Congruence of normalized eigenforms at two primes

Let $$f_i\in S_{k_i}(\Gamma_0(N_i))$$ be normalized cuspidal eigenforms for $$i=1,2$$ and let $$K$$ be the composite of the fields of Fourier coefficients generated by $$f_1$$ and $$f_2$$ and let $$\mathfrak{p}_1$$ and $$\mathfrak{p}_2$$ are distinct prime ideals in $$K$$. Under what conditions can one find $$f_3\in S_{k_3}(\Gamma_0(N_3))$$ which is a normalized cuspidal eigenform such that $$f_3\equiv f_i\mod \mathfrak{p}_i$$ for $$i=1,2$$.

This is a natural question and is most likely classical. To start, let's not have any assumptions on $$N_3$$ and $$k_3$$. Could someone point me to some references, that would be really helpful, I understand what I'm asking may be really well understood, in which case I'd like to know of what is known about all the pairs $$(k_3,N_3)$$ for which a form $$f_3$$ satisfying the simultaneous congruence exists.

• ... Let's also assume that $k_1=k_2$ to simplify matters. – user130124 Nov 1 '18 at 20:35

I assume you allow $$f_3$$ to have arbitrary Fourier coefficients (i.e., $$\mathfrak{p}_1$$ and $$\mathfrak{p}_2$$ are prime ideals in a suitable number field that contains $$K$$ and the coefficient field of $$f_3$$).

In the case of a single pair $$(f_1,\mathfrak{p}_1)=(f_2,\mathfrak{p}_2)$$, this is the famous level lowering/raising problem. Assuming the irreducibility of the residual representation, this has been solved by Ribet/Diamond-Taylor. See the following papers :

Ribet, K. A. On modular representations of $$\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$$ arising from modular forms. Invent. Math. 100 (1990), no. 2, 431–476.

Diamond, Fred ; Taylor, Richard. Nonoptimal levels of mod $$l$$ modular representations. Invent. Math. 115 (1994), no. 3, 435–462.

You may also look at Ribet's Report on mod $$l$$ representations of $${\rm Gal}(\overline{\bf Q}/{\bf Q})$$ (Proc. Sympos. Pure Math., 55, Part 2) where he explains how to adjust the weight, level and character of the form $$f_3$$. On these matters the works of Carayol and Edixhoven are also quite relevant.

In the case of distinct pairs $$(f_1,\mathfrak{p}_1)$$ and $$(f_2,\mathfrak{p}_2)$$, combining the results mentioned above already gives you necessary conditions on $$k_3$$ and $$N_3$$ to be satisfied for such a form $$f_3$$ to exist.

Finally, these notes of Mazur and Stein deal with a particular - yet quite interesting - case :

https://wstein.org/Tables/serremodpq/

• Thank you, the necessary conditions in the case in which the pairs are distinct are not known to be sufficient and this result is not known for distinct pairs when there is no requirement on $k_3$ and $N_3$ I presume? My perspective is that of lifting Galois representations. – user130124 Nov 5 '18 at 14:37
• Here is another paper where a precise "mod $pq$ Serre's conjecture" is stated: Khare, Chandrashekhar ; Kiming, Ian. Mod pq Galois representations and Serre's conjecture. J. Number Theory 98 (2003), no. 2, 329–347. – Nicolas B. Nov 5 '18 at 15:48
• Thank you, I believe now that the weak version (in which one does not make any requirement on the level and weight) does not follow from knowing Serre's conjecture and applications of the deformation theory of Galois representations. Very much related to this is my previous question mathoverflow.net/questions/314258/… . – user130124 Nov 5 '18 at 16:15