Modular points, modular curves, modular surfaces and modular forms

Abstract

This talk, instead of being a survey, will concentrate on a single example, using it to illustrate two themes, each of which has been a leitmotif of much recent work in number theory and of much of the work reported on at this Arbeitstagung (lectures of Faltings, Manin, Lang,

Computing central values of L-functions

“…The first row is D, the second m D (for typographical reasons we transposed the actual GP output). These values agree, fortunately, with Zagier's [15] formula (28) up to a global negative sign.…”

“…If f is the weight 2 eigenform associated to E then we get a map (7) Φ : One consequence of the results of Gross-Zagier is the following [15], [4], [5]. By our assumption on E(Q) we have P D = m D P 0 for some m D ∈ Z and hence…”

“…Let E be the curve y 2 + y = x 3 − x of conductor N = 37 (this is the elliptic curve over Q of positive rank with smallest conductor). This case was described in detail in [15]. It is known that E(Q) = (0, 0) .…”

Heegner points and derivatives ofL-series. II

“…The intersection number, like our height pairing, is expressed as a sum of local terms (which are calculated, in both cases, by counting representations by quadratic forms). This fact actually permits one to deduce our theorem from the results of [10] in some special cases, e.g., in the first non-trivial case N = 37 [23]. This fact actually permits one to deduce our theorem from the results of [10] in some special cases, e.g., in the first non-trivial case N = 37 [23].…”

K-Rational D-Brane Crystals

“…with generator P gen = (0, 0). This example has been discussed in some detail by Zagier [41] and much of the necessary data for this curve can also be obtained from the tables constructed by Cremona [42]. The remaining BSD ingredients of this curve are given by R E 37 = 0.051111...…”