Fundamental Discriminants Research Articles

On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

We study an infinite family of j-invariant zero elliptic curves ED:y2=x3+16D and their λ-isogenous curves ED′:y2=x3−27⋅16D, where D and D′=−3D are fundamental discriminants of a specific form, and λ is an isogeny of degree 3. A result of Honda guarantees that for our discriminants D, the quadratic number field KD=Q(D) always has non-trivial 3-class group. We prove a series of results related to the set of rational points ED′(Q)∖λ(ED(Q)), and the SL(2,Z)-equivalence classes of irreducible integral binary cubic forms of discriminant D. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of ED and the rank of its 3-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-1 curves that correspond to irreducible integral binary cubic forms of discriminant D=48035713, and we show that every curve in these classes violates the Hasse Principle.

Large Fourier coefficients of half-integer weight modular forms

abstract: This article is concerned with the Fourier coefficients of cusp forms (not necessarily eigenforms) of half-integer weight lying in the plus space. We give a soft proof that there are infinitely many fundamental discriminants $D$ such that the Fourier coefficients evaluated at $|D|$ are non-zero. By adapting the resonance method, we also demonstrate that such Fourier coefficients must take quite large values.

Value Distribution of Logarithmic Derivatives of Quadratic Twists of Automorphic L-functions

Abstract Let $d\in\mathbb{N}$ and π be a fixed cuspidal automorphic representation of $\mathrm{GL}_{d}(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. We determine the limiting distribution of the family of values $-\frac{L^{\prime}}{L}(1+it,\pi\otimes\chi_D)$ as D varies over fundamental discriminants. Here, t is a fixed real number and χD is the real character associated with D. We establish an upper bound on the discrepancy in the convergence of this family to its limiting distribution. As an application of this result, we obtain an upper bound on the small values of $\left|\frac{L^{\prime}}{L}(1,\pi\otimes\chi_D)\right|$ when π is self-dual.

Distribution of toric periods of modular forms on definite quaternion algebras

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ and $\mathcal{O}$ an Eichler order in $D$ of square-free level. We study distribution of the toric periods of algebraic modular forms of level $\mathcal{O}$. We focus on two problems: non-vanishing and sign changes. Firstly, under certain conditions on $\mathcal{O}$, we prove the non-vanishing of the toric periods for positive proportion of imaginary quadratic fields. This improves the known lower bounds toward Goldfeld's conjecture in some cases and provides evidence for similar non-vanishing conjectures for central values of twisted automorphic $L$-functions. Secondly, we show that the sequence of toric periods has infinitely many sign changes. This proves the sign changes of the Fourier coefficients $\{a(n)\}_n$ of weight 3/2 modular forms, where $n$ ranges over fundamental discriminants. In the final section, we present numerical experiments in some cases and formulate several conjectures based on them.

Effective construction of Hilbert modular forms of half-integral weight

Given a Hilbert cuspidal newform $g$ we construct a family of modular forms of half-integral weight whose Fourier coefficients give the central values of the twisted $L$-series of $g$ by fundamental discriminants. The family is parametrized by quadratic conditions on the primes dividing the level of $g$, where each form has coefficients supported on the discriminants satisfying the conditions. These modular forms are given as generalized theta series and thus their coefficients can be effectively computed. Our construction works over arbitrary totally real number fields, except that in the case of odd degree the square levels are excluded. It includes all discriminants except those divisible by primes whose square divides the level.

The distribution of values of [formula omitted

We determine the limiting distribution of the family of values L′L(1/2+ϵ,χD) as D varies over fundamental discriminants. Here, 0<ϵ<12, and χD is the real character associated with D. Moreover, we also establish an upper bound for the rate of convergence of this family to its limiting distribution. As a consequence of this result, we derive an asymptotic bound for the small values of |L′L(1/2+ϵ,χD)|.

