The period-index problem in WC-groups I: elliptic curves

$_3F_2$ hypergeometric series and periods of elliptic curves

We express the real period of a family of elliptic curves in terms of classical hypergeometric series. This expression is analogous to a result of Ono which relates the trace of Frobenius of the same family of elliptic curves to a Gaussian hypergeometric series. This analogy provides further evidence of the interplay between classical and Gaussian hypergeometric series.

$D_{2n+1}$指数的雅可比形式与水平为2的椭圆模形式之间的同构

Isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level $2$

本文研究了$D_{2n+1}$指数雅可比形式与水平为2的椭圆模形式之间的同构关系，并给出了雅可比-艾森斯坦级数的傅里叶系数公式，以及从Zagier-Eisenstein级数构造半正定模形式的方法。

$Λ$-submodules of finite index of anticyclotomic plus and minus Selmer groups of elliptic curves

Let $p$ be an odd prime and $K$ an imaginary quadratic field where $p$ splits. Under appropriate hypotheses, Bertolini showed that the Selmer group of a $p$-ordinary elliptic curve over the anticyclotomic $\mathbb Z_p$-extension of $K$ does not admit any proper $Λ$-submodule of finite index, where $Λ$ is a suitable Iwasawa algebra. We generalize this result to the plus and minus Selmer groups (in the sense of Kobayashi) of $p$-supersingular elliptic curves. In particular, in our setting the plus/minus Selmer groups have $Λ$-corank one, so they are not $Λ$-cotorsion. As an application of our ma…

Elliptic periods and primality proving

We define the ring of elliptic periods modulo an integer $n$ and give an elliptic version of the AKS primality criterion.

Period and index of genus one curves over number fields

The period of a curve is the smallest positive degree of Galois-invariant divisor classes. The index is the smallest positive degree of rational divisors. We construct examples of genus one curves with prescribed period and index over certain number fields.

On a correlation between ranks of elliptic curves and periods of continued fractions

It is proved that the rank of elliptic curves with complex multiplication introduced by B. H. Gross is one less an arithmetic complexity of the corresponding noncommutative tori with real multiplication.

Explicit sections on Kuwata's elliptic surfaces

We give explicit generators for subgroups of finite index of the Mordell-Weil groups of several families of elliptic surfaces introduced by Masato Kuwata.

Elliptic periods for finite fields

We construct two new families of basis for finite field extensions. Basis in the first family, the so-called elliptic basis, are not quite normal basis, but they allow very fast Frobenius exponentiation while preserving sparse multiplication formulas. Basis in the second family, the so-called normal elliptic basis are normal basis and allow fast (quasi linear) arithmetic. We prove that all extensions admit models of this kind.

Period Computations for Covers of Elliptic Curves

In this article, we construct algebraic equations for a curve C and a map f to an elliptic curve E, with pre-specified branching data. We do this by determining certain relations that the periods of C and E must satisfy and use these relations to approximate the coefficients to high precision. We then conjecture which algebraic numbers the coefficients are, and then we prove this conjecture to be correct.

On the index of the Heegner subgroup of elliptic curves

Let E be an elliptic curve of conductor N and rank one over Q. So there is a non-constant morphism X+0(N) --> E defined over Q, where X+0(N) = X0(N)/wN and wN is the Fricke involution of the modular curve X+0(N). Under this morphism the traces of the Heegner points of X+0(N) map to rational points on E. In this paper we study the index I of the subgroup generated by all these traces on E(Q). We propose and also discuss a conjecture that says that if N is prime and I > 1, then either the number of connected components of the real locus X+0(N)(R) is greater than 1 or (less likely) the order S of…

L-equivalence for degree five elliptic curves, elliptic fibrations and K3 surfaces

We construct nontrivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over non-algebraically closed fields) and provide a new bit of evidence for the conjectural relationship between L-equivalence and derived equivalence. The proof of the L-equivalence for curves is based on Kuznetsov's Homological Projective Duality for Gr(2,5), and L-equivalence is extended from genus one curves to elliptic surfaces using the Ogg--Shafarevich theory of twisting…

Rigidity of elliptic genera: from number theory to geometry and back

In this paper we derive topological and number theoretical consequences of the rigidity of elliptic genera, which are special modular forms associated to each compact almost complex manifold. In particular, on the geometry side, we prove that rigidity implies relations between the Betti numbers and the index of a compact symplectic manifold of dimension $2n$ admitting a Hamiltonian action of a circle with isolated fixed points. We investigate the case of maximal index and toric actions. On the number theoretical side we prove that from each compact almost complex manifold of index greater than…