On quadratic families of CM elliptic curves

Munshi, Ritabrata

doi:10.1090/s0002-9947-2011-05433-4

Cited by 8 publications

(6 citation statements)

References 12 publications

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“…Therefore applying Poisson summation for quadratic characters with conductor ≍ |t| will return a sum of the same length. This means that we are confronted with the notorious "deadlock situation" discussed for example by Munshi in [31], but in any case well recognized by experts.…”

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confidence: 99%

Quantum unique ergodicity for half-integral weight automorphic forms

Lester¹,

Radziwiłł²

2020

Duke Math. J.

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We investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for both half-integral weight holomorphic Hecke cusp forms for Γ 0 (4) lying in Kohnen's plus subspace and for halfintegral weight Hecke Maaß cusp forms for Γ 0 (4) lying in Kohnen's plus subspace. By combining the former result along with an argument of Rudnick, it follows that under GRH the zeros of these holomorphic Hecke cusp equidistribute with respect to hyperbolic measure on Γ 0 (4)\H as the weight tends to infinity.

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confidence: 99%

Quantum unique ergodicity for half-integral weight automorphic forms

Lester¹,

Radziwiłł²

2020

Duke Math. J.

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“…He uses this fact to unconditionally obtain an asymptotic formula for the first moment of higher derivatives Λ (ℓ) (1/2, f ⊗ χ d ) with ℓ ≥ 8 weighted by the number of representations of d as a sum of two squares (a situation with conductor of similar length to ours). Munshi also solves a similar problem in [15] obtaining an asymptotic for the first derivative in the special case that f corresponds to a CM elliptic curve.…”

Section: Statement Of Main Resultsmentioning

confidence: 99%

“…In this paper we study the central values of derivatives of L-functions of holomorphic GL(2) modular forms in the family of quadratic twists. The mean value of this family has been studied successfully in the past by several authors, notably Bump, Friedberg and Hoffstein [2], Murty and Murty [16], Iwaniec [10] and Munshi [14], [15].…”

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confidence: 99%

Moments of L′(½) in the Family of Quadratic Twists

Petrow

2012

International Mathematics Research Notices

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We prove the asymptotic formulae for several moments of derivatives of GL(2) L-functions over quadratic twists. The family of L-functions we consider has root number fixed to −1 and odd orthogonal symmetry. Assuming GRH we prove the asymptotic formulae for (1) the second moment with one secondary term, (2) the moment of two distinct modular forms f and g and (3) the first moment with controlled weight and level dependence. We also include some immediate corollaries to elliptic curves via the modularity theorem and the work of Gross and Zagier.

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“…In this special case Manin's conjecture [FMT89] suggests that N (S, B) should grow like B 2 . In the complex multiplication case these were studied in [Mun11] where the bound N (S, B) ≥ ǫ · B 2−η is proved for every η > 0.…”

Section: (Connection With Conic Bundles)mentioning

confidence: 99%

Quadratic families of elliptic curves and unirationality of degree 1 conic bundles

Kollár¹,

Mella²

2017

American Journal of Mathematics

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Let K be a number field and a i (t) ∈ K[t] polynomials of degree 2. We consider the family of elliptic curves E t := y 2 = a 3 (t)x 3 + a 2 (t)x 2 + a 1 (t)x + a 0 (t) ⊂ A 2 xy ( * ) parametrized by t ∈ K. Our aim is to show that there are many values t ∈ K for which the corresponding elliptic curve E t has rank ≥ 1. Conjecturally this should hold for a positive proportion of them; see the survey [RS02]. We prove rank ≥ 1 for about the square root of all t ∈ K, listed by height. We are only interested in nontrivial families, when at least two of the curves E t are smooth, elliptic and not isomorphic to each other over K. Thus a 3 (t) is not identically 0 and not all the a i (t) are constant multiples of the same square (t − c) 2 . We view the whole family as a single algebraic surface in A 3 xyt and look at the distribution of K-points. The resulting surface has degree 5 but its closure in P 3 is very singular. We prove the following. Theorem 1. Let k be any field of characteristic = 2 and a 0 (t), . . . , a 3 (t) ∈ k[t] polynomials of degree 2 giving a nontrivial family of elliptic curves. Then the surfaceis unirational over k.The proof has two parts. First assume that we know a k-point p ∈ S that is not a 6-torsion point on the corresponding elliptic curve. Then we have geometrically clear and quite explicit formulas to prove unirationality. The second, harder part is to show that there are such k-points. We have not been able to turn this part of the proof into explicit formulas; see Remark 40.From any sufficiently general k-point of S we obtain families of elliptic curves of rank ≥ 1.Corollary 2. Let k be an infinite field of characteristic = 2. Then there are infinitely many different rational functions q(u) ∈ k(u) that are quotients of degree 2 polynomials such that the rank of the elliptic curve E t as in ( * ) is ≥ 1 for all but finitely many values t = q(u) where u ∈ k.Unirationality can also be used to exhibit points of small height. For a Zariski open subset U ⊂ S, let N (U, B) be the number of K-points of height ≤ B in U . Manin's conjecture [FMT89] suggests that N (U, B) should grow at least like B 3/2 , using the naive height function ht(x, y, t) := ht(x) + ht(t). Theorem 1 implies that N (U, B) grows at least like a power of B. The proof gives an explicit value for ǫ but it seems to be small. 4 (Connection with conic bundles). We can rewritewhere A, B, C are cubics. Projection to the x-axis exhibits S as the family of conicsyt . The conic F x is singular iff x is a root of the discriminant B(x) 2 − 4A(x)C(x); in general this happens for 6 different values of x. We get a possible 7th singular fiber at infinity. After suitable birational transformations the fiber at infinity is isomorphic tois the homogenization of a 3 (t). Thus S is birational to a conic bundle with ≤ 7 singular fibers; see Definition 6.A very degenerate case is when B(x) 2 − 4A(x)C(x) ≡ 0. These are easy to enumerate by hand. The only non-rational surface occurs when a i (t) = c i (t − α) 2 for every i. Then setting z := y/(t − α)...

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On quadratic families of CM elliptic curves

Cited by 8 publications

References 12 publications

Quantum unique ergodicity for half-integral weight automorphic forms

Quantum unique ergodicity for half-integral weight automorphic forms

Moments of L′(½) in the Family of Quadratic Twists

Quadratic families of elliptic curves and unirationality of degree 1 conic bundles

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