Disparity in Selmer ranks of quadratic twists of elliptic curves. (English) Zbl 1300.11063

Klagsbrun, Zev; Mazur, Barry; Rubin, Karl

Ann. Math. (2) 178, No. 1, 287-320 (2013); erratum ibid. 198, No. 2, 879-880 (2023).

Let \(E\) be an elliptic curve defined over a number field \(K\) and, for any quadratic character \(\chi\) of \(\mathrm{Gal}(\overline{K}/K)\), let \(E^\chi\) be the associated twisted curve. The paper deals with the parity of the ranks of the 2-Selmer groups \(\mathrm{Sel}_2(E^\chi/K)\) as \(\chi\) varies. The main theorem considers the set \[ C(K,x):=\{\chi\,:\,\chi\;\text{is\;ramified\;only\;at\;primes\;} \mathfrak{q}\;\text{with\;} N^K_{\mathbb{Q}}(\mathfrak{q})\leq x\} \] (for a positive real number \(x\)) and describes the proportion of odd 2-Selmer ranks for large enough \(x\) with the formula \[ \rho(E/K):=\frac{|\{\chi\in C(K,x)\,:\,\dim_{\mathbb{F}_2}\mathrm{Sel}_2(E^\chi/K)\;\text{is\;odd}\}|}{|C(K,x)|}= \frac{1-\delta(E/K)}{2} \;,\tag{1} \] where \(\delta(E/K)\in [-1,1]\cap \mathbb{Z}\left[\frac{1}{2}\right]\) is a finite product of explicit local factors \(\delta_v\) (depending on \(K_v\) and on the reduction type of \(E\) at \(v\) when \(v|2\Delta(E)\infty\)). In particular, if \(K\) has a real embedding, then \(\delta(E/K)=0\) and the proportion is \(\frac{1}{2}\) as predicted by a conjecture of D. Goldfeld [in: Number theory, Proc. Conf., Carbondale 1979. Lect. Notes Math. 751, 108–118 (1979; Zbl 0417.14031)], but the authors provide also an explicit example of a curve \(E\) for which the values of \(\delta(E/K)\) are dense in \([-1,1]\) as \(K\) varies among the finite extensions of \(\mathbb{Q}(\sqrt{-2})\) unramified at 5 (Example 7.11).
The proof relies on the study of metabolic spaces, i.e., a pair \((V,q)\) of a finite dimensional \(\mathbb{F}_p\)-vector space \(V\) and a quadratic form \(q:V\rightarrow \mathbb{F}_p\) such that \(q\) defines a nondegenerate pairing \((\,,\,)_q\) and there is a (Lagrangian) subspace \(X\subset V\) with \(X=X^\perp\) and \(q(X)=0\). The authors consider a 2-dimensional vector space \(T\) with local quadratic forms \(q_v\) which define metabolic spaces \((H^1(K_v,T),q_v)\) for any \(v\) and, using Lagrangian subspaces to define local conditions at finitely many primes containing the ramified ones, associate to this collection a Selmer group \(\mathrm{Sel}_p(K,T)\) contained in \(H^1(K,T)\). Then the authors use a result of B. Poonen and E. Rains [J. Am. Math. Soc. 25, No. 1, 245–269 (2012; Zbl 1294.11097)] to generalize Theorem 1.4 of B. Mazur and K. Rubin [Ann. Math. (2) 166, No. 2, 579–612 (2007; Zbl 1219.11084)] to \(p=2\), and with that they compute the parity of the 2-rank of \(\mathrm{Sel}_2(K,T)\) and of its twists (defined in Section 4) and prove a formula like (1) for this general setting. Specializing at \(T=E[2]\) and \(q_v=\) the local Tate pairing, they prove that \(\mathrm{Sel}_2(K,E[2])\) (and its twists) correspond to the classical 2-Selmer group of \(E\) (and of the twisted curves \(E^\chi\,\)), and obtain the above formula for \(\rho(E/K)\).
In the final section the authors give a similar formula for the parity of \(\mathrm{Sel}_p(K,T)\) (and its twists) for \(p> 2\). In this case the specialization at \(T=E[p]\) provides Selmer groups for the abelian varieties \(E^\chi:=\mathrm{Ker}\{\mathrm{Res}^{K^\chi}_K(E)\rightarrow E\}\) of dimension \(p-1\) (where \(\chi\) is a character of order \(p\) and \(K^\chi\) is the field fixed by \(\mathrm{Ker}(\chi)\,\)).