Simultaneous indivisibility of class numbers of pairs of real quadratic fields

For a square-free integer t, Byeon (Proc. Am. Math. Soc. 132:3137–3140, 2004) proved the existence of infinitely many pairs of quadratic fields $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}(\sqrt{tD})$$ with $$D > 0$$ such that the class numbers of all of them are indivisible by 3. In the same spirit, we prove that for a given integer $$t \ge 1$$ with $$t \equiv 0 \pmod {4}$$ , a positive proportion of fundamental discriminants $$D > 0$$ exist for which the class numbers of both the real quadratic fields $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}(\sqrt{D + t})$$ are indivisible by 3. This also addresses the complement of a weak form of a conjecture of Iizuka (J. Numb. Theory, 184:122–127, 2018). As an application of our main result, we obtain that for any integer $$t \ge 1$$ with $$t \equiv 0 \pmod {12}$$ , there are infinitely many pairs of real quadratic fields $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}(\sqrt{D + t})$$ such that the Iwasawa $$\lambda $$ -invariants associated with the basic $$\mathbb {Z}_{3}$$ -extensions of both $$\mathbb {Q}(\sqrt{D})$$ and $$\mathbb {Q}(\sqrt{D + t})$$ are 0. For $$p = 3$$ , this supports Greenberg’s conjecture which asserts that $$\lambda _{p}(K) = 0$$ for any prime number p and any totally real number field K.

On class numbers, torsion subgroups, and quadratic twists of elliptic curves

The Mordell-Weil groups $E(\mathbb{Q})$ of elliptic curves influence the structures of their quadratic twists $E_{-D}(\mathbb{Q})$ and the ideal class groups $\mathrm{CL}(-D)$ of imaginary quadratic fields. For appropriate $(u,v) \in \mathbb{Z}^2$, we define a family of homomorphisms $\Phi_{u,v}: E(\mathbb{Q}) \rightarrow \mathrm{CL}(-D)$ for particular negative fundamental discriminants $-D:=-D_E(u,v)$, which we use to simultaneously address questions related to lower bounds for class numbers, the structures of class groups, and ranks of quadratic twists. Specifically, given an elliptic curve $E$ of rank $r$, let $\Psi_E$ be the set of suitable fundamental discriminants $-D 0$, we show that as $X \to \infty$, we have $$\#\, \left\{-X < -D < 0: -D \in \Psi_E \right \} \, \gg_{\varepsilon} X^{\frac{1}{2}-\varepsilon}.$$ In particular, if $\ell \in \{3,5,7\}$ and $\ell \mid |E_{\mathrm{tor}}(\mathbb{Q})|$, then the number of such discriminants $-D$ for which $\ell \mid h(-D)$ is $\gg_{\varepsilon} X^{\frac{1}{2}-\varepsilon}.$ Moreover, assuming the Parity Conjecture, our results hold with the additional condition that the quadratic twist $E_{-D}$ has rank at least 2.

Signs of Fourier coefficients of half-integral weight modular forms

Let g be a Hecke cusp form of half-integral weight, level 4 and belonging to Kohnen’s plus subspace. Let c(n) denote the nth Fourier coefficient of g, normalized so that c(n) is real for all n ge 1. A theorem of Waldspurger determines the magnitude of c(n) at fundamental discriminants n by establishing that the square of c(n) is proportional to the central value of a certain L-function. The signs of the sequence c(n) however remain mysterious. Conditionally on the Generalized Riemann Hypothesis, we show that c(n) < 0 and respectively c(n) > 0 holds for a positive proportion of fundamental discriminants n. Moreover we show that the sequence {c(n)} where n ranges over fundamental discriminants changes sign a positive proportion of the time. Unconditionally, it is not known that a positive proportion of these coefficients are non-zero and we prove results about the sign of c(n) which are of the same quality as the best known non-vanishing results. Finally we discuss extensions of our result to general half-integral weight forms g of level 4N with N odd, square-free.

A conjecture on the zeta functions of pairs of ternary quadratic forms

We consider the prehomogeneous vector space of pairs of ternary quadratic forms. For the lattice of pairs of integral ternary quadratic forms and its dual lattice, there are six zeta functions associated with the the prehomogeneous vector space. We present a conjecture which states that there are simple relations among the six zeta functions. We prove that the coefficients coincide on fundamental discriminants.

