L-functions Of Elliptic Curves Research Articles

Special values of L-functions of elliptic curves over Q and their base change to real quadratic fields

For an elliptic curve E over Q, and a real quadratic extension F of Q, satisfying suitable hypotheses, we study the algebraic part of certain twisted L-values for E / F . The Birch and Swinnerton–Dyer conjecture predicts that these L-values are squares of rational numbers. We show that this question is related to the ratio of Petersson inner products of a quaternionic form on a definite quaternion algebra over Q and its base change to F.

Maximal Galois Group of L-Functions of Elliptic Curves

We give a quantitative version of a result due to Katz about L-functions of elliptic curves over function fields over finite fields. Roughly speaking, Katz's theorem states that, on average over a suitably chosen algebraic family, the L-function of an elliptic curve over a function field becomes “as irreducible as possible” when seen as a polynomial with rational coefficients, as the cardinality of the field of constants grows. A quantitative refinement is obtained as a corollary of our main result, which gives an estimate for the proportion of elliptic curves studied whose L-functions have “big” Galois group. To do so, we make use of Kowalski's idea to apply large sieve methods in algebro-geometric contexts. Besides large sieve techniques, we use results of Hall on finite orthogonal monodromy and previous work of the author on orthogonal groups over finite fields.

Lower order terms for the one-level density of elliptic curve L-functions

It is believed that, in the limit as the conductor tends to infinity, correlations between the zeros of elliptic curve L-functions averaged within families follow the distribution laws of the eigenvalues of random matrices drawn from the orthogonal group. For test functions with restricted support, this is known to be the true for the one- and two-level densities of zeros within the families studied to date. However, for finite conductor Miller's experimental data reveal an interesting discrepancy from these limiting results. Here we use the L-functions ratios conjectures to calculate the 1-level density for the family of even quadratic twists of an elliptic curve L-function for large but finite conductor. This gives a formula for the leading and lower order terms up to an error term that is conjectured to be significantly smaller. The lower order terms explain many of the features of the zero statistics for relatively small conductor and model the very slow convergence to the infinite conductor limit. However, our main observation is that they do not capture the behaviour of zeros in the important region very close to the critical point and so do not explain Miller's discrepancy. This therefore implies that a more accurate model for statistics near to this point needs to be developed.

Discrete universality of the L-functions of elliptic curves

A discrete universality theorem is obtained in the Voronin sense for the L-functions of elliptic curves. We use the difference of an arithmetical progression h > 0 such that \( \exp \left\{ {\frac{{2\pi k}} {h}} \right\} \) is rational for some k ≠ 0. A limit theorem in the space of analytic functions plays a crucial role in the proof.

Computations in non-commutative Iwasawa theory

We study special values of L-functions of elliptic curves over Q twisted by Artin representations that factor through a false Tate curve extension $Q(\mu_p^\infty,\sqrt[p^\infty]{m})/Q$. In this setting, we explain how to compute L-functions and the corresponding Iwasawa-theoretic invariants of non-abelian twists of elliptic curves. Our results provide both theoretical and computational evidence for the main conjecture of non-commutative Iwasawa theory.

ON THE NON-VANISHING OF ELLIPTIC CURVE $L$-FUNCTIONS AT THE CENTRAL POINT

We show that a large number of elliptic curve L-functions do not vanish at the central point, conditionally on the Generalized Riemann Hypothesis and on an assumption on the regular distribution of the root number.

A weighted discrete universality theorem for L-functions of elliptic curves

There is not abstract.

The Joint Universality for L-Functions of Elliptic Curves

A joint universality theorem in the Voronin sense for L-functions of elliptic curves over the field of rational numbers is proved.

Discrete value: Distribution of L-functions of elliptic curves

A discrete universality theorem in the Voronin sense for L-functions of elliptic curves is proved.

On the Vanishing of Twisted L-Functions of Elliptic Curves

Let E be an elliptic curve over Q with L-function LE (S). We use the random matrix model of Katz and Sarnak to develop a heuristic for the frequency of vanishing of the twisted Lfunctions LE (l, X), as X runs over the Dirichlet characters of order 3 (cubic twists). We also compute explicitly the conjecture of Keating and Snaith about the moments of the special values LE (l, X) in the family of cubic twists. Finally, we present experimental data which is consistent with the conjectures for the moments and for the vanishing in the family of cubic twists of LE (S).

Determining L-functions by twists

Suppose that L 1( s) and L 2( s) are two L-functions whose twists by a set of Dirichlet characters simultaneously vanish (vanish mod p) at a critical point. We examine the extent to which this property determines the L-functions in the cases of L-functions of elliptic curves, of number fields, and of curves over finite fields.

