### Abstract

The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions, L^{♯} _{p}. (f, T) and L^{♭} _{p}. (f, T), for a weight-two modular form ∑ a_{n}q^{n} and a good prime p. This generalizes work of Pollack who worked in the supersingular case and also assumed a_{p} = 0. These Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: we bound the rank and estimate the growth of the Šafarevič-Tate group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.

Original language | English (US) |
---|---|

Pages (from-to) | 885-928 |

Number of pages | 44 |

Journal | Algebra and Number Theory |

Volume | 11 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2017 |

Externally published | Yes |

### Fingerprint

### Keywords

- Birch and Swinnerton-Dyer
- Elliptic curve
- Iwasawa Theory
- Modular form
- P-adic L-function
- Šafarevič-Tate group

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**On pairs of P-Adic L-Functions for weight-two modular forms.** / Sprung, Florian.

Research output: Contribution to journal › Article

*Algebra and Number Theory*, vol. 11, no. 4, pp. 885-928. https://doi.org/10.2140/ant.2017.11.885

}

TY - JOUR

T1 - On pairs of P-Adic L-Functions for weight-two modular forms

AU - Sprung, Florian

PY - 2017/1/1

Y1 - 2017/1/1

N2 - The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions, L♯ p. (f, T) and L♭ p. (f, T), for a weight-two modular form ∑ anqn and a good prime p. This generalizes work of Pollack who worked in the supersingular case and also assumed ap = 0. These Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: we bound the rank and estimate the growth of the Šafarevič-Tate group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.

AB - The point of this paper is to give an explicit p-adic analytic construction of two Iwasawa functions, L♯ p. (f, T) and L♭ p. (f, T), for a weight-two modular form ∑ anqn and a good prime p. This generalizes work of Pollack who worked in the supersingular case and also assumed ap = 0. These Iwasawa functions work in tandem to shed some light on the Birch and Swinnerton-Dyer conjectures in the cyclotomic direction: we bound the rank and estimate the growth of the Šafarevič-Tate group in the cyclotomic direction analytically, encountering a new phenomenon for small slopes.

KW - Birch and Swinnerton-Dyer

KW - Elliptic curve

KW - Iwasawa Theory

KW - Modular form

KW - P-adic L-function

KW - Šafarevič-Tate group

UR - http://www.scopus.com/inward/record.url?scp=85021355727&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021355727&partnerID=8YFLogxK

U2 - 10.2140/ant.2017.11.885

DO - 10.2140/ant.2017.11.885

M3 - Article

VL - 11

SP - 885

EP - 928

JO - Algebra and Number Theory

JF - Algebra and Number Theory

SN - 1937-0652

IS - 4

ER -