Extremal primes for elliptic curves with complex multiplication

Let $X_0^6(1)/W_6$ be the Atkin-Lehner quotient of the Shimura curve $X_0^6(1)$ associated to a maximal order in an indefinite quaternion algebra of discriminant $6$ over $\mathbb Q$. By realizing modular forms on $X_0^6(1)/W_6$ in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer's formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over $\overline\mathbb Q$ with complex multiplication.

In this paper, we prove that if every supersingular elliptic curve over Fp can be lifted to a CM elliptic curve by an imaginary order OD for some D ? pθ, then θ ≥ 1 2 . We also prove that if every supersingular elliptic curve over Fp can be lifted to a CM elliptic curve by an imaginary order OD for some D ? pθ, then θ ≥ 23 as suggested by Elkies.

We prove several results on torsion points and Galois representations for complex multiplication (CM) elliptic curves over a number field containing the CM field. One result computes the degree in which such an elliptic curve has a rational point of order $N$, refining results of Silverberg. Another result bounds the size of the torsion subgroup of an elliptic curve with CM by a nonmaximal order in terms of the torsion subgroup of an elliptic curve with CM by the maximal order. Our techniques also yield a complete classification of both the possible torsion subgroups and the rational cyclic isogenies of a $K$-CM elliptic curve $E$ defined over $K(j(E))$.

Let be a CM elliptic curve defined over a number field K, with Weiestrass form y^3=x^3+bx or y^2=x^3+c. For every positive integer m, we denote by the m-torsion subgroup of and by the m-th division field, i.e. the extension of K generated by the coordinates of the points in . We classify all the fields K_7. In particular we give explicit generators for K_7/K and produce all the Galois groups textrm{Gal}(K_7/K). We also show some applications to the Local–Global Divisibility Problem and to modular curves.

The transcendental Brauer group of a product of CM elliptic curves

We define an adelic version of a CM elliptic curve $E$ which is equipped with an action of the profinite completion of the endomorphism ring of $E$. The adelic elliptic curve so obtained is provided with a natural embedding into the adelic Heisenberg group. We embed into the adelic Heisenberg group the set of commensurability classes of arithmetic $1$-dimensional $\mathbb{K}$-lattices (here and subsequently, $\mathbb{K}$ denotes a quadratic imaginary number field) and define theta functions on it. We also embed the groupoid of commensurability modulo dilations into the union of adelic Heisenberg groups relative to a set of representatives of elliptic curves with $R$-multiplication ($R$ is the ring of algebraic integers of $\mathbb{K}$). We thus get adelic theta functions on the set of $1$-dimensional $\mathbb{K}$-lattices and on the groupoid of commensurability modulo dilations. Adelic theta functions turn out to be acted by the adelic Heisenberg group and behave nicely under complex automorphisms (Theorems 6.12 and 6.14).

We study the low‐lying zeros of a family of ‐functions attached to the complex multiplication elliptic curve , for each odd and square‐free integer . Specifically, upon writing the ‐function of as for the appropriate Grössencharakter of conductor , we consider the collection of ‐functions attached to , , where for each integer , denotes the primitive character inducing . We observe that of the ‐functions in have negative root number. is thus not one of the essentially homogeneous families of the universality conjecture of Sarnak, Shin and Templier [33], with unitary, symplectic or orthogonal (odd or even) symmetry type. By computing the one‐level density in the family of ‐functions in with conductor at most , we find that naturally decomposes into subfamilies: more specifically, a collection of symplectic ( for , even) and orthogonal ( for , odd) subfamilies. For each such subfamily, we moreover compute explicit lower order terms in decreasing powers of .

Let $E/\mathbb{Q}$ be a fixed elliptic curve. For each prime $p$ of good reduction, write $E(\mathbb{F}_p) \cong \mathbb{Z}/d_p \mathbb{Z} \oplus \mathbb{Z}/e_p \mathbb{Z}$, where $d_p \mid e_p$. Kowalski proposed investigating the average value of $d_p$ as $p$ runs over the rational primes. For CM curves, he showed that $x\log\log{x}/\log{x} \ll \sum_{p \le x} d_p \ll x\sqrt{\log{x}}$. It was shown recently by Felix and Murty that in fact $\sum_{p \le x} d_p$ exceeds any constant multiple of $x\log\log{x}/\log{x}$, once $x$ is sufficiently large. In the opposite direction, Kim has shown that the expression $x\sqrt{\log{x}}$ in the upper bound can be replaced by $x\log\log{x}$. In this paper, we obtain the correct order of magnitude for the sum: $\sum_{p \le x} d_p \asymp x$ for all large $x$.

