The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at 2 and 3, K/Q is abelian and K^\infty is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of Gal(K^\infty/Q). We also give analogous results when K/Q is non-abelian, the base field is not Q and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their "regulator constants", and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.
We present a conjectural formula for the principal minors and the characteristic polynomial of Gross's regulator matrix associated to a totally odd character of a totally real field. The formula is given in terms of the Eisenstein cocycle, which was defined and studied earlier by the authors and collaborators. For the determinant of the regulator matrix, our conjecture follows from recent work of Kakde, Ventullo and the first author. For the diagonal entries, our conjecture overlaps with the conjectural formula presented in our prior work. The intermediate cases are new and provide a refinemen…
In 1979 Goldfeld conjectured: 50\% of the quadratic twists of an elliptic curve defined over the rationals have analytic rank zero. In this expository article we present a few recent developments towards the conjecture, especially its first instance - the congruent number elliptic curves.
The Birch and Swinnerton--Dyer conjecture famously predicts that the rank of an elliptic curve can be computed from its $L$-function. In this article we consider a weaker version of this conjecture called the parity conjecture and prove the following. Let $E_1$ and $E_2$ be two elliptic curves defined over a number field $K$ whose 2-torsion groups are isomorphic as Galois modules. Assuming finiteness of the Shafarevich-Tate groups of $E_1$ and $E_2$, we show that the Birch and Swinnerton-Dyer conjecture correctly predicts the parity of the rank of $E_1\times E_2$. Using this result, we complet…
Let K/Q be Galois and let eta in K* be such that the multiplicative Z[G]-module generated by eta is of Z-rank n.We define the local theta-regulators Delta\_p^theta(eta) in F\_p for the Q\_p-irreducible characters theta of G=Gal(K/Q). Let V\_theta be the theta-irreducible representation. A linear representation L^theta=delta.V\_theta is associated withDelta\_p^theta(eta) whose nullity is equivalent to delta$\ge$1 (Theorem 3.9). Each Delta\_p^theta(eta) yields Reg\_p^theta(eta) modulo p in the factorization $\prod$\_theta (Reg\_p^theta(eta))^phi(1) of Reg\_p^G(eta) := Reg\_p(eta)/p^[K : Q] (norm…
For a prime $\ell$ and an abelian variety $A$ over a global field $K$, the $\ell$-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the $\mathbb{Z}_{\ell}$-corank of the $\ell^{\infty}$-Selmer group and the analytic rank agree modulo $2$. Assuming that $\mathrm{char} K > 0$, we prove that the $\ell$-parity conjecture holds for the base change of $A$ to the constant quadratic extension if $\ell$ is odd, coprime to $\mathrm{char} K$, and does not divide the degree of every polarization of $A$. The techniques involved in the proof include the étale cohomo…
We lower bound the Faltings height of an abelian variety over a number field by the sum of its injectivity diameter and the norm of its bad reduction primes. It leads to an unconditional bound on the rank of Mordell-Weil groups. Assuming the height conjecture of Lang and Silverman, we then obtain a Northcott property for the regulator on the set of simple abelian varieties defined over a fixed number field, of fixed dimension $g$, bounded rank and with dense rational points over a number field. We remove the simplicity assumption in the principally polarized case by giving a refined version of…
We prove the local epsilon-isomorphism conjecture of Fukaya and Kato [FK06] for certain crystalline families of G_Qp-representations. This conjecture can be regarded as a local analogue of the Iwasawa main conjecture for families. Our work extends earlier work of Kato for rank-1 modules (cf. [Ven13]), of Benois and Berger for crystalline G_Qp-representations with respect to the cyclotomic extension (cf. [BB08]), as well as of Loeffler, Venjakob, and Zerbes (cf. [LVZ13]) for crystalline G_Qp- representations with respect to abelian p-adic Lie extensions of Qp. Nakamura [Nak13, Nak14] has also f…
Let $X$ be a variety defined over a local field $K$ of mixed characteristic $(0,p)$ with a totally degenerate reduction in the sense of Raskind and Xarles. Generalizing earlier work of Raskind and Xarles and relying on some conjectures we define a map, which we call the toric regulator, from the various motivic cohomology groups of $X$ to certain $p$-adically uniformized tori over $K$. This construction captures the part of the étale regulators on $X$ that land in the Galois cohomology of the submodules of cohomology which are extensions of $\mathbb{Z}_l$ by $\mathbb{Z}_l(1)$, simultaneously f…
The parity conjecture has a long and distinguished history. It gives a way of predicting the existence of points of infinite order on elliptic curves without having to construct them, and is responsible for a wide range of unexplained arithmetic phenomena. It is one of the main consequences of the Birch and Swinnerton-Dyer conjecture and lets one calculate the parity of the rank of an elliptic curve using root numbers. In this handbook, we explain how to use local root numbers of elliptic curves to realise some of these phenomena, with an emphasis on explicit calculations. The text is aimed at…
We study the 2-parity conjecture for Jacobians of hyperelliptic curves over number fields. Under some mild assumptions on their reduction, we prove the conjecture over quadratic extensions of the base field. The proof proceeds via a generalisation of a formula of Kramer and Tunnell relating local invariants of the curve, which may be of independent interest. A new feature of this generalisation is the appearance of terms which govern whether or not the Cassels--Tate pairing on the Jacobian is alternating, which first appeared in work of Poonen--Stoll. We establish the local formula in many ins…