Reviewer: Andrea Bandini (Parma)

MSC:

11G05	Elliptic curves over global fields
14G25	Global ground fields in algebraic geometry

Keywords:

elliptic curves; parity; quadratic twists; Selmer groups

Citations:

Zbl 0417.14031; Zbl 1294.11097; Zbl 1219.11084

Software:

ecdata

References:

[1]	J. W. S. Cassels, ”Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer,” J. Reine Angew. Math., vol. 217, pp. 180-199, 1965. · Zbl 0241.14017
[2]	J. E. Cremona, Algorithms for Modular Elliptic Curves, Cambridge: Cambridge Univ. Press, 1992. · Zbl 0758.14042
[3]	J. E. Cremona, T. A. Fisher, C. O’Neil, D. Simon, and M. Stoll, ”Explicit \(n\)-descent on elliptic curves. I. Algebra,” J. Reine Angew. Math., vol. 615, pp. 121-155, 2008. · Zbl 1242.11039
[4]	T. Dokchitser and V. Dokchitser, ”Elliptic curves with all quadratic twists of positive rank,” Acta Arith., vol. 137, iss. 2, pp. 193-197, 2009. · Zbl 1275.11097
[5]	T. Dokchitser and V. Dokchitser, ”Regulator constants and the parity conjecture,” Invent. Math., vol. 178, iss. 1, pp. 23-71, 2009. · Zbl 1219.11083
[6]	D. Goldfeld, ”Conjectures on elliptic curves over quadratic fields,” in Number Theory, Carbondale 1979, New York: Springer-Verlag, 1979, vol. 751, pp. 108-118. · Zbl 0417.14031
[7]	D. R. Heath-Brown, ”The size of Selmer groups for the congruent number problem. II,” Invent. Math., vol. 118, iss. 2, pp. 331-370, 1994. · Zbl 0815.11032
[8]	B. Howard, ”The Heegner point Kolyvagin system,” Compos. Math., vol. 140, iss. 6, pp. 1439-1472, 2004. · Zbl 1139.11316
[9]	D. M. Kane, On the ranks of the \(2\)-Selmer groups of twists of a given elliptic curve. · Zbl 1300.11061
[10]	Z. Klagsbrun, B. Mazur, and K. Rubin, Distribution of Selmer ranks of quadratic twists of elliptic curves. · Zbl 1300.11063
[11]	K. Kramer, ”Arithmetic of elliptic curves upon quadratic extension,” Trans. Amer. Math. Soc., vol. 264, iss. 1, pp. 121-135, 1981. · Zbl 0471.14020
[12]	B. Mazur and K. Rubin, ”Finding large Selmer rank via an arithmetic theory of local constants,” Ann. of Math., vol. 166, iss. 2, pp. 579-612, 2007. · Zbl 1219.11084
[13]	B. Mazur and K. Rubin, ”Ranks of twists of elliptic curves and Hilbert’s tenth problem,” Invent. Math., vol. 181, iss. 3, pp. 541-575, 2010. · Zbl 1227.11075
[14]	B. Mazur, K. Rubin, and A. Silverberg, ”Twisting commutative algebraic groups,” J. Algebra, vol. 314, iss. 1, pp. 419-438, 2007. · Zbl 1128.14034
[15]	J. S. Milne, Arithmetic Duality Theorems, Boston, MA: Academic Press, 1986, vol. 1. · Zbl 0613.14019
[16]	P. Monsky, ”Generalizing the Birch-Stephens theorem. I. Modular curves,” Math. Z., vol. 221, iss. 3, pp. 415-420, 1996. · Zbl 0853.11048
[17]	B. Poonen and E. Rains, ”Random maximal isotropic subspaces and Selmer groups,” J. Amer. Math. Soc., vol. 25, iss. 1, pp. 245-269, 2012. · Zbl 1294.11097
[18]	D. E. Rohrlich, ”Galois theory, elliptic curves, and root numbers,” Compositio Math., vol. 100, iss. 3, pp. 311-349, 1996. · Zbl 0860.11033
[19]	K. Rubin, Euler Systems, Princeton, NJ: Princeton Univ. Press, 2000, vol. 147. · Zbl 0977.11001
[20]	K. Rubin and A. Silverberg, ”Families of elliptic curves with constant mod \(p\) representations,” in Elliptic Curves, Modular Forms, & Fermat’s Last Theorem, Int. Press, Cambridge, MA, 1995, vol. I, pp. 148-161. · Zbl 0856.11027
[21]	. J-P. Serre, Cohomologie Galoisienne, Fifth ed., New York: Springer-Verlag, 1994, vol. 5. · Zbl 0812.12002
[22]	. J-P. Serre, ”Propriétés galoisiennes des points d’ordre fini des courbes elliptiques,” Invent. Math., vol. 15, iss. 4, pp. 259-331, 1972. · Zbl 0235.14012
[23]	P. Swinnerton-Dyer, ”The effect of twisting on the 2-Selmer group,” Math. Proc. Cambridge Philos. Soc., vol. 145, iss. 3, pp. 513-526, 2008. · Zbl 1242.11041
[24]	J. Tate, ”Duality theorems in Galois cohomology over number fields,” in Proc. Internat. Congr. Mathematicians, Djursholm: Inst. Mittag-Leffler, 1963, pp. 288-295. · Zbl 0126.07002
[25]	J. Tate, ”Algorithm for determining the type of a singular fiber in an elliptic pencil,” in Modular Functions of One Variable, IV, New York: Springer-Verlag, 1975, vol. 476, pp. 33-52. · Zbl 1214.14020