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For an elliptic curve $$E/{\mathbb {Q}}$$ , let $$a_p$$ denote the trace of its Frobenius endomorphism over $${\mathbb {F}}_p$$ , where p is a prime of good reduction for E. Hasse’s theorem asserts that $$|a_p| \le 2\sqrt{p}$$ . In this paper we establish average asymptotics for primes p for which $$a_p \in \left( 2\sqrt{p} - f(p), 2\sqrt{p}\right) $$ or $$a_p \in \left( c\sqrt{p} - f(p), c\sqrt{p}\right) $$ , where $$f(x) = o(\sqrt{x})$$ is a function satisfying mild growth conditions and $$0< c < 2$$ is a constant.

For an elliptic curve E / Q , Hasse's theorem asserts that # E ( F p ) = p + 1 − a p , where | a p | ≤ 2 p . Assuming that E has complex multiplication, we establish asymptotics for primes p for which a p is in subintervals of the Hasse interval [ − 2 p , 2 p ] of measure o ( p ) . In particular, given a function f = o ( 1 ) satisfying some mild conditions, we provide counting functions for primes p where | a p | ∈ ( 2 p ( 1 − f ( p ) ) , 2 p ) , and for primes where a p ∈ ( 2 p ( c − f ( p ) ) , 2 c p ) , where c ∈ ( 0 , 1 ) is a constant.

Fix an elliptic curve E / Q . For each prime p of good reduction, let a p = p + 1 − # E ( F p ) . A well-known theorem of Hasse asserts that | a p | ≤ 2 p . We say that p is a champion prime for E if a p = − ⌊ 2 p ⌋ , that is, # E ( F p ) is as large as allowed by the Hasse bound. Analogously, we call p a trailing prime if a p = ⌊ 2 p ⌋ . In this note, we study the frequency of champion and trailing primes for CM elliptic curves. Our main theorem is that for CM curves, both the champion primes and trailing primes have counting functions ∼ 2 3 π x 3 / 4 / log ⁡ x , as x → ∞ . This confirms (in corrected form) a recent conjecture of James–Tran–Trinh–Wertheimer–Zantout.

For an elliptic curve E / Q , we define an extremal prime for E to be a prime p of good reduction such that the trace of Frobenius of E at p is ± ⌊ 2 p ⌋ , i.e., maximal or minimal in the Hasse interval. Conditional on the Riemann Hypothesis for certain Hecke L -functions, we prove that if End ( E ) = O K , where K is an imaginary quadratic field of discriminant ≠ − 3 , − 4 , then the number of extremal primes ≤ X for E is asymptotic to X 3 / 4 / log ⁡ X . We give heuristics for related conjectures.

Let $E/\mathbb{Q} $ be an elliptic curve. Silverman and Stange define primes $p$ and $q$ to be an elliptic, amicable pair if $\#E(\mathbb{F} _p) = q$ and $\#E(\mathbb{F} _q) = p$. More generally, they define the notion of aliquot cycles for elliptic curves. Here, we study the same notion in the case that the elliptic curve is defined over a number field~$K$. We focus on proving the existence of an elliptic curve~$E/K$ with aliquot cycle $(\mathfrak{p} _1, \ldots , \mathfrak{p} _{n})$ where the $\mathfrak{p} _{i}$ are primes of~$K$ satisfying mild conditions.

It has been shown in several settings that the product of two eigenforms is rarely an eigenform. In this paper we consider the more general question of when the product of an eigenform with any modular form is again an eigenform. We prove that this can occur only in very special situations. We then relate the divisibility of eigenforms to linear independence of vectors of Rankin-Selberg L-values.

We consider a specific family of elliptic curves with rational 3-torsion subgroup. We arithmetically define 3-Selmer groups through isogeny and 3-descent maps, then associate the image of the 3-descent maps to solutions of homogeneous cubic polynomials affiliated with the elliptic curve E and an isogenous curve E′. Thanks to the work of Cohen and Pazuki, we have solubility conditions for the homogeneous polynomials. Using these conditions, we give a graphical approach to computing the size of 3-Selmer groups. Finally, we translate the conditions on graphs into a question concerning ranks of matrices and give an upper bound for the rank of the elliptic curve E by calculating the size of the Selmer groups.

