is non-trivial in general and is related to certain congruences of $f$ with eigenforms of positive analytic rank. We use the techniques of visibility to show that, under certain hypotheses (which includes the first part of the Birch and Swinnerton-Dyer conjecture on rank), if an odd prime $q$ divides this factor, then $q$ divides either the order of the Shafarevich-Tate group or the order of a component group of $A$. Suppose $p$ is an odd prime such that $p^2$ does not divide $N$, $p$ does not divide the order of the rational torsion subgroup of $A$, and $f$ is congruent modulo a prime ideal over $p$ to an eigenform whose associated abelian variety has positive Mordell-Weil rank. Then we show that $p$ divides the factor mentioned above; in particular, $p$ divides the numerator of the ratio of the special $L$-value to the real period of $A$. Both of these results are as implied by the second part of the Birch and Swinnerton-Dyer conjecture, and thus provide theoretical evidence towards the conjecture.

Motivation & Objective

• To understand the arithmetic significance of the numerator in the ratio of the special L-value to the real period for abelian varieties A_f attached to newforms.
• To identify an integer factor in this ratio that arises from congruences between the newform f and eigenforms of positive Mordell-Weil rank.
• To use the theory of visibility to show that primes dividing this factor must also divide the order of the Shafarevich-Tate group or component group of A_f under certain hypotheses.
• To provide theoretical evidence for the second part of the Birch and Swinnerton-Dyer conjecture by linking visibility to the conjectural order of the Shafarevich-Tate group.

Proposed method

• Derives a formula for L(A_f,1)/Ω_A_f as a rational number using the theory of modular parametrizations and the structure of J₀(N).
• Identifies a visible integer factor in the numerator of this rational ratio, which is related to congruences between the newform f and eigenforms of positive analytic rank.
• Applies the theory of visibility to show that if an odd prime q divides this factor, then q divides either |Ш(A_f)| or the order of a component group c_p(A_f), assuming the first part of the Birch and Swinnerton-Dyer conjecture on ranks.
• Uses the action of Frobenius on ℓ-adic Tate modules and Galois cohomology to relate the order of the component group to the valuation of Euler factors at p.
• Employs the isomorphism between H⁰(ℚ_ℓ, V_ℓ^I_p / T_ℓ^I_p) and the cokernel of (1 - Frob_p^{-1}) on T_ℓ^I_p to compute the p-adic valuation of the Euler factor P_p(p^{-1}).
• Establishes that the prime-to-p part of the component group order c_p(A_f) matches the valuation of the Euler factor P_p(p^{-1}) via a Galois cohomological argument involving the fixed points of Frobenius.

Experimental results

Research questions

• RQ1 Can the numerator of the ratio L(A_f,1)/Ω_A_f be factored into an integer that reflects arithmetic congruences of the newform f?
• RQ2 If an odd prime q divides this integer factor, does q necessarily divide the order of the Shafarevich-Tate group or a component group of A_f?
• RQ3 Does the theory of visibility imply that the Birch and Swinnerton-Dyer conjectural order of Ш(A_f) is non-trivial when f is congruent to an eigenform of positive Mordell-Weil rank?
• RQ4 How is the component group order c_p(A_f) related to the p-adic valuation of the Euler factor P_p(p^{-1})?

Key findings

• The ratio L(A_f,1)/Ω_A_f is a rational number, and its numerator contains an integer factor that is non-trivial in general and tied to congruences between f and eigenforms of positive analytic rank.
• If an odd prime q divides this integer factor and the first part of the Birch and Swinnerton-Dyer conjecture holds, then q divides either |Ш(A_f)| or c_p(A_f) for some p dividing N.
• When f is congruent modulo a prime ideal over p to an eigenform whose associated abelian variety has positive Mordell-Weil rank, and under additional hypotheses (p²∤N, p∤|A_f(Q)_{tors}|), then p divides the numerator of L(A_f,1)/Ω_A_f.
• The p-adic valuation of the Euler factor P_p(p^{-1}) equals the order of the Galois-fixed points in V_ℓ^I_p / T_ℓ^I_p, which is isomorphic to the component group order c_p(A_f) up to p-power factors.
• The component group order c_p(A_f) is determined by the Galois cohomology of the Tate module, specifically via the fixed points of Frobenius on the quotient V_ℓ^I_p / T_ℓ^I_p.
• The theory of visibility implies that if f is congruent to a form of positive rank, then Ш(A_f) is non-trivial, and this is reflected in the numerator of the L-value ratio, supporting the second part of the Birch and Swinnerton-Dyer conjecture.