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14
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The radius of convergence of the p-adic exponential function.

As every number theorist learns, the radius of convergence of $exp(x)$, defined by the usual power series in a neighborhood of zero, is $$\rho = p^{-1/(p-1)}.$$ This is typically proven by computing ...
5
votes
2answers
854 views

Direct proof of special case of Hasse's theorem for elliptic curves

Consider the elliptic curve $y^2 = x^3 + x$ over $\mathbb{F}_p$, where $p \equiv 1 \pmod 4$. If memory serves correctly, the number of points (excluding the point at infinity) is $p - a$ where $a$ is ...
0
votes
0answers
261 views

Does it exist a p-adic L function which interpolates the values of the complex one at positive integers?

I known that there are classical ways to construct $p$-adic $L$ functions for Dirichlet characters through $p$-adic integrals. We fix a character $\chi$ modulo $Np^r$ with $N$ and $p$ coprime and an ...
7
votes
3answers
2k views

What is the p-adic valuation of a number?

There seem to be two conflicting definitions for p-adic valuation in the literature. Firstly, for any non-zero integer n, we have $\nu=\nu_p(n)$ is the greatest non-negative integer such that $p^\nu$ ...
7
votes
1answer
524 views

Can an etale (phi, Gamma) module be an extension of non-etale ones?

This question is about p-adic representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p / \mathbb{Q}_p)$ and $(\varphi, \Gamma)$-modules. By theorems of Fontaine, Cherbonnier-Colmez and Kedlaya, the ...
8
votes
1answer
595 views

Fields of definition for p-adic overconvergent modular eigenforms

If we consider the action of the $U_p$ operator on overconvergent $p$-adic modular forms, then we can get some information about the field over which the eigenforms are defined by looking at the ...
7
votes
4answers
1k views

Extension of valuation

Fix a prime number $p$. Suppose that I have a valuation $v_p: \mathbb{Q} \to \mathbb{Q}$ on the rationals $\mathbb{Q}$. That is, $v_p( p^n(\frac{a}{b})) = p^{-n}$ where each of $a,b$ is not divisible ...
11
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1answer
495 views

Are centrally extended p-adic groups defined over F_1?

Let G be a semisimple algebraic group. Following work of Matsumoto [1], Brylinski and Deligne [2] constructed a central extension of the functor G : Rings → Groups by the second algebraic ...
4
votes
1answer
249 views

Classifying continuous characters $X \to \mathbb{Z}_p^*$, $X=\mathbb{Z}_p^*$ or $(1+p\mathbb{Z}_p)^{\times}$ ?

Question : are the continuous characters of the form $\eta : \mathbb{Z}_p^* \to \mathbb{Z}_p^*$, or $\eta : (1+p\mathbb{Z}_p)^{\times} \to \mathbb{Z}_p^*$ (i.e., on the principal units in ...
10
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2answers
2k views

p-adic L-functions

For modular forms, it is known that you can construct p-adic L-functions by interpolating (p-power conductor) twists of their associated L-functions at special values. Similarly, Kubota-Leopoldt's ...
5
votes
1answer
622 views

bibl. q.s on Dwork's “p-adic cycles”, Mazur's “p-adic variations”:

Matthew Emerton mentioned recently the relevance of Dwork's "p-adic cycles". As I wonder if I should read that, reviews of it are ambiguous, I'd be happy on remarks and possible further bibl. hints. ...
12
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3answers
2k views

Dwork's use of p-adic analysis in algebraic geometry

Using p-adic analysis, Dwork was the first to prove the rationality of the zeta function of a variety over a finite field. From what I have seen, in algebraic geometry, this method is not used much ...
14
votes
1answer
1k views

Stark's conjecture and p-adic L-functions

Not long back I asked a question about the existence of p-adic L-functions for number fields that are not totally real; and I was told that when the number field concerned has a nontrivial totally ...
16
votes
3answers
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An unfamiliar (to me) form of Hensel's Lemma

In his very nice article Peter Roquette, History of valuation theory. I. (English summary) Valuation theory and its applications, Vol. I (Saskatoon, SK, 1999), 291--355, Fields Inst. Commun., ...
10
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5answers
1k views

When does a p-adic function have a Mahler expansion?

Let $f: \mathbb{Z}_p \rightarrow \mathbb{C}_p$ be any continuous function. Then Mahler showed there are coefficients $a_n \in \mathbb{C}_p$ with $$ f(x) = \sum^{\infty}_{n=0} a_n {x \choose n}. $$ ...
7
votes
1answer
441 views

how do you evaluate the p-adic modular form E_p-1 in the region |j|<1

background/motivation let Ek denote the modular form of level one and weight k with q-expansion given by $E_k(q)=1- \frac{2k}{b_k}\sum_n \sigma_{k-1}(n)q^n$ where σi is the divisor sum and bk ...
14
votes
4answers
2k views

Special values of $p$-adic $L$-functions.

This is a very naive question really, and perhaps the answer is well-known. In other words, WARNING: a non-expert writes. My understanding is that nowadays there are conjectures which essentially ...
7
votes
2answers
976 views

Free subquotient of Galois representations coming from Hida theory

Let $\mathbf{T}$ be the reduced nearly ordinary Hecke algebra of level $N$ of Hida theory for $\operatorname{GL}_{2}$ over $\mathbb{Q}$ (or more generally over a totally real field $F$). Then ...
15
votes
3answers
1k views

2-adic Coefficients of Modular Hecke Eigenforms.

Suppose that N is prime, and consider the normalized cuspidal Hecke eigenforms of weight 2 and level Gamma_0(N). For such an eigenform f, the coefficients generate ...