An Analogue of Kida's Formula for the p-adic L-functions of Modular Elliptic Curves

In this paper, we give a formula which describes the change of the λ-invariant of the p-adic L-function of a modular elliptic curve in a p-extension of Abelian fields. This formula is an analogue of Kida's formula.

Iwasawa L-Functions of Elliptic Curves with Additive Reduction

In this paper we investigate the Iwasawa L-functions of elliptic curves with potentially multiplicative reduction and prove analogues of the conjectures of Birch and Swinnerton-Dyer for them.

OnL-functions of elliptic curves and cyclotomic towers

In this paper we shall strengthen a recent theorem of Ralph Greenberg [1] concerning L-functions of elliptic curves and anticyclotomic towers. Let K be an imaginary quadratic field and M an abelian extension of K, possibly of infinite degree. We say that the extension M/K is anticyclotomic if M is Galois over Q and if the nontrivial element of Gal(K/Q) acts on Gal(M/K) by inversion. Now fix a finite set P of rational primes and consider the compositum (in an algebraic closure of K) of all anticyclotomic extensions of K which are unramified outside P. We denote this compositum by L and refer to it as the maximal anticyclotomic extension of K unramified outside P. The field L may also be described as the union of all ring class fields of K with conductor divisible only by primes in P. Class field theory shows that Gal(L/K) is isomorphic to the product of a finite group and the group

The modular symbol and continued fractions in higher dimensions

The modular symbol has proved a very useful tool for dealing with holomorphic automorphic forms on the upper half-plane. It serves, for instance, to compute the action of Hecke operators on automorphic forms, to give the value of twisted L-functions of certain elliptic curves in terms of a finite sum of modular symbols, and to prove that divisors of degree zero supported on the cusps of a modular curve have finite order in the Jacobian of that curve. For these matters, see [1], Chaps. IV and V, and the references cited there. Here is a definition of the modular symbol. Let H be the upper half-plane, that is, the set of all complex numbers with positive imaginary part. A cusp of H is defined to be any rational number (considered as a point of the real line) or oc. The cusps belong to the boundary of H in the extended complex plane. Let SL(2,1R) act on H in the usual way. For any subgroup F of finite index in SL(2, 9 ) we can form the quotient H/F. There is a closed, compact Riemann surface X(F) such that H/F is biholomorphic to the complement of some finite set of points x 1 . . . . ,x~ in X(F). These points x i are called the cusps of X(F). Let ~?(F) denote the holomorphic differential l-forms on X(F). Integration of forms over cycles yields a canonical N-linear map from H~(X(F);IR) to the complex dual space of ~(F). The theory of Riemann surfaces tells us that this map is an isomorphism. Now choose any two rational cusps a and b on the boundary of H. Let l be any reasonable arc beginning at a, ending at b, and staying in H in the interim for instance, l could be the geodesic in H from a to b. Then one easily shows that integration over the image of l in X(F) gives a linear map from ~?(F) to I12. This linear map gives an element of HI(X(F); IR) in the canonical way described above. We denote this homology class by [a, hi. We call [ ] the modular symbol. In summary, the "classical" modular symbol is a map from the set of pairs of cusps of H into H~(X(F);IR). This symbol enjoys properties which are the

Determination of L -Functions of Elliptic Curves Parametrized by Modular Functions

Proceedings of the London Mathematical SocietyVolume s3-28, Issue 3 p. 439-456 Articles Determination of L-Functions of Elliptic Curves Parametrized by Modular Functions J. B. Slater, J. B. Slater Brasenose College, OxfordSearch for more papers by this author J. B. Slater, J. B. Slater Brasenose College, OxfordSearch for more papers by this author First published: May 1974 https://doi.org/10.1112/plms/s3-28.3.439Citations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volumes3-28, Issue3May 1974Pages 439-456 RelatedInformation

L-functions of elliptic curves with complex multiplication, II

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Dirichlet $L$-functions, elliptic curves, hypergeometric functions, and rational approximation with partial sums of power series

Abstract. We consider the diophantine approximation of exponential generating functions at rational arguments by their partial sums and by convergents of their (simple) continued fractions. We establish quantitative results showing that these two sets of approximations coincide very seldom. Moreover, we offer many conjectures about the frequency of their coalescence. In particular, we consider exponential generating functions with real Dirichlet characters and with coefficients of L-functions of elliptic curves, where calculational data provide striking examples showing agreement for certain partial quotients of high index and gargantuan heights. Lastly, we similarly examine hypergeometric functions; note that e is a special case of the latter.