Let E be an elliptic curve defined over an abelian number field K of degree m. For a prime ideal p of OK of good reduction we consider E over the finite field OK/p and let ap(E) be the trace of the Frobenius morphism. If E does not have complex multiplication, a generalization of the Lang-Trotter Conjecture asserts that given r, f ∈ Z with f > 0 and f | m, there exists a constant CE,r,f ≥ 0 such that #{p : N(p) ≤ x, degK(p) = f and ap(E) = r} ∼ CE,r,f ·  √ x log x if f = 1, log log x if f = 2, 1 if f ≥ 3. We prove that this conjecture holds on average in certain families of elliptic curves defined over K.

In this paper we discuss efficient algorithms for com- puting the values of the partition function and implement these algorithms in order to conduct a numerical study of some conjec- tures related to the partition function. We present the distribution of p(N) for N ≤ 10 9 for primes up to 103 and small powers of 2 and 3.

Let En:y2=x3−n2x denote the family of congruent number elliptic curves. Feng and Xiong (2004) equate the nontriviality of the Selmer groups associated with En to the presence of certain types of partitions of graphs associated with the prime factorization of n. In this paper, we extend the ideas of Feng and Xiong in order to compute the Selmer groups of En.

In this paper, we examine the Lang–Trotter conjecture for elliptic curves which possess rational 3-torsion points. We prove that if one averages over all such elliptic curves then one obtains an asymptotic similar to the one predicted by Lang and Trotter.

Let T p,k (x) be the characteristic polynomial of the Hecke operator Tp acting on the space of level 1 cusp forms S k (1). We show that T p,k (x) is irreducible and has full Galois group over Q for k < 2000 and p < 2000, p prime.

(1.1) E : y + a1xy + a3y = x 3 + a2x 2 + a4x+ a6 where a1, a2, a3, a4, a6 ∈ Z. Let N(E) denote the conductor of E, j(E) the j-invariant of E, and L(E, s) = ∑∞ n=1 a(n)n −s the Hasse-Weil L-function of E. If E is modular, then let FE(z) = ∑∞ n=1 aE(n)q n ∈ S2(N(E), χ1) be the associated weight 2 cusp form. Here χ1 denotes the trivial Dirichlet character. Throughout, D will denote a square-free integer, and χD shall denote the Kronecker character for the field Q( √ D). Let h(D) denote the order of Cl(D), the ideal class group of Q( √ D). If l is prime, then define h(D)l by

Let T 2k p (x) denote the characteristic polynomial of the action of T 2k p on S2k . It is well known that T 2k p (x) # Z[x] and has degree dk where dk := dim(S2k). However, much more is conjectured to be true. Maeda has conjectured that the Hecke algebra of S2k over Q is simple, and that its Galois closure over Q has Galois group Sdk . Recently there have been numerous investigations regarding the irreducibility of characteristic polynomials of Hecke operators on S2k . The existence of such polynomials have proven to be useful in proving nonvanishing theorems for central values of level 1 modular L-functions, and in constructing base changes to totally real number fields for level 1 eigenforms (see [C-F, H-M, Ko-Z]). In this note we show that a ``positive proportion'' of the Hecke polynomials T 2k p (x) are irreducible if there are two distinct primes l and q for which T 2k q (x) is irreducible over Fl , the finite field with l elements. Throughout p will denote a prime number.

Suppose that f Zn>1 an(f)qTn is a cusp form of weight 2k (k E N). We denote by L(f, s) the L-function of f. For Re(s) sufficiently large, the value of L(f, s) is given by L(f, s) = En>1 an), and one can show that L(f, s) has analytic continuation to the entire complex plane. The value of L(f, s) at s = k will be of particular interest to us, and we will refer to this value as the central critical value of L(f, s). Let XD denote the Dirichlet character associated to the extension Q(\D)/Q, that is, XD (n) (va), where AD denotes the discriminant of Q( D)/Q. Define the Dth quadratic twist of f to be fxD = En>1 an(f)XD (n)qn. For any integer D, the L-function of fx, is the Dth quadratic twist of L(f, s), that is, L(fxD , s) Zn> aan(f )XD(n). We will be interested in determining how often L(fxD,s) has nonzero central critical value as D varies over all integers. Since XD 2 = XD we will restrict our attention to the square-free integers D. We expect that as we let D vary over all of the square-free integers, a positive proportion of the L-functions L(fxD , s) will have nonzero central critical value. Indeed, Goldfeld [7] conjectures that for newforms f of weight 2, L(fxD 1) 78 0 for 1 of the square-free integers. Given an elliptic curve E: y2 = x3 +Ax2 +Bx+C (A, B, C E Z) with conductor NE and an integer D, we define the Dth quadratic twist of E to be the curve ED: Y2 = x3+ADx2+BD2x+CD3. Let L(ED, s) denote the L-function associated to ED. For square-free D coprime to 2NE, L(ED, s) is simply the Dth quadratic twist of L(Ei, s). If f E S2(N) is a newform with integer coefEcients, we know via the theory of Eichler and Shimura that there is an elliptic curve E over Q having conductor N so that L(E, s) = L(f, s). Thus if D is coprime to 2N, then L(ED, s) L(fXD, s). Also, one knows from the work of Kolyvagin [13], as supplemented by the work of Murty and Murty [17] or that of Bump, Friedberg and Hoffstein [3] (see also [10] for a shorter proof), that if E is a modular elliptic curve and if L(E, 1) 7 0, then the rank of E is 0. Thus, if f has the property that a positive proportion of the twists of L(f, s) have nonzero central critical value, then this implies that a positive density of the quadratic twists ED have rank 0. There have been many papers which have proved results in this direction. For example, in [2], [3], [6], [10], [16], [17], [19], [28] one can find general theorems on

