An introduction to p-adic L-functions — Lean blueprint

2.2. Special values and arithmetic🔗

There are deep results and conjectures relating special values of L-functions to arithmetic invariants. A prototype is the class number formula.

Theorem2.2.1
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Let F be a number field with r_1 real embeddings, r_2 pairs of complex embeddings, w roots of unity, discriminant D, regulator R and class number h_F. Then \zeta_F has a simple pole at s=1 with residue \Res_{s=1}\zeta_F(s) = \frac{2^{r_1}(2\pi)^{r_2}R}{w\sqrt{\abs{D}}}\,h_F. This rests on Definition 2.1.2. (Mathlib formalises this as the statement that (s-1)\zeta_F(s) tends to the displayed residue as s \to 1^{+} along the reals.)

Lean code for Theorem2.2.1●1 declaration, 1 missing
Proof for Theorem 2.2.1
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This is a classical analytic class number formula; we record it for motivation rather than prove it. The left-hand side is an analytic invariant of the meromorphic function \zeta_F, while the right-hand side is a product of arithmetic invariants of F. The bridge between them comes from comparing the Dirichlet series \zeta_F(s) with a sum over lattice points in the ideal class groups: the residue at s=1 counts ideals of bounded norm, which by the geometry of numbers is governed by the covolume of the unit lattice (the regulator R), the number of roots of unity w, the discriminant D, and the number of ideal classes h_F.

Theorem2.2.2
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(Birch–Swinnerton-Dyer conjecture.) Let E/\Q be an elliptic curve. Then the Mordell–Weil group E(\Q) is finitely generated, and \ord_{s=1}L(E,s) = \mathrm{rank}_{\Z}\,E(\Q). Moreover the leading Taylor coefficient of L(E,s) at s=1 is given by an explicit product of arithmetic invariants of E. This concerns Definition 2.1.4.

Proof for Theorem 2.2.2
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This is an open conjecture, included to motivate the p-adic theory; no proof is claimed. As with the class number formula, the left-hand side is analytic and the right-hand side arithmetic, and the two worlds are so different that even the existence of the left-hand side — that L(E,s) is defined at s=1 — requires the analytic continuation supplied (over \Q) by the modularity theorem. The known low-rank cases proceed through Heegner points and Euler systems, which give the two inequalities \ord_{s=1}L(E,s)\le\mathrm{rank}\,E(\Q) and \ge, respectively, when the relevant Tate–Shafarevich group \Sha(E/\Q)[p^\infty] is finite. These same Iwasawa-theoretic tools are what the present notes develop in the simpler setting of the Riemann zeta function.

Iwasawa theory seeks p-adic analogues of such statements, replacing complex analysis (poorly suited to arithmetic) by p-adic analysis (where arithmetic arises naturally). For an elliptic curve E and each prime p there is an Iwasawa Main Conjecture equating a p-adic analytic L-function with p-adic arithmetic invariants of E; the bottom row of the BSD/IMC square is far more tractable than the top, and the IMC for elliptic curves is known much more widely (Kato; Skinner–Urban) than BSD itself. In these notes we treat the simplest instance of the picture: the Main Conjecture for the p-adic Riemann zeta function, formulated by Iwasawa, which is known completely for every prime.