Computing intersections of closed geodesics on the modular curve

In a recent work of Duke, Imamoḡlu, and Tóth, the linking number of certain links on the space SL(2,Z)\\SL(2,R) is investigated. This linking number has an alternative interpretation as the intersection number of closed geodesics on the modular curve, which is the focus of this paper. By relating the intersection number to a combinatorial computation involving rivers of Conway topographs, an efficient algorithm for computing intersection numbers is produced. A formula for the total intersection of a pair of positive coprime fundamental discriminants is also derived, which can be thought of as a real quadratic analogue of a classical result of Gross and Zagier. The paper ends with numerical computations and distribution questions relating to intersection numbers.

A generalization of a theorem of Hecke for [formula omitted] to fundamental discriminants

Let p>3 be an odd prime, p≡3mod4 and let π+,π− be the pair of cuspidal representations of SL2(Fp). It is well known by Hecke that the difference mπ+−mπ− in the multiplicities of these two irreducible representations occurring in the space of weight 2 cusps forms with respect to the principal congruence subgroup Γ(p), equals the class number h(−p) of the imaginary quadratic field Q(−p). We extend this result to all fundamental discriminants −D of imaginary quadratic fields Q(−D) and prove that an alternating sum of multiplicities of certain irreducibles of SL2(Z/DZ) is an explicit multiple, up to a sign and a power of 2, of either the class number h(−D) or of the sums h(−D)+h(−D/2), h(−D)+2h(−D/2); the last two possibilities occur in some of the cases when D≡0mod8. The proof uses the holomorphic Lefschetz number.

Joint equidistribution of CM points

We prove the mixing conjecture of Michel and Venkatesh for toral packets with negative fundamental discriminants and split at two fixed primes; assuming all splitting fields have no exceptional Landau-Siegel zero. As a consequence we establish for arbitrary products of indefinite Shimura curves the equidistribution of Galois orbits of generic sequences of CM points all whose components have the same fundamental discriminant; assuming the CM fields are split at two fixed primes and have no exceptional zero. The joinings theorem of Einsiedler and Lindenstrauss applies to the toral orbits arising in these results. Yet it falls short of demonstrating equidistribution due to the possibility of intermediate algebraic measures supported on Hecke correspondences and their translates. The main novel contribution is a method to exclude intermediate measures for toral periods. The crux is a geometric expansion of the cross-correlation between the periodic measure on a torus orbit and a Hecke correspondence, expressing it as a short shifted convolution sum. The latter is bounded from above generalizing the method of Shiu and Nair to polynomials in two variables on smooth domains.

A CERTAIN DIRICHLET SERIES OF RANKIN–SELBERG TYPE ASSOCIATED WITH THE IKEDA LIFT OF HALF‐INTEGRAL WEIGHT

In this article we obtain an explicit formula for certain Rankin-Selberg type Dirichlet series associated to certain Siegel cusp forms of half-integral weight. Here these Siegel cusp forms of half-integral weight are obtained from the composition of the Ikeda lift and the Eichler-Zagier-Ibukiyama correspondence. The integral weight version of the main theorem had been obtained by Katsurada and Kawamura. The result of the integral weight case is a product of $L$-function and Riemann zeta functions, while half-integral weight case is a infinite summation over negative fundamental discriminants with certain infinite products. To calculate explicit formula of such Rankin-Selberg type Dirichlet series, we use a generalized Maass relation and adjoint maps of index-shift maps of Jacobi forms.

The distribution of class numbers in a special family of real quadratic fields

We investigate the distribution of class numbers in the family of real quadratic fields Q ( d ) \mathbb {Q}(\sqrt {d}) corresponding to fundamental discriminants of the form d = 4 m 2 + 1 d=4m^2+1 , which we refer to as Chowla’s family. Our results show a strong similarity between the distribution of class numbers in this family and that of class numbers of imaginary quadratic fields. As an application of our results, we prove that the average order of the number of quadratic fields in Chowla’s family with class number h h is ( log ⁡ h ) / 2 G (\log h)/2G , where G G is Catalan’s constant. With minor modifications, one can obtain similar results for Yokoi’s family of real quadratic fields Q ( d ) \mathbb {Q}(\sqrt {d}) , which correspond to fundamental discriminants of the form d = m 2 + 4 d=m^2+4 .