We generalise works of Kobayashi to give a formulation of the Iwasawa main conjecture for modular forms at supersingular primes. In particular, we give analogous definitions of the plus and minus Coleman maps for normalised new forms of arbitrary weights and relate Pollack’s p-adic L-functions to the plus and minus Selmer groups. In addition, by generalising works of Pollack and Rubin on CM elliptic curves, we prove the ‘main conjecture’ for CM modular forms.

We prove that some fields generated by j j -invariants of CM elliptic curves (of infinite dimension over Q \mathbb {Q} ) satisfy the Property (B). The singular moduli are chosen so as to have supersingular reduction simultaneously above a fixed prime q q , which provides strong q q -adic estimates leading to an explicit lower bound for the height.

Let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, with ring of integers $\mathcal{O}$ and Hilbert class field $H$. Suppose $p\nmid [H:K]$ is a prime number which splits in $K$, say $p\mathcal{O}=\mathfrak{p}\mathfrak{p}^*$. Let $H_\infty=HK_\infty$ where $K_\infty$ is the unique $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. Write $M(H_\infty)$ for the maximal abelian $p$-extension of $H_\infty$ unramified outside the primes above $\mathfrak{p}$, and set $X(H_\infty)=\mathrm{Gal}(M(H_\infty)/H_\infty)$. In this paper, we establish the main conjecture of Iwasawa theory for the Iwasawa module $X(H_\infty)$. As a consequence, we have that if $X(H_\infty)=0$, the relevant $L$-values are $\mathfrak{p}$-adic units. In addition, the main conjecture for $X(H_\infty)$ has implications toward (a) the BSD Conjecture for a class of CM elliptic curves; (b) weak $\mathfrak{p}$-adic Leopoldt conjecture.

Bruin and Najman [LMS J. Comput. Math. 18 (2015), pp. 578–602], Ozman and Siksek [Math. Comp. 88 (2019), pp. 2461–2484], and Box [Math. Comp. 90 (2021), pp. 321–343] described all the quadratic points on the modular curves of genus 2 ≤ g ( X 0 ( n ) ) ≤ 5 2\leq g(X_0(n)) \leq 5 . Since all the hyperelliptic curves X 0 ( n ) X_0(n) are of genus ≤ 5 \leq 5 and as a curve can have infinitely many quadratic points only if it is either of genus ≤ 1 \leq 1 , hyperelliptic or bielliptic, the question of describing the quadratic points on the bielliptic modular curves X 0 ( n ) X_0(n) naturally arises; this question has recently also been posed by Mazur. We answer Mazur’s question completely and describe the quadratic points on all the bielliptic modular curves X 0 ( n ) X_0(n) for which this has not been done already. The values of n n that we deal with are n = 60 n=60 , 62 62 , 69 69 , 79 79 , 83 83 , 89 89 , 92 92 , 94 94 , 95 95 , 101 101 , 119 119 and 131 131 ; the curves X 0 ( n ) X_0(n) are of genus up to 11 11 . We find all the exceptional points on these curves and show that they all correspond to CM elliptic curves. The two main methods we use are Box’s relative symmetric Chabauty method and an application of a moduli description of Q \mathbb {Q} -curves of degree d d with an independent isogeny of degree m m , which reduces the problem to finding the rational points on several quotients of modular curves.

One of the most fruitful developments in the last couple of decades, has been the use of K -theory in the study of special values of L -functions of modular forms. This link is manifested in several ways. The first is through the work of Beilinson, who constructed zeta-elements living in K 2 of a modular curve. He managed to prove that their image under the regulator map yielded the residue of the L -series at s = 0. In an orthogonal direction, Coates and Wiles found a non-archimedean approach to relating the L -function of a CM elliptic curve at s = 1, with a certain Euler system of elliptic units. They were then able to prove the first concrete theorems in the direction of the Birch and Swinnerton-Dyer conjecture. It was Kato who realised the approach of Beilinson and that of Coates-Wiles could be combined, with spectacular results. He noticed that the zeta-elements considered by Beilinson satisfied norm-compatibility relations, reminiscent of those satisfied by elliptic units. Moreover, he devised a purely p -adic method whereby these elements could be used to study L -values. Underlying his discoveries was the ground-breaking work of Perrin-Riou in the early nineties, which extended the local Iwasawa theory used by Coates and Wiles to a very general framework (required for modular forms and non-CM elliptic curves). We shall review some of the highlights of their work, and in Chapters III, V and VI we will generalise it to modular symbols and Λ-adic families of modular forms.

The rank of a CM elliptic curve and a recurrence formula

We give a new characterisation of elliptic curves of Shimura type in terms commuting families of Frobenius lifts and also strengthen an old principal ideal theorem for ray class fields. These two results combined yield the existence of global minimal models for elliptic curves of Shimura type, generalising a result of Gross. Along the way we also prove a handful of small but new results regarding elliptic curves with complex multiplication.