An inequality proved firstly by Remak and then generalized by Friedman shows that there are only finitely many number fields with a fixed signature and whose regulator is less than a prescribed bound. Using this inequality, Astudillo, Diaz y Diaz, Friedman and Ramirez-Raposo succeeded to detect all fields with small regulators having degree less or equal than 7. In this paper we show that a certain upper bound for a suitable polynomial, if true, can improve Remak-Friedman's inequality and allows a classification for some signatures in degree 8 and better results in degree 5 and 7. The validity…
Let F be a totally real field of degree d and let p be an odd prime which is totally split in F. We define and study one-dimensional partial eigenvarieties interpolating Hilbert modular forms over F with weight varying only at a single place v above p. For these eigenvarieties, we show that methods developed by Liu, Wan and Xiao apply and deduce that, over a boundary annulus in weight space of sufficiently small radius, the partial eigenvarieties decompose as a disjoint union of components which are finite over weight space. We apply this result to prove the parity version of the Bloch--Kato c…
We compare general inequalities between invariants of number fields and invariants of abelian varieties over number fields. On the number field side, we remark that there is only a finite number of non-CM number fields with bounded regulator. On the abelian side, assuming the height conjecture of Lang and Silverman, we obtain a Northcott property for the regulator on the set of abelian varieties with dense rational points over a number field. This amounts to say that the arithmetic of CM fields is similar, with respect to the invariants considered here, to the arithmetic of abelian varieties o…
We consider questions in Galois cohomology which arise by considering mod $p$ Galois representations arising from automorphic forms. We consider a Galois cohomological analog for the standard heuristics about the distribution of Wieferich primes, i.e. prime $p$ such that $2^{p-1}$ is 1 mod $p^2$. Our analog relates to asking if in a compatible system of Galois representations, for almost all primes $p$, the residual mod $p$ representation arising from it has unobstructed deformation theory. This analog leads in particular to formulating a mod $p$ analog for almost all primes $p$ of the classic…
A proof of Andrews-El Bachraoui's conjecture on the parity of coefficients of a $q$-series
School of Statistics and Information, Shanghai University of International Business and Economics（上海对外经贸大学统计与信息学院）；School of Mathematical Sciences, Suzhou University of Science and Technology（苏州科技大学数学系）
This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erdős-Mirksy conjecture which states that there are infinitely many positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number of divisors of $x$. This conjecture was first proven by Heath-Brown in 1984, but the method did not reveal the nature of the set of values $d(x)$ for such $x$. In particular, one could not conclude that there was any particular value $A$ for which $d(x)=d(x+1)=A$ infinitely often. In the previous paper …
We construct a period regulator for motivic cohomology of an algebraic scheme over a subfield of the complex numbers. For the field of algebraic numbers we formulate a period conjecture for motivic cohomology by saying that this period regulator is surjective. Showing that a suitable Betti--de Rham realization of 1-motives is fully faithful we can verify this period conjecture in several cases. The divisibility properties of motivic cohomology imply that our conjecture is a neat generalization of the classical Grothendieck period conjecture for algebraic cycles on smooth and proper schemes. Th…
We investigate the parity of the coefficients of certain eta-quotients, extensively examining the case of $m$-regular partitions. Our theorems concern the density of their odd values, in particular establishing lacunarity modulo 2 for specified coefficients; self-similarities modulo 2; and infinite families of congruences in arithmetic progressions. For all $m \leq 28$, we either establish new results of these types where none were known, extend previous ones, or conjecture that such results are impossible. All of our work is consistent with a new, overarching conjecture that we present for ar…
Recently, Hirschhorn and the first author considered the parity of the function $a(n)$ which counts the number of integer partitions of $n$ wherein each part appears with odd multiplicity. They derived an effective characterization of the parity of $a(2m)$ based solely on properties of $m.$ In this note, we quickly reprove their result, and then extend it to an explicit characterization of the parity of $a(n)$ for all $n\not\equiv 7 \pmod{8}.$ We also exhibit some infinite families of congruences modulo 2 which follow from these characterizations. We conclude by discussing the case $n\equiv 7 …
A 2015 conjecture of Codesido-Grassi-Mariño in topological string theory relates the enumerative invariants of toric CY 3-folds to the spectra of operators attached to their mirror curves. We deduce two consequences of this conjecture for the integral regulators of $K_2$-classes on these curves, and then prove both of them; the results thus give evidence for the CGM conjecture. (While the conjecture and the deduction process both entail forms of local mirror symmetry, the consequences/theorems do not: they only involve the curves themselves.) Our first theorem relates zeroes of the higher norm…