Let $K$ be a fixed number field, assumed to be Galois over $\mathbb Q$. Let $r$ and $f$ be fixed integers with $f$ positive. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree $f$ prime ideals of $K$ with trace of Frobenius equal to $r$. Except in the case $f=2$, we show that "on average," the number of such prime ideals with norm less than or equal to $x$ satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang-Trotter conjecture and extends the work of several authors., Preprint of an old paper

The majority of asset pricing theories relate expected returns on assets to their conditional variances and covariance’s. Since conditional variances and covariance’s are not observable, researchers have to estimate conditional second moments relying on models. An important concern is the accuracy of these models and how researchers may estimate them more accurately. In this paper, various measures of volatility have been examined ranging from time invariant to time variant measures. In the former case one of the simplest measures examined was the standard deviation. A weakness of this measure is the assumption that volatility is constant, this being due to the standard deviation of returns increasing with the square root of the length of the period. Empirical evidence, however, shows us that the behavior of asset returns in the real world changes randomly over time. This led us to an examination of time variant models for measuring volatility.

We prove that the product of two nearly holomorphic Hecke eigenforms is again a Hecke eigenform for only finitely many choices of factors., Comment: 8 pages

According to some recently published results, counter-based compaction outperforms compaction by linear feedback shift registers. These results, however, are based on oversimplified assumptions. In this paper, we discuss an error model to describe the behavior of a faulty circuit under test. We study the three most popular counter-based compaction schemes, (i.e., one's counting, transition counting and edge counting). Using Markov processes we derive equations for iterative computations of exact aliasing probability for any test session length and determine the asymptotic probability of aliasing. For one's counting, we also present a closed form expression that, for any test session length, gives the exact aliasing probability. Finally, we present some examples to compare the aliasing in the counter-based compaction and compaction by a linear feedback shift register. These examples indicate that aliasing by LFSRs is more “predictable” than aliasing by counters.

This chapter provides an overview of scaling and linearization. Also discusses scaling of linear response curves; the simplest and the most common type of response curve is a straight line . Linearization is the term applied to the process of correcting the output of an ADC in order to compensate for non-linearities present in the response curve of a measuring system. Non-linearities can arise from a number of different components, but it is often the sensors themselves that are the primary sources. There are several linearization methods to choose from and what -ever method is selected, it must suit the peculiarities of the system's response curve. Polynomials can be used for linearizing smooth and slowly varying functions, but are less suitable for correcting irregular deviations or sharp `corners' in the response curve. They can be adapted to closely match a known functional form or they can b e used in cases where the form of the response function is indeterminate.

If p is prime, then let E p denote the elliptic curve over the rationals given by $${E_p}:{y^2} = {x^3} - 32{p^3}$$ . We prove that E P has only the trivial point (at infinity) for at least \(\frac{1}{3}\) of the primes p. In fact, we will show that for 1/3 of the primes p, L(E P ,1) ≠0.

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Finite difference equations of motion of the sixth order and the associated boundary conditions for principal planes are formulated for a f.c.c. lattice of mass particles. The equations are solved for face-shear and thickness-twist waves in a plate with free faces. Computations of mode-shapes and the frequency spectrum are presented for copper and the results are compared with a previous solution for a simple cubic material and with the solution of the classical equations of elasticity.

In this paper, we give examples of elliptic curves for which a positive proportion of the quadratic twists satisfy a weak form of the Birch and Swinnerton–Dyer conjecture modulo 3.