Deep Transfers of p-Class Tower Groups

Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G from the maximal subgroups H of index (G：H) = p in G to the derived subgroup G ' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels ud(G) =(ker(T H,G ')) (G： H) = p. For all finite 3-groups G of coclass cc(G) = 1, the family ud(G) is determined explicitly. The results are applied to the Galois groups G =Gal(F3 (∞)/ F) of the Hilbert 3-class towers of all real quadratic fields F = Q(√d) with fundamental discriminants d > 1, 3-class group Cl3(F) □ C3 × C3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1 d 7, and a few exceptional cases are pointed out for 1 d 8.

On correlations between class numbers of imaginary quadratic fields

Let $h(-n)$ be the class number of the imaginary quadratic field with discriminant $-n$. We establish an asymtotic formula for correlations involving $h(-n)$ and $h(-n-l)$, over fundamental discriminants that avoid the congruence class $1\pmod{8}$. Our result is uniform in the shift $l$, and the proof uses an identity of Gauss relating $h(-n)$ to representations of integers as sums of three squares. We also prove analogous results on correlations involving $r_Q(n)$, the number of representations of an integer $n$ by an integral positive definite quadratic form $Q$.

Missing Class Groups and Class Number Statistics for Imaginary Quadratic Fields

ABSTRACTThe number of imaginary quadratic fields with class number h is of classical interest: Gauss’ class number problem asks for a determination of those fields counted by . The unconditional computation of for h ⩽ 100 was completed by Watkins, using ideas of Goldfeld and Gross–Zagier; Soundararajan has more recently made conjectures about the order of magnitude of as h → ∞ and determined its average order. In the present paper, we refine Soundararajan’s conjecture to a conjectural asymptotic formula for odd h by amalgamating the Cohen–Lenstra heuristic with an archimedean factor, and obtain an adelic, or global, refinement of the Cohen–Lenstra heuristic. We also consider the problem of determining the number of imaginary quadratic fields with class group isomorphic to a given finite abelian group G. Using Watkins’ tables, one can show that some abelian groups do not occur as the class group of any imaginary quadratic field (for instance, does not). This observation is explained in part by the Cohen–Lenstra heuristics, which have often been used to study the distribution of the p-part of an imaginary quadratic class group. We combine heuristics of Cohen–Lenstra together with our prediction for the asymptotic behavior of to make precise predictions about the asymptotic nature of the entire imaginary quadratic class group, in particular addressing the above-mentioned phenomenon of “missing” class groups, for the case of p-groups as p tends to infinity. Furthermore, conditionally on the Generalized Riemann Hypothesis, we extend Watkins’ data, tabulating for odd h ⩽ 106 and for G a p-group of odd order with |G| ⩽ 106. (In order to do this, we need to examine the class numbers of all negative prime fundamental discriminants − q, for q ⩽ 1.1881 × 1015.) The numerical evidence matches quite well with our conjectures, though there appears to be a small “bias” for class number divisible by powers of 3.

Average values of L-functions in even characteristic

Let k=Fq(T) be the rational function field over a finite field Fq, where q is a power of 2. In this paper we solve the problem of averaging the quadratic L-functions L(s,χu) over fundamental discriminants. Any separable quadratic extension K of k is of the form K=k(xu), where xu is a zero of X2+X+u=0 for some u∈k. We characterize the family I (resp. F, F′) of rational functions u∈k such that any separable quadratic extension K of k in which the infinite prime ∞=(1/T) of k ramifies (resp. splits, is inert) can be written as K=k(xu) with a unique u∈I (resp. u∈F, u∈F′). For almost all s∈C with Re(s)≥12, we obtain the asymptotic formulas for the summation of L(s,χu) over all k(xu) with u∈I, all k(xu) with u∈F or all k(xu) with u∈F′ of given genus. As applications, we obtain the asymptotic mean value formulas of L-functions at s=12 and s=1 and the asymptotic mean value formulas of the class number hu or the class number times regulator huRu.

Indivisibility of class numbers of imaginary quadratic fields

We quantify a recent theorem of Wiles on class numbers of imaginary quadratic fields by proving an estimate for the number of negative fundamental discriminants down to -X whose class numbers are indivisible by a given prime ell and whose imaginary quadratic fields satisfy any given set of local conditions. This estimate matches the best results in the direction of the Cohen–Lenstra heuristics for the number of imaginary quadratic fields with class number indivisible by a given prime. This general result is applied to study rank 0 twists of certain elliptic curves.